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Hindawi Publishing CorporationJournal of GravityVolume 2013 Article ID 549824 28 pageshttpdxdoiorg1011552013549824

Review ArticleSpin Foam Models with Finite Groups

Benjamin Bahr1 Bianca Dittrich2 and James P Ryan2

1 DAMTP University of Cambridge Wilberforce Road Cambridge CB3 0WA UK2MPI for Gravitational Physics Albert Einstein Institute Am Muhlenberg 1 14476 Potsdam Germany

Correspondence should be addressed to Bianca Dittrich bdittrichperimeterinstituteca

Received 30 March 2013 Accepted 3 June 2013

Academic Editor Kazuharu Bamba

Copyright copy 2013 Benjamin Bahr et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Spin foam models loop quantum gravity and group field theory are discussed as quantum gravity candidate theories and usuallyinvolve a continuous Lie group We advocate here to consider quantum gravity-inspired models with finite groups firstly as a testbed for the full theory and secondly as a class of new lattice theories possibly featuring an analogue diffeomorphism symmetryTo make these notes accessible to readers outside the quantum gravity community we provide an introduction to some essentialconcepts in the loop quantum gravity spin foam and group field theory approach and point out themany connections to the latticefield theory and the condensed-matter systems

1 Introduction

Spin foam models [1ndash8] arose as one possible quantizationmethod for gravityThese models can be seen in the traditionof lattice approaches to quantum gravity [9] in which gravityis formulated as a statistical system on either a fixed orfluctuating lattice In a rather independent fashion latticemethods are also ubiquitous in condensed-matter and (Yang-Mills) gauge theories It is interesting to note that the latticestructures appearing in the strong coupling expansion ofQCD or in the high-temperature expansion for the Isinggauge [10] systems are of spin foam type

In any lattice approach to gravity one has to discuss thesignificance of the (choice of) lattice for physical predictionsAlthough this point arises also for other lattice field theoriesit carries extreme importance for general relativity When itcomes to matter fields on a lattice we can interpret the latticeitself as providing the background geometry and thereforebackground space timeThis could not contrastmorewith thesituation in general relativity where geometry and thereforespace time itself is a dynamical variable and has to be anoutcome rather than an ingredient of the theory This isone aspect of background independence for quantum gravity[11 12]

One possibility to alleviate the dependence of physicalresults on the choice of lattice is to turn the lattice itself

into a dynamical variable and to sum over (a certain classof) lattices This is one motivation for the dynamical trian-gulation approach [13ndash15] causal dynamical triangulations[16 17] and tensor modeltensor group field theory [18ndash26] See also [27 28] for another possibility to introducea dynamical lattice Once one decides to sum over latticestructure one must provide a prescription to do so The classof triangulations to be summed over can be restricted forinstance in order to implement causality [16 17 29] or tosymmetrymdashreduce models [30ndash32]1

Another possibility is to ask for models which are perse lattice or discretization independent Such models mightarise as fixed points of a Wilsonian renormalization flowand represent the so-called perfect discretizations [33ndash38]For gravity this concept is intertwined with the appearanceof discrete representation of diffeomorphism symmetry inthe lattice models [36ndash41] Such a discrete notion of dif-feomorphism symmetry can arise in the form of vertextranslations [42ndash46] Discrete geometries can be representedby a triangulation carrying geometric data for instance inthe Regge calculus [47 48] the lengths of the edges inthe triangulation Vertex translations are then symmetriesthat act (locally) on the vertices of the triangulations bychanging the geometric data of the adjacent building blocksIf the symmetry is fully implemented into the model thenthis action should not change the weights in the partition

2 Journal of Gravity

function Such vertex translation symmetries are realized in3D gravity where general relativity is a topological theory[49] This means that the theory has no local degrees offreedom only a topology-dependent finite number of globalones The first-order formalism of 3D gravity coincides witha 3D version of the BF-theory [50] The BF-theory is a gaugetheory and can be formulated with a gauge group given bya Lie group The 3D gravity is obtained by choosing 119878119880(2)However one can also choose finite groups [51ndash53] In thiscase one obtains models used in quantum computing [54] aswell as models describing topological phases of condensed-matter systems so called string-net models [55] Moreoverthe BF-systems can be seen as Yang-Mills-like systems fora special choice of the coupling parameter (correspondingto zero temperature) Hence in 3D we have Yang-Mills-liketheories with the usual (lattice) gauge symmetry as well asthe topological BF-theories with the additional translationsymmetries

The BF-theory (in any dimension) is a topological theorybecause it has a large gauge group of symmetries whichreduces the physical degrees of freedom to a finite numberIt is characterized not only by the usual (lattice) gaugesymmetry determined by the gauge group but also by theso-called translational symmetries based on the 3-cells (iecubes) of the lattice and parametrized by the Lie algebraelements (if we work with the Lie groups) For 3D gravitythese symmetries can be interpreted as being associatedto the vertices of the dual lattice (for instance given bya triangulation) and are hence termed vertex translationsHowever for 4D the symmetries are still associated to the 3-cells of the lattice and hence to the edges of the dual lattice ortriangulation As in general there aremore edges than verticesin a triangulation we obtainmuchmore symmetries than thevertex translations onemight look for in 4D gravity To obtainvertex translations of the dual lattice in 4D one would ratherneed a symmetry based on the 4-cells of the lattice

In 4D gravity is not a topological theoryThere is howevera formulation due to Plebanski [56] that starts with the BF-theory and imposes the so-called simplicity constraints on thevariablesThese break the symmetries of the BF-theory downto a subgroup that can be interpreted as diffeomorphismsymmetry (in the continuum) This Plebanski formulation isalso the one used in many spin foam models Here one ofthe main problems is to implement the simplicity constraints[57ndash64] into the BF-partition function The status of thesymmetries is quite unclear in these models however thereare indications [40] that in the discrete the reduction of theBF-translation symmetries does not leave a sufficiently largegroup to be interpretable as diffeomorphisms Neverthelessif these models are related to gravity then there is a pos-sibility that diffeomorphism symmetry as the fundamentalsymmetry of general relativity arises as a symmetry forlarge scales In the abovementioned method of regainingsymmetries by coarse graining one can then expect thatthis diffeomorphism symmetry arises as vertex translationsymmetry (on the vertices of the dual lattice)

Hence we want to emphasize that in 4D spin foammodels are candidates for a new class of lattice models inaddition to Yang-Mills-like systems and the BF-theory This

new class would be characterized by a symmetry group inbetween those for the Yang-Mills theory (usual lattice gaugesymmetry) and BF-symmetry (usual lattice gauge symmetryand translation symmetry associated to the 3-cells) Here wewould look for a symmetry given by the usual lattice gaugesymmetry and translation symmetries associated to the 4-cells

As we will show in this work the 4D spin foam modelswhich as gravity models are based on 119878119874(4) 119878119874(3 1) or119878119880(2) can be easily generalized to finite groups (or moregenerally tensor categories [65 66]) Although the immediateinterpretation as gravity models is lost one can neverthelessask whether translation symmetries are realized and if nothow these could be implemented This question is mucheasier to answer for finite groups than in the full gravitycase One can therefore see these models as a test bed forthe full theory This also applies for renormalization andcoarse graining techniques which need to be developed toaccess the large-scale limit of spin foams With finite groupmodels it might be in particular possible to access themany-particle (that is many simplices or building blocks inthe triangulation) and small-spin (corresponding to smallgeometrical size of the building blocks) regime2 This is incontrast to the few-particle and large-spin (semiclassical)regime [67ndash69] which is accessible so far

In the emergent gravity approaches [70ndash72] one attemptsto construct models which do not necessarily start fromgravitational or even geometrical variables but neverthelessshow features typical of gravity The models proposed herecan also be seen as candidates for an emergent gravityscenario A related proposal is the truncated Regge modelsin [73ndash76] in which the continuous length variables of theRegge calculus are replaced by values in Z119902 with 119902 = 2 forldquothe Ising quantum gravityrdquo

Apart fromproviding awealth of testmodels for quantumgravity researchers another intention of this work is to givean introduction to some of the spin foam concepts and ideasto researchers outside quantum gravity Indeed spin foamsarise as graphical tools in high-temperature expansions oflattice theories or more generally in the construction ofdual models We will review these ideas in Section 2 wherewe will discuss the Ising-like systems and introduce spinnets as a graphical tool for the high-temperature expansionNext in Section 3 we will discuss lattice gauge theories withthe ldquoAbelianrdquo finite groups Here spin foams arise in thehigh-temperature expansion The zero-temperature limit ofthese theories gives topological BF-theories which we willdiscuss inmore detail in Section 4 In Section 5 we detail spinfoammodels with the non-Abelian finite groups Following astrategy from quantum gravity we will also discuss the so-called constrained models which arise from the BF-modelsby implementing simplicity constraints here in the formof edge projectors into the partition function This will ingeneral change the topological character of the BF-models tonontopological ones

We will then discuss in Section 6 a canonical descriptionfor lattice gauge theories and show that for the BF-theoriesthe transfer operators are given by projectors onto the so-called physical Hilbert space These are known as stabilizer

Journal of Gravity 3

conditions in quantum computing and in the description ofstring nets Here we will discuss the possibility to alter theprojectors in order to break some subset of the translationsymmetries of the BF-theories

Next in Section 7 we give an overview of group fieldtheories for finite groups Group field theories are quantumfield theories on these finite groups and generate spin foamsas the Feynman diagrams Finally in Section 8 we discussthe possibility of nonlocal spin foams We will end with anoutlook in Section 9

2 The Ising-Like Models

We will review the construction of the models [77 78]that are dual to the Ising-like models Henceforth theywill be referred to as dual models As we will see similartechniques lead to the spin foam representation for (Ising-like) gauge theories The main ingredient comprises of aFourier expansion of the couplings which can be understoodas functions of group variables We will concentrate on thegroups Z119902 with 119902 = 2 for the Ising models proper

21 Spin Foam Representation Consider a generic lattice ofregular or random type3 in which the edges are oriented(In principle this last condition is not necessary for the Isingmodels that is 119902 = 2) One defines an Ising model on such alattice with spins 119892V associated to the vertices and nearest-neighbour interactions by utilizing the following partitionfunction

119885 =1

2♯Vsum

119892V

prod

119890

exp (120573119892119904(119890)119892minus1

119905(119890)) (1)

where 119892V isin 0 1 120573 is a coupling constant which is inverselyproportional to temperature and ♯V is the number of verticesin the lattice Note that the group product is the standardproduct on Z2 that is addition modulo 24 The action 119878 =

120573sum119890119892119904(119890)119892

minus1

119905(119890)describes the coupling between the spin 119892119904(119890) at

the source or starting vertex and the spin 119892119905(119890) at the target orterminating vertex of every edge

More generally one can assume that 119892V isin 0 119902 minus 1

(giving the various vector Pottsmodels)The group product isthe standard one onZ119902 that is addition modulo 119902 A furthergeneralization of (1) is to allow the edge weights 119908119890 to be(even locally varying) functions of the two group elements(119892119904(119890) and 119892119905(119890)) associated to the edge 119890

119885 =1

119902♯Vsum

119892V

prod

119890

119908119890 (119892119904(119890)119892minus1

119905(119890)) (2)

Now these group-valued functions 119908119890 can be expanded withrespect to a basis given by the irreducible representations ofthe group [79] For the groupsZ119902 this is just the usual discreteFourier transform

119908 (119892) =

119902minus1

sum

119896=0

119908 (119896) 120594119896 (119892)

119908 (119896) =1

119902

119902minus1

sum

119892=0

119908 (119892) 120594119896(119892)

(3)

where 120594119896(119892) = exp((2120587119894119902)119896 sdot 119892) are the group charactersCharacters for the Abelian groups are multiplicative that is120594119896(1198921 sdot1198922) = 120594119896(1198921) sdot120594119896(1198922) and 120594119896(119892

minus1) = [120594119896(119892)]

minus1= 120594

119896(119892)

Applying the character expansion to the weights in(2) one obtains (using the multiplicative property of thecharacters)

119885 =1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890) 120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890))(prod

119890

120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119896119890

(prod

119890

119908119890 (119896119890))prod

V(sum

119892V

prod

119890supV

120594119896119890

(119892119900(V119890)V ))

(4)

where 119900(V 119890) = 1 if V is the source of 119890 and minus1 if V is the targetof 119890 Here we simply resorted the product so that in the lastline the factorprod

119890supV120594119896119890

(119892119900(V119890)V ) involves all of the appearances

of 119892V We can now sum over 119892V which according to theFourier inversion theoremgives a delta function involving the119896119890 of the edges adjacent to V

119885 = sum

119896119890

(prod

119890

119908119890 (119896119890))prod

V120575(119902)

(sum

119890supV

119900 (V 119890) 119896119890) (5)

where 120575(119902)(sdot) is the 119902-periodic delta function5We have represented the partition function in a form

where the group variables are replaced by variables takingvalues in the set of representation labels This set inherits itsown group product via this transform and is known as thePontryagin dual of 119866 For Z119902 the Pontryagin dual and thegroup itself are isomorphic as both 119892 119896 isin 0 119902 minus 1 Wewill call a representation of the form of (5) where the sumover group elements is replaced by a sum over representationlabels a spin foam representation6 Generically speakingrepresentations meeting at vertices must satisfy a certaincondition For the Abelian models above this conditionstated that the oriented sum of the representation labelsmeeting at a given vertex had to vanish In other words thetensor product of the representations meeting at an edge hasto be trivial In other models one finds other characteristicsas the following (i) For the non-Abelian groups this trivialitycondition generalizes to the choice of a projector from thetensor product of all representations meeting at a vertex intothe trivial representationThis choice is encoded as additionalinformation attached to the vertices (ii) For the gauge theorymodels wewill examine later the representations are attachedto faces while the triviality condition involves faces sharing agiven edge

Note that the number of representation labels 119896119890 differsfrom the number of group variables 119892V since the numberof edges is generally larger than the number of vertices Inaddition however these representation labels are subject toconstraints These are referred to as the Gauss constraintssince they demand that the analogue of the electric field(the representation labels) be divergence free (the sum of


the representation labels meeting at a vertex has to be zeromodulo 119902)

22 Dual Models To finalize the construction of the dualmodels [78] one solves theGauss constraints and replaces therepresentation labels associated to the edges of the lattice byvariables defined on the dual lattice

(2d) A particularly clear example arises in two dimensionsOn a differentiable manifold a divergence-free vectorfield V can be constructed from a scalar field 120601 via V1 =1205972120601 V2 = minus1205971120601 On the lattice a similar constructionleads to a dual model with variables associated tothe vertices of the dual lattice7 Indeed for Z2 onefinds again the Ising model just that the high- (low-)temperature regime of the original model is mappedto the low- (high-) temperature regime of the dualmodel

(3d) In three dimensions a divergence-free vector field Vcan be constructed from another vector field bytaking its curl V119894 = 120598119894119895119896 120597119895119908119896 However adding agradient of a scalar field to does not change V rarr + 120601 leaves V rarr V This translates into alocal gauge symmetry for the dual lattice theory Forthe dual lattice model variables reside on the edgesof the dual lattice and the weights define couplingsamong the edges bounding the faces (that is the two-dimensional cells) of the dual lattice Moreover theaforementioned local gauge symmetry is realized atthe vertices of the dual lattice (since the continuumgauge symmetry is parametrized by a scalar field)As a result the dual model is a gauge theory a classof theories that we will discuss in more detail inSection 3

(4d) In four dimensions a similar argument to that justmade in three dimensions leads to a class of dualmodels that are ldquohigher-gauge theoriesrdquo [80 81] thevariables are associated to the faces of the dual latticeand the weights define couplings among the facesbounding three-dimensional cells

(1d) Finally let us examine one-dimensional modelsEvery vertex is adjacent to two edges Hence theGauss constraints expressed in (5) force the rep-resentation labels to be equal (we assume periodicboundary conditions) We can easily find that

119885 = 119902♯Vsum

119896

(prod

119890

119908119890 (119896)) = 119902♯Vsum

119896

119908(119896)♯119890 (6)

For the second equality above we are restricted tocouplings that are homogenous on the lattice that isthey do not depend on the position or orientation ofthe edge

23 High-Temperature Expansion and Spin Nets This spinfoam representation is well adapted to describe the high-temperature expansion [82] Indeed if one considers the Ising

model with couplings as in (1) one obtains the followingcoefficients in the character expansion

119908 (0) = 1198901205732 cosh120573

119908 (1) = 1198901205732 cosh120573 tanh120573

(7)

One can now expand the partition function (5) in powers ofthe ratio 119908(1)119908(0) that is into terms distinguished by thenumber of times the representation labelled 119896119890 = 1 appears

The ldquogroundrdquo state is the configuration with 119896119890 = 0

for all edges Edges with 119896119890 = 1 are said to be ldquoexcitedrdquoThe Gauss constraints in (5) enforce that the number ofldquoexcitedrdquo edges incident at each vertex must be even On acubical lattice therefore the lowest-order contribution arisesat (119908(1)119908(0))4 from those configurations with excited edgeson the boundary of a single plaquette The full amplitudeat this order involves a combinatorial factor namely thenumber of ways one can embed the square (with four edges)into the lattice

This reasoning extends to all Ising-like models and tohigher-order terms in the perturbation series which can berepresented by closed graphs known as polymers embeddedinto the lattice The expansion in terms of these graphs canthen be organized in different ways [82] Consider one par-ticular contribution to the expansion that is a configurationwhere a some subset of edges carries nontrivial representationlabels In general one may decompose such a subset intoconnected components where one defines two sets of edgesas disconnected if these sets do not share a vertex

We will call such a connected component a restrictedspin net where restricted refers to the condition that alledges must carry a nontrivial representationThe Kronecker-120575s in (5) forbid the spin net to have open ends that is allvertices are bivalent or higher (For the Ising model with itslone nontrivial representation only even valency is allowed)Furthermore edges meeting at a bivalent vertex whetherthere is a ldquochange of directionrdquo or not must carry the samerepresentation label (On this point we have assumed thatthe couplings 119908119890 depend neither on the positions nor on thedirections of the edges otherwise one must equip the spinnets with more embedding data) The representation labelsmay only change at vertices that are trivalent or higher socalled branching points This allows one to define geometric(restricted) spin nets8 which are specified by two quantities(i) the connectivity of the spin net vertices that is the valencyof the branching points within the spin net and (ii) the lengthof the edges of the spin net that is the number of latticeedges that make up a spin net edge The contribution of sucha spin net to the free energy9 can be split into two parts oneis specific to the spin net itself and from (4) one records thatit is given by

prod

119890119896119890= 0

119908119890 (119896119890) (8)

The other part is a combinatorial factor determined by thenumber of embeddings of the spin net into the ambientlattice This could be summarized as a measure (or entropic)factor which carries information about the lattice in questionincluding its dimension


24 Some Considerations for Quantum Gravity To reiterateone notes from (8) that the spin net-specific contributionfactorizes with respect to the edges of the spin net for anedge with representation 119896 it is given by 119908(119896)119897 where 119897 is thelength of the spin net edge Meanwhile the combinatorialfactors are determined by the ambient lattice and its structureTo describe quantum gravity one is interested rather morein ldquobackground-independentrdquo models where such factorsshould not play a role In this context one can define analternative partition function over geometric spin nets whoseedges carry both a representation label and an edge-lengthlabelThis length variable encodes the geometry of space timeas a dynamical variable on the geometric spin net

From that point of view ldquobackground independencerdquomotivates the search for models within which the spin netweights are independent of these edge lengths These can betermed abstract spin nets [83] (Note that with this definitionthe abstract spin nets generally still remember some featuresof the ambient lattice for instance the valency of the spinnet cannot be higher than the valency of the lattice) Forthe Ising model such abstract spin nets arise in the limittanh120573 = 1 that is for zero temperature In contrast tothe high-temperature expansion which is a sum orderedaccording to the length of the excited edges abstract spinnetworks arise in a regime where this length does not playa role

A similar structure arises in string-net models [55]which were designed to describe condensed phases of (scale-free) branching strings Indeed these string-net models areclosely related to (the canonical formulation of) topolog-ical field theories such as the BF-theories which we willdescribe in Section 6 In this canonical formulation spinnet(work)s appear as one choice of basis for the Hilbertspace of theBF-theories String-netmodels assign amplitudes(corresponding to the so-called physical wave functions inthe BF-theory) to spin networks that satisfy certain (gauge)symmetry properties This implies that these spin networkscan be freely deformed on the lattice without changing theiramplitude

In the following section we will consider gauge systemsfor which we will devise once again a spin foam representa-tion For these gaugedmodels rather than labelling the termsin the expansion using spin nets we will utilize a different yetsimilar structure known as a spin foam These have a similarstructure to spin nets except that the various roles are playedby simplices one dimension higher To clarify the roles ofthe vertices and edges are assumed by the edges and facesrespectively With this proviso other concepts introducedabove translate nicely Spin foam edges and faces are generallycomprised of several edges and faces in the ambient latticeMoreover one can define both geometric and abstract spinfoams [83] Both are branched surfaces [1ndash6 84] whoseconstituent faces carry a nontrivial representation label10 Theamplitudes for geometric spin foams will most often dependon the area of their surfaces that is the number of plaquettesin the ambient lattice constituting the spin foam surface inquestion as well as the lengths of their branches that is thenumber of edges in the ambient lattice constituting the spin

foam branch in question For instance this is the case in theYang-Mills-like theories where geometric spin foams arise inthe strong coupling or high-temperature expansion

The conditions for abstract spin foams that is amplitudesindependent of the surface area may be understood asrequiring that the amplitudes be invariant under trivial (faceand edge) subdivision This can be used to specify measurefactors for the spin foam models [1ndash5 83 85ndash87] Abstractspin foams are also generated by the tensor models discussedin Section 7 however the spin foams thus generated are notldquorestrictedrdquo in the manner we defined earlier that is thetrivial representation label is allowed Furthermore the spinfoams need not be embeddable into a given lattice See also[88] for one possible connection between partition functionswith restricted and unrestricted spin foams

This discussion on geometric and abstract spin nets andspin foams assumes that the couplings do not depend on theposition or direction of the edges or plaquettes in the latticeotherwise one would need to introducemore decorations forthe spin nets and foams However one can show [89] forinstance that the Z2 Ising gauge model in 4D with varyingcouplings is universal In other words one can encode allother Z119901 lattice models Here it would be interesting todevelop the equivalent spin foam picture and to see how aspin foam with additional decorations could encode otherspin foam models

3 Lattice Gauge Theories

The spin foam representation originated as a representationof partition functions for gauge theories [1ndash6 11] Such gaugetheories can also be formulated with discrete groups [10 5290] and for Z2 they give the Ising gauge models [10] Aspin foam representation for the Yang-Mills theories withcontinuous Lie groups has been presented in [91] see also[92 93] and the results can be easily adapted to discretegroups For the Abelian (discrete and continuous) groupsthese are again closely related to the construction of dualmodels [78] and the high-temperature (or strong coupling)expansion [84]

Given a lattice one associates group variables to its(oriented) edges To be more precise a lattice in this contextis an orientable 2-complex 120581 within which one can uniquelyspecify the 2-cells called faces or plaquettes The aim isto construct models with gauge symmetries at the verticeswhere the gauge action is given by 119892119890 rarr 119892119904(119890)119892119890119892

minus1

119905(119890)and the

119892V are the gauge symmetry parameters They are associatedto the vertices of the lattice remember that 119904(119890) is the sourcevertex of 119890 and 119905(119890) is the target vertex Gauge-invariantquantities are given by the Wilson loops11 Expressing theaction as a function of these quantities ensures that it in turn isgauge invariant as can be easily checked AWilson loop is thetrace (in somematrix representation) of the oriented productof group variables associated to a loop of edges in the lattice12The ldquosmallestrdquoWilson loops are those constructed from edgesaround a single lattice face The associated face variables arethen ℎ119891 = prod119892119890 where prod denotes the oriented product


Given a unitary (finite-dimensional) matrix representa-tion of a group 119892 997891rarr 119880(119892) a Wilson-styled action is givenby

119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)

Here120572 is a coupling constant and119880 is amatrix representationof Z119902 As we saw earlier for the groups Z119902 the irreducibleunitary representations are 1 dimensional and labelled by 119896 isin

0 119902 minus 1 so that the representations matrices are realizedby 119892 997891rarr exp((2120587119894119902)119896 sdot 119892)

As before let us generalize to a class of partition functionsof the form

119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)

where the weights 119908119891 are class functions that is just likethe trace they are invariant under conjugation 119908119891(119892ℎ119892

minus1) =

119908119891(ℎ) Moreover we included a measure normalizationfactor 1119902 for every summation over the group Z119902

31 Spin Foam Representation The face weights 119908119891 as classfunctions can be expanded in characters For the Abeliangroups Z119902 this again just amounts to the discrete Fouriertransform Thus the expansion takes the initial form

119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)

while the derivation of the spin foam representation proceedsas in Section 2 as follows

119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)

where 119900(119891 119890) = 1 if the orientation of the edge coincides withthe one induced by the ldquoadjacentrdquo face and minus1 otherwise Inthe last step we simply replace the Kronecker-120575 weightingeach edge by its square and associate one factor to each vertexThe reason is that in the spin foam representation one usuallyworks with vertex amplitudes which in this case is 119860V =

prod119890supV120575

(119902)(sum

119891sup119890119900(119891 119890) 119896119891) Having said that these amplitudes

and the splitting are more complicated for both the non-Abelian groups (see Section 5) and continuous groups

32 Dual Models The construction of the dual models [78]proceeds by solving for the Gauss constraints associatedto the edges in (12) that is by solving the Kronecker-120575sThese enforce that the oriented sum of representation labelsassociated to the faces incident at a given edge vanishes

(2d) For the lowest-dimensional case namely two dimen-sions one obtains a ldquotrivialrdquo dual theory Since everyedge lies on just two faces all of the representationlabels coincide 119896119891 equiv 119896 and one obtains (assumingthat the 119908119891(119896119891) do not depend on the position andorientation of the plaquettes 119908119891(119896119891) = 119908(119896))

119885 = sum

119896

(119908119896)♯119891 (13)

(3d) For the three-dimensional case we have already seenthat lattice models without any gauge symmetry aredual to lattice gauge models For example the Isinggauge model in 3119889 is dual to the 3119889 Ising model

(4d) In four dimensions lattice gauge models are dual tolattice gauge models In the case that the cohomologyof the lattice is nontrivial topological models mightappear as the dual model [94]

33 High-Temperature Expansion and Spin Foams As for theIsing models there is a high-temperature expansion whichfor the Ising gauge model is also an expansion in powersof 119908(1)119908(0) = tanh120573 The discussion is very similar tothe one for the Ising models with the difference being thatthe perturbative contributions are now represented by closedsurfaces rather than closed polymers

The ldquoground staterdquo is the configuration with 119896119891 = 0 forall faces The Gauss constraints demand that the numberof excited plaquettes (those with 119896119891 = 0) incident at everyedge must be even In particular this means that the excitedplaquettes must form closed surfaces In the case of theIsing gauge model on a hypercubical lattice the lowest-ordercontribution arises at (119908(1)119908(0))6 from those configurationswith excited faces on the boundary of a single 3119889 cube As onemight expect there is a combinatorial factor arising from thenumber of embeddings of this cube into the ambient lattice

More generally the perturbative contributions are nowdescribed by closed possibly branched surfaces (For thefree energy one has only to consider connected components[95]) Here a proper branching is a sequence of edges whereat least three excited plaquettes meet Excited plaquettesmake up the elementary surfaces or spin foam faces Inother words at the inner edges of these spin foam facesthere are always only two excited plaquettes meeting Againthe Gauss constraints demand that the representation labelsof all of the plaquettes in a given spin foam face agreeFor homogeneous and isotropic couplings the spin foamamplitude (ignoring the combinatorial embedding factors)depends on the number of plaquettes 119886 inside each spinfoam face but not on how the spin foam is embedded intothe lattice The contribution of a spin foam face with label119896 is sim (1199081198961199080)

119886 (In general for models with the non-Abelian groups the amplitudes might also depend on the


length of the branching or spin foam edges)This definition ofgeometric and abstract spin foams [83] parallels the one givenfor spin nets Note that the condition for abstract spin foamsinvalidates the conditions of the high-temperature expansionwhich is an expansion in the number of excited plaquettes ofthe underlying lattice

In Section 4 we will discuss the BF-theory This is atopological model that satisfies the conditions necessary toclass it as an abstract spin foam In other words the amplitudedoes not depend on the areas 119886 of the spin foam faces Thereason is that119908(119896)119908(0) equiv 1 for the BF-models For the Isinggauge model this is the case at zero temperature13

34 Expansion around a Topological Sector If one starts fromthe first line in (11) and keeps the summation over both groupand representation labels one obtains

119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)

This corresponds to a first-order representation of the Yang-Mills theory where the 119892119890 are the connection variables andthe 119896119891 represent the dual (electric field or flux) variablesThe model where all 119908119891(119896119891) = 1 is a topological BF-modelwhose partition function can be solved exactly Hence theYang-Mills theory can be understood as a deformed BF-theory [92 93 96 97] The deformation appears here as theln(119908119891(119896119891)) term in the continuum it is 119861 and ⋆11986114

The expansion of models with propagating degrees offreedom for instance those modelling 4119889 gravity and theYang-Mills theories around topological theories has beendiscussed for instance in [83 98 99]

In the case of the Ising model one expands around theBF-theory partition function by introducing an expansionparameter 120572 = tanh120573 minus 1 which is small for low tempera-tures

119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (1 + 120572sum

1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912

minus 1) + sdot sdot sdot + 120572♯119891

times sum

1198911ltsdotsdotsdotlt119891

♯119891

120575(2)

(1198961198911

minus 1) sdot sdot sdot 120575(2)

(119896119891♯119891

minus 1))

(15)

Note that to get to the second equality we have used the factthat 120575(2)(119896119891) + (1 + 120572) 120575

(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus

for a given configuration 119896119891119891 the coefficient of 1205721 recordsthe number of excited faces The coefficient of 1205722 gives thenumber of ordered pairs of excited faces The next coefficientrecords the number of ordered triples of faces and so onThelast coefficient of 120572♯119891 is one for the configuration 119896119891 equiv 1 andzero for all other configurations The terms in this expansioncan be seen as expectation values of observables in a BF-model where the observables are the numbers of ordered 119899-tuples of excited plaquettes for each 0 le 119899 le ♯119891

4 Topological Models

In the previous section we saw that theories of the Yang-Mills type can be understood as a deformation of the BF-theories which we will discuss here in more detail Fromthe conventional standpoint 119865 represents the curvature of aconnection usually taking values in the Lie group while119861 is aLie algebra-valued (119863minus2)-formHence in constructing thesemodels for discrete groups the following question arises withwhat should one replace the 119861 variables We have seen thatthe119861-field corresponds to the representation labels Examplesfor topological models with discrete groups are discussed inseveral texts [51 53 100 101] See also [102 103] for the relationbetween the Chern-Simons-like theories and the BF-liketheories in 3119889 with discrete groupZ119902 In this section we willrestrict to the Abelian groups in particular Z119902 The modelsfor the non-Abelian groups will be presented in Section 5

The BF-theory is a topological field theory in which theequations of motion demand that the ldquolocalrdquo curvature van-ishes One often starts with the partition function encodingthis requirement It is defined in amanner very similar to thatof the previous sections (see for instance [6 104]) as follows

119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)

where again ℎ119891(119892119890) = prod119890sub119891

119892119890 means the oriented product15

of edges around a face Note that this is just a special case ofthe gauge theory partition functions (10) obtained by setting119908119891 = 120575

(119902)The partition function can be easily evaluated

119885 = ♯[

[

configurations 119892119890119890

prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)


For the groups Z119902 an expansion of the type detailed in (14)yields

119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)

One may interpret the term in the exponential as a BF-theory action where the representation labels 119896119891 representthe 119861-field on the lattice while the products ℎ119891(119892119890) =

sum119890sub119891

119900(119891 119890) 119892119890 represent curvatureThe spin foam representation can be derived in the same

way as for the gauge theories in Section 31

119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)

This recasts the partition function as

119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)

that is as the number of assignments 119896119891119891 satisfying theGauss constraint for every edge In the following section wewill discuss the local symmetries of the BF-theory partitionfunction We will find that on the set of all configurations119896119891119891

there acts a translation symmetry This is realized atthe 3-cells of the lattice Hence 119885 reflects the orbit volumeof this translational gauge symmetry It is this symmetry thatleads to divergences in the BF-theory partition functions forthe ldquocontinuousrdquo Lie groups [105 106]16

41 Symmetries of the Partition Function Wewill nowdiscussthe gauge symmetries of the partition function (16) Asits form coincides with that of the gauge theories (10)the partition function is invariant under the usual gaugetransformations 119892119890 rarr 119892119904(119890)119892119890119892

minus1

119905(119890) where 119892V isin Z119902 are gauge

parameters associated to the vertices This is manifest in therepresentation given in (16)

There is a further symmetry the so-called translationsymmetry which is easiest to see in the spin foam represen-tation This symmetry is based on the 3-cell 119888 of the latticethat is the cubes for a hypercubical lattice17 Consider a field119888 997891rarr 119896119888 of gauge parameters associated to the cubes and definethe gauge transformations

1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)

where 119900(119888 119891) = 1 if the orientation of 119891 agrees with thatinduced by 119888 and minus1 otherwise If 119896119891119891 is a configuration

that satisfies all of the Gauss constraints then so does 1198961015840119891119891

and vice versa The reason stems from the following fact ifan edge 119890 is in the boundary of a 3-cell 119888 then there areexactly two faces say1198911 and1198912 that are both in the boundaryof 119888 and adjacent to 119890 The orientation factors are such thatthe contributions of 119896119888 to the two faces 1198911 and 1198912 are ofopposite signs in the Gauss constraint associated to 119890 Hencethe stated result follows and the contributions of these twoconfigurations 119896119891119891 and 119896

1015840

119891119891 to the partition function are

equalThe above symmetry is a realization of the well-known

cohomological principle that ldquothe boundary of a boundary iszerordquo (120597 ∘ 120597 = 0) and underlies the Bianchi identity for thecurvature With the Bianchi identity one can also explain thetranslation symmetry see [39 49]

In 3119889 the BF-theory with 119878119880(2) as its gauge grouphas a gravitational interpretation This translation symmetrycorresponds to the diffeomorphism symmetry of the theoryThe gauge field 119896119888 corresponds in the dual lattice to a fieldassociated to the dual vertices and one can interpret thegauge transformation as a translation18 of these dual vertices(which in the gravitymodels are the vertices in a triangulationof space time)

In 4119889 the 3-cells are dual to edges in the dual latticeso this symmetry translates the dual edges rather than thedual vertices For gravity-like models on the other handone would like to break down this translation symmetrywhere it is based on the dual edges to symmetries based onthe dual vertices (Correspondingly in 4119889 diffeomorphismsymmetry of the action leads to theNoether charges encodingthe contracted Bianchi identities and not the Bianchi identityitself) In the case of gravity one strategy is to impose the so-called simplicity constraints [56ndash61 64] within the partitionfunction For models with finite groups the question arisesas to whether there is a class of 4119889 gauge models withldquotranslation-likerdquo symmetries based on the dual vertices Suchmodels would be nontopological and feature propagatingdegrees of freedom as a symmetry group based on dualvertices is much smaller than that for BF where it is basedon dual edges See also the discussion in Sections 5 and 6

5 Gauge Theories for the Non-Abelian Groups

In the following we will consider how to generalize theconstruction performed so far for the finite but non-Abeliangroups119866 Details about representations can be found in [79]

Again we denote by 120588 the irreducible unitary represen-tations of 119866 on C119899

≃ 119881120588 where 119899 = dim 120588 Note that for thenon-Abelian groups 119899 can be larger than 1 Every function119891 119866 rarr C can be decomposed into matrix elements ofrepresentations that is

119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)


where the sum ranges over all ldquoequivalence classesrdquo ofirreducible representations of 119866 The 119891120588 are given by

119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)

where |119866| denotes the number of elements in the group 119866Introducing an inner product between functions 1198911 1198912

119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)

then the matrix elements of 120588 are orthogonal that is

⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)

Furthermore we denote by

120594120588 (119892) = tr (120588 (119892)) (26)

the character of 120588 then the 120575-function on the group119866 is givenby

120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)

51 The 119866-Gauge Theory on a Two-Complex 120581 Consider anoriented two-complex19 120581 Denote the set of edges 119864 and theset of faces 119865 then by a connection we mean an assignment119864 rarr 119866 of group elements 119892119890 to edges 119890 isin 119864 For every face119891 isin 119865 we denote the curvature of this connection by

ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)

which is the ordered product of group elements of edges inthe boundary of 119891 where we take 119900(119891 119890) = plusmn1 depending onthe relative orientation of 119890 and 119891 In the non-Abelian casethe ordering of the group elements around a faceplaquetteactually matters and we choose here the convention asdepicted in Figure 1

As before we define a gauge-invariant partition functionby choosing a collection of weight-functions 119908119891 119866 rarr

C invariant under conjugation These encode the actionldquoand the path integral measurerdquo of the system The partitionfunction is defined as

119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5

Figure 1 A face 119891 bordered by edges 1198901 119890

5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)

where |119866| is the number of elements in 119866 Since 119908119891 can beexpanded into characters via (22) the path integral can bewritten as

119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943

sdot sdot sdot 120588119891(119892plusmn1

119890119899

)1198941198991198941

(31)

In sum there is for each edge 119890 a representation 120588119891(119892119890)

appearing for every face 119891 adjacent to 119890 119891 sup 119890 Thecorresponding sum results in (assuming that the orientationsof all 119891 agree with those of 119890 for the moment in order not tooverburden the notation)

(119875119890)11989411198942sdotsdotsdot11989411989911989511198952sdotsdotsdot119895119899

=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)

It is not hard to see that the operator 119875 called the Haar-intertwiner given by

(119875119890120595)11989411198942sdotsdotsdot119894119899

= (119875119890)11989411198942sdotsdotsdot11989411989911989511198952sdotsdotsdot119895119899

12059511989511198952sdotsdotsdot119895119899

(33)

which maps the edge-Hilbert space

H119890 = 1198811205881

otimes 1198811205882

otimes sdot sdot sdot otimes 119881120588119899

(34)

to itself is actually an orthogonal projector on the gauge-invariant subspace of H119890

20 Note that while in the Abeliancase one has to sum over all representations over faces suchthat at all edges the ldquoorientedrdquo sum adds up to zero (as in(12)) in the non-Abelian generalization one has to sum overall representations such that at each edge 119890 the representationsof the faces 119891 sup 119890 meeting at 119890 have to contain in theirtensor product the trivial representation Also one obtainsa nontrivial tensor 119875119890 for each edge which in the case ofthe Abelian gauge groups is just theC-number 120575plusmn119896

1plusmn1198962plusmnsdotsdotsdotplusmn119896

1198990

Since the representations for the non-Abelian groups can bemore than 1 dimensional in general the tensor 119875119890 has indiceswhich are contracted at each vertex V in the two-complex 120581

Choosing an orthonormal base 120580(119896)119890 119896 = 1 119898 for the

invariant subspace ofH119890 we get

119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2

Figure 2 The relative orientations of both 1198901to 119891 and 119890

2to 119891

agree so in both Hilbert spaces1198671198901and119867

1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894

1198902| have indices belonging to 119881

120588119891in opposite

positions so they may be contracted

In case the orientations of 119890 and 119891 do not agree for some119890 then 119892119890 is essentially replaced by 119892

minus1

119890in (32) which

leads to the appearance of the dual21 representation in thetensor product (34) of the H119890 In this case 120580(119896)

119890labels a

basis of intertwiner maps between the tensor product of allrepresentations associated to faces119891with 119900(119891 119890) = minus1 and thetensor product of all representations associated to the faceswith 119900(119891 119890) = 1

In (35) one regards |120580119890⟩ and ⟨120580119890| as attached to respec-tively the endpoint and the beginning of 119890 For each vertexV in 120581 the associated |120580119890⟩ and ⟨120580119890| can be contracted in acanonical way For every face 119891 which touches V there areexactly two edges 1198901 1198902 in the boundary of 119891 that meet at VThe definitions above are exactly such that if one chooses abasis in each119881120588 and the dual basis in each119881

lowast

120588 in 120580119890

1

and 1205801198902

thetwo indices associated to 119891 are in opposite position so theycan be contracted See Figure 2 for an example

Therefore contracting all the appropriate 120580119890 at one ver-tex leaves one with the vertex-amplitude AV(120588119891 120580119890) whichdepends on the representations 120588119891 and intertwiners 120580119890 associ-ated to the faces and edges thatmeet at V (and the orientationsof these) The vertex amplitude can be computed by evaluat-ing the so-called neighbouring spin network function livingon graph which results from a dimensional reduction of theneighbourhood of V Construct a spin network where thereis a vertex for each edge 119890 touching V and a line between anytwo vertices for each face119891 between two edges Assigning the120588119891 to the lines of the spin network and the intertwiners |120580119890⟩ or⟨120580119890| to the vertices depending on whether the correspondingedge 119890 is incoming or outgoing of V results in a spin networkwhose evaluation givesAV

With this the state sum can using (31) and (32) bewritten in terms of vertex amplitudes via

119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)

52 Examples The first example we consider is the 119866 BF-theory which corresponds to an integral over only flat

connections that is the choice 119908119891(ℎ) = 120575119866(ℎ) Since the 120575-function can be decomposed into characters with (119908119891)120588 =

dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)

If 120581 is the two-complex dual to a triangulation of a 3-dimensional manifold then the vertex amplitude is essen-tially the analogue of the 6119895-symbol for 119866

22

The second example that one usually considers is theYang-Mills theoryTheWilson action can be specified similarto theAbelian case by choosing a unitary representation 120588 sothat

119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)

where 120572 is the coupling constant The weights are then givenby 119908119891(ℎ) = exp(minus119878119884119872(ℎ))

53 Constrained Models There is a generalization of thestate-sum models (36) coming from the desire to obtainnontopological generalizations of the BF-theory It amountsto choosing the intertwiners 120580119890 not to span all of the invariantsubspace but only a proper subspace 119881119890 sub Inv(H119890) Thisoriginates in the attempt to define a state-sum model forgeneral relativity which can be written as a constrained BF-theoryThe subspace119881119890 is viewed as the space of intertwinerssatisfying the so-called simplicity constraints see for exam-ple [56 64 107] Examples for such models are the Barrett-Crane model [57] and the EPRL and FK models [58ndash61]Thesemodels can also bewritten in the formof a path integralover connections [91 108 109] but we will not concernourselves with this here

In the following we will introduce a class of modelswhich can be seen as the generalization of the Barrett-Crane models to finite groups Given a finite group 119867 themodel is a state-sum model for the group 119866 = 119867 times 119867Irreducible representations of 119866 are then pairs of irreduciblerepresentations (120588

+

119891 120588

minus

119891) of 119867 In the edge-Hilbert space

H119890 there is a specific element 120580119861119862 of the space of gauge-invariant elements called the Barrett-Crane intertwiner It isonly nonzero if and only if the representations 120588+

119891and 120588minus

119891are

dual to each other In that case for an edge 119890 with attachedfaces 119891119896 consider the Hilbert space

H119890 = 119881120588+1198911

otimes sdot sdot sdot otimes 119881120588+119891119896

(39)

Again we have assumed that the orientations of all faces119891119896 and 119890 agree in order not to overburden the notationotherwise replace 119881120588+

119891

by its dual or equivalently exchange120588plusmn

119891 If we consider the projector 119890 H119890 rarr H119890 onto the

subspace of elements invariant under the action of119867 then 119890can be seen as an element of H119890 otimes Hlowast

119890 which is naturally

isomorphic to H119890 because 120588plusmn

119891are dual to each other The

corresponding element 120580119861119862 inH119890 is invariant under119867times119867 asone can readily see and it is our distinguished element ontowhich 119875119890 is projecting


Since the projection space for each edge is (at most) 1dimensional the vertex amplitude for the BC-model does notdepend on intertwiners but only on the representations 120588+

119891=

(120588119891)lowast associated to the faces In fact since the representations

are unitary every representation is dual to itself so weeffectively have only one representation 120588119891 attached to eachface Using diagrammatical calculus [110] it is not hard toshow that the vertex amplitude for the BCmodel is given by asum over squares of the BF-theorymodel on the same vertexthat is

A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)

where the sum ranges over an orthonormal basis 120580119890 of 1198783-intertwiners in H119890 for every edge 119890 at the vertex V

Let us shortly discuss the case of 119867 = 1198783 the group ofpermutations in three elements For 1198783 which is generatedby (12) the permutation of the first two elements and (123)which is the cyclic permutation subject to the relation (12)2 =(123)

3= 1 the representations theory is well known [79]

There are three irreducible representations called [1] [minus1]and [2] The first is the trivial representation and the secondone maps a permutation 120590 to its sign (minus1)

120590 The third one istwo dimensional and

120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)

The nontrivial tensor products of the representations decom-pose as

[minus1] otimes [minus1] = [1] [minus1] otimes [2] = [2]

[2] otimes [2] = [1] oplus [minus1] oplus [2]

(42)

Therefore the intertwiner space in H119890 is for example threedimensional when there are four faces attached to 119890 carryingthe representation [2] Hence for 119867 = 1198783 the BC-model isan example for a constrained version of an119867 times119867-state-summodel as defined in the last section because 119875119890 projects ontoa proper subspace of the invariant subspace ofH119890

This class of models is an example for an abstract spinfoam [83] since its amplitudes only depend on combinatorialdata and the topology of the two-complex 120581 In particularit leads to a model which is invariant under refinement ofthe two-complex 120581 by trivial subdivisions of edges or facesHence these models provide potential examples for modelswhich are background independent but not topological (asis the case in the full theory)

Note that there is a wealth of different models whichcome from different choices of nontrivial subspaces of H119890onto which 119875119890 is projecting In particular one could considerEPRL-like models where 119866 = 1198784 times 1198784 or 119866 = 1198604 times 1198604 andconsider the subspace of intertwiners for 120588+

119891= 120588

minus

119891 which

are in the image of a boosting map 119887 119881120588119891

rarr 119881+

120588119891

otimes 119881minus

120588119891

given by the fusion coefficients (see eg [58ndash61] for 119867 =

119878119880(2)) Here 1198784 (1198604) is the group of ldquoevenrdquo permutation offour elements Another possibly more geometric way wouldbe to consider embeddings 1198784 rarr 1198785 and thus constructproper subspaces of 1198785-intertwiners

23 Such models can beconsidered as truncations of the EPRLmodels as 1198784 (1198604) is theldquochiralrdquo symmetry group of the tetrahedron and a subgroupof the rotation group Here it will be interesting to investigatethe relation to the full models in more detail

The models can serve to test many proposals for thefull theory for instance how the different choices of edgeprojectors 119875119890 determine the physical degrees of freedom orparticle content of the given theory This is related to theproblem of implementing the simplicity constraints into spinfoam models We are planning to investigate the features ofthese models in particular their symmetries [65a 65b] andbehaviours under coarse graining in future work

6 The Hilbert Space the Transfer Operatorsand Constraints

We intend to derive the transfer operators [111 112] for theYang-Mills-like gauge theories having partition functions ofthe forms (10) and (30) and incorporating a generic finitegroup 119866 of cardinality |119866| We will permit 119866 to be non-Abelian The first part of the discussion is of a similar natureto that found in [112] for the Yang-Mills theory However wewill also include the BF-theory as a special case From thetransfer operators one can obtain via a limiting procedurethe Hamiltonian operators However for the BF-theory wewill see that this limiting procedure is not necessary Ratherthe transfer operators can be understood as projectors ontothe space of the so-called physical statesThese can be charac-terized as lying in the kernel of the ldquoHamiltonianrdquo constraintsThis illustrates the general principle that a path integral withlocal symmetries acts a projector onto a constrained subspace[113 114] Conversely one can construct a projector onto theconstrained subspace as a path integral See [115] in whichthis is performed for 3119889 gravity in its formulation as an119878119880(2) BF-theory

The transfer operator is defined on a Hilbert space Hso that the partition function can be written as

119885 = trH119873 (43)

We assume for simplicity24 that the lattice is hypercubicalOne lattice direction is designated as a time direction inwhich there are periodic boundary conditions The trace trHis the trace over states in the following Hilbert space It isassociated to the spatial lattice and is built as a direct productof the Hilbert spaces associated to the spatial edges 119890119904 Moresuccinctly written as H = ⨂

119890119904

H119890119904

the ldquoconfigurationrdquovariable attached to any given edge is a group element so theassociated Hilbert space is the space of complex functions onthe group equipped with an inner product25 that is H119890 =

C[119866]We will work in the representation in which a basis of

eigenstates is given by |119892119890⟩This is known as the holonomy or


connection representation For any unitary matrix represen-tation 119892119890 997891rarr 120588(119892119890)119886119887 of the group 119866 these are simultaneouseigenstates of the so-called edge holonomy operators 120588(119892119890)119886119887

120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)

These basis states are orthonormal

⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)

where 120575119866 is the delta function on the group such that(1|119866|) sum

119892120575(119866)

(119892 1198921015840) = 1 We have also the completeness

relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)

We use this resolution of identity 119873 times in (43) to rewritethe partition function as

119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)

where we introduced 119899 = 0 119873 to label the ldquoconstanttime hypersurfacesrdquo in the lattice On the other hand we willassume that our partition function is of the form

119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)

Here we introduced the weight functions 119908119891 of theholonomies around plaquettes ℎ119891 which in the form ofthe right-hand side can also be chosen to differ for spatialplaquettes 119891119904 and timelike plaquettes 119891119905 (This is necessaryif one wants to obtain the Hamiltonian in the limit of smalllattice constant in timelike direction) Since we are dealingwith a gauge theory the weights 119908119891 are class functions thatis invariant under conjugation furthermore we will assumethat 119908119891(ℎ) = 119908119891(ℎ

minus1) that is the weights are independent of

the orientation of the face A weight function can thereforebe expanded into characters which are linear combinationsof matrix elements of representations (in fact characters arejust the trace) Hence the weight functions will be quantizedas edge holonomy operators

On the right-hand side of (48) we have just split theproduct into factors labelled by the time parameter 119899 Herewe assume that the plaquettes (119891119905)119899 are the ones betweentime 119899 and 119899 + 1 Comparing the two forms of the partitionfunctions (47) and (48) we can conclude that

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4

Figure 3 A timelike plaquette in a cubical lattice The groupelements at the timelike edges can be understood to act as gaugetransformations on the vertices either at time steps 119899 or 119899 + 1Integration over the group elements assoicated to the timelike edgesenforces therefore (via group averaging) a projector onto the gaugeinvariant Hilbert space

The two outer factors can be easily quantized as multiplica-tion operators so that we can write

= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)

To tackle the operator note that the group elementsassociated to the timelike edges appearing in the plaquetteweights

119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)

can be understood as acting as gauge transformations associ-ated to the vertices of the spatial lattice see Figure 3

That is we define operators Γ(120574) 120574 isin 119866♯V119904 by

Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1

119905(119890)and 119904(119890) 119905(119890) are the source

and target vertices of the edge 119890 respectively The operatorsΓ generate gauge transformations as

⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)

Now we can see the plaquette weight in (52) as either awave function at time 119899 or a wave function at time 119899 + 1

on which the group elements associated to the timelikeedges act as gauge transformations The sum in (51) over thegroup elements 119892119890

119905

then induces an averaging over all gaugetransformations that is a projection onto the space of gauge-invariant states Hence we can write

= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)


The operator 119866 is a projector that is 2119866

= 119866 One caneasily show that Γ(120574)119866 = 119866Γ(120574) = 119866 for any 120574 isin 119866

♯V119904

hence it projects onto the space of gauge-invariant functionsAs the operator is made up from gauge-invariant plaquettecouplings it commutes with 119866 so that we can write

= 1198660119866 (57)

Here we see an example for a general mechanism namelythat the transfer operator for a partition function with localgauge symmetries acts as a projector onto the space of gauge-invariant states We can characterize such gauge-invariantstates as being annihilated by constraint operators In the caseof the usual lattice gauge symmetries these constraints are theGauss constraints which we will discuss later on

To discuss the action of 1198700 we introduce first the spinnetwork basis in which 0 will be diagonal

61 SpinNetwork Basis So far we encountered the holonomyoperators that is the multiplication operators 120588(119892119890)119886119887 in theconfiguration representation The conjugated operators arethe electric fields or fluxes which act as ldquomatrixrdquo multiplica-tion operators in the spin network basis This is a convenientbasis to discuss the remaining part1198700 of the transfer operator

To obtain the spin network basis [11 12] we just need touse that every function on the group can be expanded as asum over the matrix elements 120588(sdot)119886119887 of all irreducible unitaryrepresentations 120588 For the Abelian groups all irreduciblerepresentations are one dimensional hence 119886 119887 = 0 andwe obtain the discrete Fourier transform (3) In the generalcase of the non-Abelian groups we define spin network states|120588 119886 119887⟩ by

prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)

which are orthonormal that is ⟨120588 119886 119887 | 1205881015840 119886

1015840 119887

1015840⟩ =

120575120588120588101584012057511988611988610158401205751198871198871015840 In the spin network basis we can easily express the left

119871119890(ℎ) and right translation operators 119877119890(ℎ) These are theconjugated operators to the holonomy operators (44) Tosimplify notation we will just consider the Hilbert space andstates associated to one edge and omit the edge subindex Wedefine

(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)

In the spin network basis

⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)

where we sum over repeated indices That is

(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)

The remaining factor 1198700 of the transfer operator factorizesover the edges and it is straightforward to check that it actson one edge as

0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)

(Here one has to use that119908119891119905

is a class function and invariantunder inversion of the argument) On a spin network statewe obtain

0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)

where in the second line we used that a class function canbe expanded into characters 120594120588 and in we used the third theorthogonality relation

1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)

The expansion coefficients 119888120588 are given by

119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)

That is1198700 acts as a diagonal in the spin network basis and asthe eigenvalues only depend on the representation labels 120588 itcommutes with left and right translations For the Yang-Millstheory (with Lie groups) in the limit of continuous time oneobtains for1198700 the Laplacian on the group [111 112]

62 Gauge-Invariant Spin Nets Given a spin network func-tion 120595 which can be regarded as function 120595(ℎ119890) ofholonomies associated to edges where ℎ isin 119866

119890 is a connec-tion and a gauge transformation 120574 isin 119866

V an assignmentof group elements to vertices of the network then the gaugetransformed spin network function is given by

Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)

where 119904(119890) and 119905(119890) are the source and the target vertices ofthe edge 119890 A gauge transformation can therefore be expressedas a linear operator in terms of and as shown in thelast section Decomposing the spin network function into thebasis |120588119890 119886119890 119887119890⟩ that is

120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)

one can readily see that the condition for a function 120595 to beinvariant under all gauge transformations 120572119892 translates into acondition for the coefficients 120588

119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)


where the product ranges over vertices V and each tensor 120580(V)which has the indices 119886119890 for each edge 119890 outgoing of and 119887(119890)for each edge 119890 incoming V is an invariant element of thetensor representation space

120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)

that is an intertwiner between the tensor product ofincoming and outgoing representations for each vertex Anorthonormal basis on the space of gauge-invariant spinnetwork functions therefore corresponds to a choice oforthonormal intertwiner in each tensor product representa-tion space for each vertex where the inner product is thetensor product of the inner products on each 119881120588 119881

lowast

120588

Note that neither the holonomies 120588(ℎ119890)119886119887 nor left andright translations are gauge invariant therefore they mapgauge-invariant functions to nongauge-invariant ones How-ever they can be combined into gauge-invariant combina-tions such as the holonomy of a Wilson loop within a net

63 Local Constraint Operators and Physical Inner ProductWe have seen that for a general gauge partition function ofthe form of (48) the transfer operator (57)

= 1198660119866 (70)

incorporates a projection 119866 onto the space of gauge-invariant statesThis is a general feature of partition functionswith gauge symmetries [113 114] Indeed in quantum gravityone attempts to construct a projector onto gauge-invariantstates by constructing an appropriate partition function

As has been discussed in Section 41 theBF-models enjoya further gauge symmetry the translation symmetries (21)(A generalization of these symmetries holds also for the non-Abelian groups) Hence we can expect that another projectorwill appear in the transfer operator Indeed it will turnout that the transfer operator for the BF-models is just acombination of projectors

This is straightforward to see as for the BF-models thatthe plaquette weights are given by 119908119891(ℎ) = 120575119866(ℎ) so that

= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)

Here one might be worried by taking the square roots ofthe delta functions however we are on a finite group Theoperator 2 is diagonal in the basis |119892⟩ and has only twoeigenvalues 119902♯119891119904 on states where all plaquette holonomiesvanish and zero otherwise The square root of 2 is definedby taking the square root of these eigenvalues

Hence is proportional to another projector 119865 whichprojects onto the states forwhich all the plaquette holonomiesare trivial that is states with zero (local) curvature

The final factor in the transfer operator is 0 whichaccording to the matrix element of 0 in (56) (putting119908119891(ℎ) = 120575119866(ℎ)) is proportional to the identity

0 = |119866|♯119890119904IdH (72)

where the numerical factor arises due to our normalizationconvention for the group delta functions which amounts to120575119866(id) = |119866| Hence the transfer operator for the BF-theoryis given by

= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)

and since for the projectors we have 119873119866

= 119866 119873

119865= 119865 the

partition function simplifies to

119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)

The projection operators in the transfer operator ensure thatonly the so-called physical states 120595 isin Hphys are contributingto the partition function 119885 These are states in the imageof the projectors (we will call the corresponding subspaceHphys) and can be equivalently described to be annihilatedby constraint operators which we will discuss below

The objective in canonical approaches to quantum gravity[11 12] is to characterize the space of physical states accordingto the constraints given by general relativity which followfrom the local symmetries of the theory (The quantizationof these constraints in a consistent way is highly complicated[116ndash118]) Indeed also in general relativity as well as in anytheory which is invariant under reparametrizations of thetime parameter one would expect that the transfer operator(or the path integral) is given by a projector so that theproperty

2sim would hold26 This would also imply a

certain notion of discretization independence namely theindependence of transition amplitudes on the number of timesteps used in the discretization see also the discussion in [38]on the relation between reparametrization symmetry in thetime parameter and discretization independence For a latticebased on triangulations one can introduce a local notion ofevolution the so-called tent moves [41 119ndash121]Thesemovesevolve just one vertex and the adjacent cells Local transferoperators can be associated to these tent moves These wouldevolve the spatial hypersurface not globally but locally andwould allow to choose some arbitrary order of the verticesto evolve In this way one can generate different slicings of atriangulation aswell as different triangulationsThe conditionthat the transfer operator associated to a tentmove is given bya projector would then implement an even stronger notion oftriangulation independence

However reparametrization invariance which wouldlead to such transfer operators is usually broken by dis-cretizations already for one-dimensional toy models Forldquocoarse graining and renormalizationrdquo methods to regainthis symmetry see [36ndash38 122] In the case of gravity ormore generally nontopological field theories these methodswill however lead to nonlocal couplings for the partitionfunctions [122] for which one would have to modify thetransfer operator formalism

Furthermore one has to construct a physical innerproduct on the space of the physical sates This process isquite trivial in the case considered here as we are consideringfinite groups and all of the projectors are proper projectors


(a) (b)

Figure 4 Two states in the spin network basis for Z2which are

equivalent under the physical inner product The thick edges carrynontrivial representations The right state can be obtained from theleft state by applying a plaquette stabilizer

Hence the physical states are proper states in the ldquoso-calledrdquokinematical Hilbert space H and the inner product of thisspace can be taken over for the physical states In contrastalready for the BF-theory with ldquocompactrdquo Lie groups thetranslation symmetries lead to noncompact orbits This leadsto physical states which are not normalizable with respect tothe inner product of the kinematical Hilbert space For theintricacies of this case (with 119878119880(2)) see for instance [115] Afurther complication for gravity is the very complicated non-commutative structure of the constraints [123ndash125]

Let us summarize the projectors onto = 119866 119865 Also inthe case of generalized projectors a physical inner productcan be defined [12 126] between equivalence classes ofkinematical states labelled by representatives 1205951 1205952 through

⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)

Kinematical states which project to the same (physical) statedefine the same equivalence class This leads for instanceto an identification of the two states (for the group Z2)in Figure 4 This is analogous to the action of the spatialdiffeomorphism group in ldquo3119863 and 4119863rdquo loop quantumgravitysee the discussion in [115] and reflects the concept ofabstract spin foams whose amplitude does not depend on theembedding into the lattice in Section 2 Indeed any two stateswhich can be deformed into each other by applying the so-called stabilizer operators introduced below in (76) and (82)are equivalent to each other The reason is that the physicalHilbert space is the common eigenspace to the eigenvalue 1with respect to all of the stabilizer operators Hence any partof a state that undergoes a nontrivial change if a stabilizer isapplied will be projected out in the physical inner product

Another problem in quantum gravity is then to findldquoquantumrdquo observables [127ndash131] that are well defined on thephysical Hilbert space In the case of gravity these Diracobservables are very hard to obtain explicitly even classicallythere are almost none known In particular such Diracobservables have to be nonlocal [132] If one interprets aquantum gravity model as a statistical system a related task

is to find a well-defined order parameter characterizing thephases Expectation values of gauge-variant observables maybe projected to zero under the physical inner product [133]As in our case we work with finite-dimensional Hilbertspaces and the physical Hilbert space is just a subspace ofthe kinematical one there is a construction principle forphysical observables For any operator 119874 on the kinematicalHilbert space one can consider 119874 which preserves thephysicalHilbert space and corresponds to a ldquogauge averagingrdquoof 119874 However many observables constructed in this waywill turn out to be constants For the BF-theory observablescan be constructed by considering a product of the stabilizeroperators discussed below along non-contractable loops [54134ndash136] The resulting observables are nonlocal and thecardinality of a set of independent operators (equivalently thedimension of the physical Hilbert space) just depends on thetopology of the lattice and not on its size

We will now discuss the stabilizer operators whichcharacterize the physical states In topological quantumcomputing these are known as stabilizer conditions [54]

We have considered the operators Γ(120574) in (53) that imple-ment a gauge transformation according to the assignments(120574)V119904

of group elements to the vertices of the ldquospatial sub-rdquolattice These can be easily localized to the vertices V119904 ofthe spatial lattice by choosing the gauge parameters 120574 (anassignment of one group element to every vertex in the spatiallattice) such that all group elements are trivial except forone vertex V119904 We will call the corresponding operators ΓV

119904

(120574)

where now 120574 denotes an element in the gauge group 119866 (andnot in the direct product 119866♯V

119904) These can be expressed by theleft and right translation operators (59)

ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)

Physical states |120595⟩ have to satisfy27

ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)

for all vertices V119904 and all group elements 120574 As the operatorsΓ define a representation of the group we can restrict to theelements of a generating set of the group

This suggests considering the action of these ldquostarrdquo oper-ators in the spin network basis

ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)

where on the right-hand side edges 119890 are the ones starting atV119904 119890

1015840 are those ending at V119904 and 11989010158401015840 are all of the remaining

edges in the spatial lattice From this expression one canconclude that the spin network states transform at V119904 in atensor product of representations

120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)


where 120588lowast is the dual representation to 120588 Gauge-invariant

states can be constructed by considering the projections ofthese representations to the trivial ones or equivalently bycontracting the matrix indices of the states with the inter-twiners between the tensor product of ldquoincomingrdquo represen-tations and the tensor product of ldquooutgoingrdquo representationsat the vertices see Section 62

For the Abelian groups Z119902 the irreducible representa-tions are one-dimensional hence we can omit the indices 119886 119887in the spin network basis The tensor product of represen-tations (79) is also one dimensional and equal to the trivialrepresentation provided that

sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)

for the representation labels 119896119890 1198961198901015840 of the outgoing andincoming edges at V119904 respectively Hence we recover theGauss constraints from the spin foam representation (12)Here these Gauss constraints appear as based on verticesas opposed to edges as in (12) But the spatial vertices V119904represent the timelike edges and the Gauss constraints in thecanonical formalism are just the Gauss constraints associatedto the timelike edges in the spin foam representation (12)

Let us turn to the other projector 119865 It maps to states forwhich all of the spatial plaquettes 119891119904 have trivial holonomiesHence the conditions on physical states can be written as

119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)

where we have introduced the plaquette operators (corre-sponding to the flatness constraints)

119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)

Here 120588 should be a faithful representation otherwise onemight find that also states with local non-trivial holonomiessatisfy the conditions (81) see for instance the discussion in[137] On the other hand this can be taken as one possiblegeneralization of the model For the non-Abelian groups theplaquette operators additionally depend on the choice of avertex adjacent to the face at which the holonomy aroundthe face starts and ends However the conditions (81) do notdepend on this choice of vertex if a holonomy around a faceis trivial for one choice of vertex then it will be trivial for allother vertices in this face as these holonomies just differ by aconjugation

The BF-theory in any dimension is a topological fieldtheory that is there are only finitelymany physical degrees offreedom depending on the topology of space Consequentlythe number of physical states or equivalently of equivalenceclasses of kinematical states in the sense of (75) is finite anddoes not scale with the lattice volume

There are a number of generalizations one could considerFirst one can couple matter in the form of defects to themodels In 3D the 119878119880(2) BF-theory corresponds to a first-order formulation of 3D gravity Point-like particles can becoupled to the model see for instance [134ndash136] and lead

to the violation of the flatness constraint (82) (coupling tothe mass of the particles) and to the violation of the Gaussconstraints (77) (coupling to the spin of the particles) at theposition of the particle The defects can be understood aschanging the topology of the spatial lattice that is changingits first fundamental group To have the same effect in sayfour dimensions one needs to couple strings see [138ndash140]

In the context of quantum computing [54] one doesnot necessarily impose the conditions (77) and (82) asconstraints but as characterizations for the ground statesof the system Elementary excitations or quasi-particles arestates in which either the flatness or the Gauss constraintsare violated These excitations appear in pairs (at least for theAbelian models [141]) and can be created by so-called ribbonoperators [54 141ndash143] which in the context of the properparticle models in 3D gravity are related to gauge-invariantldquoDiracrdquo observables [144] For further generalizations basedon symmetry breaking from 119866 to a subgroup of 119866 see forinstance [141]

In 4D gravity is not topological rather there are propa-gating degrees of freedom Similar to the discussion for thepartition functions in Section 41 one can try to constructnewmodels which are nearer to gravity by breaking down thesymmetries of the BF-theory In 3D the flatness constraintsare defined on the plaquettes of the lattice Constraintsact also as generators of gauge transformations (indeed theconditions (77) and (82) impose that physical states haveto be gauge invariant) and the flatness constraints can beinterpreted as translating the vertices of the dual lattice inspace time [39 40 145 146] which can be seen as an action ofa diffeomorphism In 4D the same interpretation holds onlyfor some exceptional cases corresponding to special latticesthat do not lead to space time curvature see the discussionin [107] In general the flatness constraints rather generatetranslations of the edges of the dual lattice [147]

A possible generalization leading to gravity-like modelsis therefore to replace the flatness conditions (81) based onplaquettes with some contractions of these conditions suchthat these new conditions are based on the 3-cells thatis cubes in a hyper-cubical lattice This would correspondto the contractions of the form 119865119864119864 (the Hamiltonianconstraints) and 119865119864 (diffeomorphism constraints) in theldquocomplexrdquo Ashtekar variables of the curvature 119865 with theelectric flux variables 119864 [11 12 148ndash150] The difficulty inconstructing such models is consistency of the constraintalgebra that is to find constraints that form a closed algebraHowever to consider such models for finite groups should bemuch simpler than in the case of full gravity Even if suchconsistent constraint algebras cannot be found the physicalHilbert spaces could be constructed for instance with thetechniques in [123ndash125 151]

Furthermore it will be illuminating to derive the trans-fer operators for the constrained models introduced inSection 53 as in the full theory the relation between thecovariant models and the Hamiltonians in the canonicalformalism is still open [152 153] To this end a connectionrepresentation [91 109] can be employed


7 Tensor Models and Tensor GroupField Theories

Tensor models are theories of random topological spaces inthe fashion of matrix models [154 155] of which they may beseen as a superset

Matrix models are a well-studied theory that can describea multitude of different scenarios including 2119889 gravity withcosmological constant [156ndash159] (optionally coupled to the2119889 IsingPotts matter [160 161] or the 2119889 Yang-Mills matterstheories [162]) certain string theories [163] the enumerationof virtual knots and links [164ndash167] and the list goes onMoreover they have inspired the development of a plethoraof useful techniques to solve and analyse statistical ensemblesof random matrices [168] such as the topological expansion[169ndash172] the eigenvalue method [173] the double-scalinglimit the method of orthogonal polynomials [174] and thecharacter expansion [175 176] to name just a few Tensormodels are a natural extension of this idea to higher dimen-sions The ambition is to develop a similar technology withsimilar applications for tensor models in general

Despite some initial pessimism [177] a recent spikeof optimism occurred within the growing tensor modelcommunity when it was discovered that a large class ofsuch theories [18 178ndash182] possessed a 1119873-expansion [183ndash189] that is their Feynman graphs could be partitionedinto manageable subsets using some parameter 119873 In factit was shown that the leading order sector in the large-119873 limit contained an infinite but manageable number ofthe Feynman graphs with the topology of the 119863-sphere (119863being the rank of the tensor) This leading order sector wasanalysed for a variety of models which included the plainmodel [190 191] placing the IsingPotts [192] and dimer [193194] matter interactions on these 119863-dimensional topologicalspaces as well as dually weighing the spaces [195] Criticalexponents were extracted indicating that this sector of thetheory displayed identical physical properties to those ofthe branched polymer phase of the Euclidean dynamicaltriangulations [196 197] This likelihood was further con-firmedwhen it was shown that their respectiveHausdorff andspectral dimensions coincided [198] Interestingly aspectsof universality have also been displayed by this leadingorder sector [199] while the quantum symmetries have beencatalogued at all orders and take the suggestive form of aVirasoro-like algebra [200ndash202]

While the majority of the results stated above has beenexplicitly detailed for the simplest class of tensor models theindependent identically distributed IID tensor models theinitial result on the 1119873-expansion applies to many moreclasses of tensor models including certain topological andquantum-gravity-inspired tensormodelsThus a current aimis to develop the above formalism for these broader tensormodel scenarios As stated in the introduction spin foamsfor quantum gravity generically involved the Lie groups andso more structure than that provided by tensor models isneeded to describe them

Tensor group field theories (TGFTs) [19ndash21 203] areto quantum field theory as tensor models are to quantum

mechanics Spin foam diagrams naturally arise as the Feyn-man graphs of TGFTs [1ndash5] and indeed various quantum-gravity-inspired TGFTs exist in the literature such as theBoulatov-Ooguri theories [204 205] the EPRL and FKtheories [22ndash24 206] and the Baratin-Oriti theories [2526] As a result they have garnered increasing attentionfrom the quantum gravity community since inherently allcurrent spin foam models for 4d quantum gravity involve atruncation of the degrees of freedom (due to the use of a singlelattice structure) and a manifest loss of diffeomorphism sym-metry With their explicit summation over 119863-dimensionaltopological spaces the hope is that TGFTs recapture theselost degrees of freedom Indeed some evidence has alreadybeen found at the level of the TGFT action for a seed of dif-feomorphism symmetry [207] Moreover many techniquesmay be applied to these field theories that are idiosyncratic toquantum field theories (as opposed to quantum mechanics)Perhaps themost striking is the study of their renormalisationgroup properties [208ndash218] but also work has commencedon the study of mean field theory properties [219] mattercoupling [220ndash222] various symmetry analyses [223 224]and instantonic field theory solutions [225ndash228] (includingcosmological applications [229])

Having said all that the aim of this paper is not to detailspin foam models with the Lie groups but rather spin foammodels with finite groups This places us back under thepurview of tensor models and it is these that we will describein the coming section

To commence let us detail some of the general setup

71 General Setup Let us construct the class of independentidentically distributed (IID) models We will attempt to beprecise without being especially detailed We refer the readerto [230] for a more thorough explanation The fundamentalvariable is a complex rank-119863 tensor (119863 ge 2) which maybe viewed as a map 119879 1198671 times sdot sdot sdot times 119867119863 rarr C where the119867119894 are complex vector spaces of dimension 119873119894 It is a tensorso it transforms covariantly under a change of basis of eachvector space independently Its complex-conjugate 119879 is itscontravariant counterpart

One refers to their components in a given basis by 119879119892 and119879119892 where 119892 = 1198921 119892119863 119892 = 119892

1 119892

119863 and the bar (minus)

distinguishes contravariant from covariant indices We havepurposefully denoted the indices by 119892119894 as they may be viewedas elements of respective Z119873

119894

As one might imagine with these ingredients one can

build objects that are invariant under changes of basesTheseso-called trace invariants are a subset of (119879 119879)-dependentmonomials that are built by pairwise contracting covariantand contravariant indices until all indices are saturatedIt emerges readily that the pattern of contractions for agiven trace invariant is associated to a unique closed 119863-colored graph in the sense that given such a graph onecan reconstruct the corresponding trace invariant and viceversa28

In Figure 5 we illustrate the graphB1 the unique closed119863-colored graph with two vertices which represents theunique quadratic trace invariant trB

1

(119879 119879) = 119879119892 120575119892119892 119879119892


More generally we denote the trace invariant correspond-ing to the graphB by trB(119879 119879)

From now on we will make two restrictions (i) all of thevector spaces have the same dimension119873 and (ii) we consideronly connected trace invariants that is trace invariants corre-sponding to graphs with just a single connected component

Given these provisos the most general invariant actionfor such tensors is

119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896

is the set of connected closed 119863-colored graphswith 2119896 vertices 119905B is the set of coupling constants and120596(B) ge 0 is the degree of B (see [18] for its definition andproperties) This defines the IID class of models

72 1119873-Expansion The central objects for further investi-gation are the free energy (per degree of freedom) associatedto these models

119864 (119905B) =1

119873119863log(int119889119879119889119879119890

minus119873119863minus1

119878(119879119879)) (84)

along with the various other 119899-point Green functions Whenfacing such a quantity the standard procedure is to expandit in a Taylor series with respect to the coupling constants 119905Band to evaluate the resultingGaussian integrals in terms of theWick contractions It transpires that the Feynman graphs Gcontributing to 119864(119905B) are none other than connected closed(119863 + 1)-colored graphs with weight

119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)

SYM(G) is a symmetry factor and B(120588) runs over thesubgraphs with colors 1 119863 of G while VG and FG

are the vertex and face sets of G respectively For 119863 =

2 the degree 120596(G) coincides with the genus of a surfacespecified by G A few more words of explanation are mostdefinitely in order here The graphs G isin Γ

(119863+1) arise in thefollowing manner One knows that a given term in the Taylorexpansion is a product of trace invariants upon which oneperforms Wick contractions For such a term one indexesthese trace invariants by 120588 isin 1 120588

max that is we

index their associated 119863-colored graphsB(120588) A single Wickcontraction pairs a tensor119879 lying somewhere in the productwith a tensor 119879 lying somewhere else One represents sucha contraction by joining the black vertex representing 119879 tothe white vertex representing 119879 with a line of color 0 Thus acomplete set of the Wick contractions results in a connected(as one is dealing with the free energy) closed (119863+1)-coloredgraph A particular Wick contraction is drawn in Figure 6

1

2

3

ℬ1

trℬ1(T T) = Tg1g2g3 120575g1g1 120575g2g2 120575g3g3 Tg1g2g3

Figure 5 A closed 119863-colored graph and its associated traceinvariant (for119863 = 3)

0 00

0

0

1

11

1

12

2

2

2 23

3

3

33

Figure 6 A Wick contraction of two trace invariants (for 119863 = 3)The contraction of each (119879 119879) pair is represented by a dashed lineof color 0

It requires a bit more work to reconstruct the amplitudeexplicitly see [230] Importantly 120596(G) is a nonnegativeinteger and so one can order the terms in the Taylorexpansion of (84) according to their power of 1119873 Quiteevidently therefore one has a 1119873-expansion

73 Interpretation Strikingly (119863 + 1)-colored graphs repre-sent119863-dimensional simplicial pseudomanifoldsWewill givea rough presentation here One distinguishes the 119896-bubbles ofspecies 1198941 119894119896 as those maximally connected subgraphswith the colors 1198941 119894119896These 119896-bubbles are identifiedwiththe (119863 minus 119896)-simplices of the associated simplicial complexesMoreover one can see clearly that the 119896-bubbles of species1198941 119894119896 are nested within (119896 + 1)-bubbles of species1198941 119894119896 119895 where 119895 isin 0 119863 1198941 119894119896 This setof nested relationships encodes the gluing of the simpliceswithin the simplicial complex This argument may be maderigorously Thus at the very least rank-119863 tensor modelscapture a sum over119863-dimensional topological spaces

Moreover a geometrical interpretation may be attachedmost readily to the amplitudes of the IID model by setting119905B = 119892

|VB|2SYM(B) for all B where 119892 is some couplingconstant Then one may rewrite the amplitudes as

SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)

whereN119896 denotes the number of 119896 simplices in the simplicialcomplex associated toG and

120581119863minus2 = log119873

120581119863 =1

2(1

2119863 (119863 minus 1) log119873 minus log119892)

(88)


Interestingly the discretization of the Einstein-Hilbert actionon an equilateral119863-dimensional simplicial complex takes theform

119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)

= Λ vol (120590119863)N119863 minusvol (120590119863minus2)16120587119866Newton

times (2120587N119863minus2 minus119863 (119863 + 1)

2arccos( 1

119863)N119863)

(89)

where 119866Newton and Λ are the bare Newton and cosmologicalconstants respectively and vol(120590119896) = (119886

119896119896)radic(119896 + 1)2119896

is the volume of the equilateral 119896-simplex while 120575(120590119863minus2)

denotes the deficit angles associated to the (119863 minus 2)-simplicesIf one further associates (119873 119892) to (119866Newton Λ) in the followingfashion

log119873 =vol (120590119863minus2)8119866Newton

log119892

=119863

16120587119866Newtonvol (120590119863minus2)

times (120587 (119863 minus 1) minus (119863 + 1) arccos 1119863)

minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)

then one finds that the Feynman amplitudes for the IIDmodel are coincide with those in a Euclidean dynamicaltriangulations approach We will return to this later

74 Large-119873 Limit In the large-119873 limit only one subclassof graphs survives containing those graphs with 120596(G) = 0At this point there is a marked difference between two andhigher dimensions

(i) In the 2-dimensional model 120596(G) is the genus of thegraph in question Thus the graphs surviving in thislimit are all graphs with the topology of the 2-sphere

(ii) In higher dimensions that is 119863 ge 3 it was shown in[190 191] that the only graphs surviving this limitingprocedure are the melonic graphs (with this namestemming from their distinctive structure) Whilethese have the topology of the 119863-sphere they do notconstitute all possible (119863+1)-colored graphs with thistopology

The series constituting the leading order sector

119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)

has a finite radius of convergence This indicates that thetheory displays critical behaviour for some values of thecoupling constants characterised by some critical exponentsThere are various scenarios that one might investigate Onesimple yet interesting case is to set 119905B = 119892

|VB|2SYM(B) forallB where 119892 is some coupling constant In this scenario theseries display the following critical behaviour

119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)

Multicritical behaviour can be extracted by tuning severalcoupling constants independently

Given the description provided in the previous sub-section one notices that the large-119873 limit corresponds to119866Newton rarr 0 Moreover tuning 119892 rarr 119892119888 one enters theregime dominated by graphs with large numbers of vertices(or equivalently simplicial complexes with large numbers ofsimplices) As it stands this corresponds to a large-volumelimit However with some more work one can interpret it asa continuum limit To begin the average volume is

⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)

To obtain a continuum limit one should tune 119892 rarr 119892119888 whilekeeping the average volume finite This may be achieved inthe following fashion

119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)

where Λ 119877 is a renormalized cosmological constant

75 Coupling to the Ising and PottsMatter Thecoupling to theIsingPotts matter in tensor models is a direct generalisationof that scenario in matrix models [231 232] For the Ising


matter one considers a model with two complex tensors andaction

119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)

and 119888 is a coupling constant This class of models hasbeen examined in [192] where critical exponents have beenextracted which coincide with those of branched polymers

This matter model may be extended to the 119902-state Pottsmatter by increasing the number of tensors along with asuitable alteration of the propagator

76 Non-IID Models The IID models are but the simplest ofa much larger class of models possessing a 1119873-expansiondefined by actions of the form

119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)

The difference resides in the propagator and the scalings120572(B) of the coupling constant which can be chosen toreproduce any local (quantum-gravity-inspired) spin foammodel (in this case with the finite rather than the Liegroup information) As an example let us briefly detail thechoice that yields the Boulatov-Ooguri class of tensormodelsFirstly the propagator is chosen to be

119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)

The 120572(B) can be specified so that the resulting amplitudes forthe free energy take the form

119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)

Given that EG and FG are the edge and face sets of Grespectively while 119900(119890 119891) is the respective orientation of theedge 119890 lying in the boundary of the face 119891 it is clear that119867119891

is the holonomy around the face 119891 As a result the amplitudewithin square brackets is the BF-theory amplitude associated

to the topological manifold represented by G It may beevaluated to obtain some G-dependent factor of 119873 (actuallydirectly dependent on the second Betti number of G [233ndash235]) which allows the graphs to be organised into a 1119873-expansion

A more in-depth analysis has yet to be completedalthough these preliminary results are very promising Onemay modify the propagator further to obtain the EPRLFK [22ndash24] and BO [25 26] quantum gravity spin foamamplitudes

8 Nonlocal Spin Foams

We now turn briefly to one of the main open problems facingquantum gravity practitioners the study of the large-scalebehaviour emerging from various models We will restrictourselves here to two points the first motivates the use of thesimple models considered here in this endeavour the secondcomments on how the study of large-scale behaviour maynecessitate the treatment of nonlocality

To pass from small to large scales it is clear that renor-malization group methods could be useful However theapplication of such techniques at least to those spin foammodels currently discussed as quantum gravity candidates[57ndash61] is hindered not only by many conceptual issues butalso by the tremendously complicated amplitudes emergingfrom these models Here our ldquotoy spin foam modelsrdquo couldhelp both to develop new techniques and to investigate thestatistical field theory aspects of the models in question

As a matter of fact it may be the case that certainproperties are independent of the specifics of the amplitudesFor instance consider the questions of whether and how tosum over topologies within spin foammodelsThismight notdepend on the chosen group underlying the model

On top of that note that there are a number of quantumgravity approaches which rather work with quite simple(vertex) amplitudes [13ndash17 73ndash76] (we have in mind here thework of [16 17] in particular) in which the question of thelarge-scale limit can be addressed The models proposed inourwork here can be seen as small spin cut-off versions of thefull theoryThe hope is that for thesemodels one can replicatethe successes of [16 17] by probing the many-particle (andsmall-spin) regime in contrast to the very few-particle andlarge-spin regime considered up to now

There has been some very interesting work exploringcoarse graining in the spin foam context [236ndash238] Inde-pendently the related concept of tensor networks [239ndash242]a generalization of the spin nets introduced here has beendeveloped as a tool for coarse graining In most frameworksthe local form of the spin foams does not change It maybe necessary to accommodate also for nonlocal couplings29in particular if one attempts to regain diffeomorphism sym-metry which is broken by the discretizations employed inmany quantum gravity models [36 37 122] As explained in[243] such nonlocal couplings can be accommodated intothe tensor network renormalization framework In this casethe nonlocal couplings are compensated by introducingmore


and more complicated building blocks (carrying higher-order variables)

Indeed after applying block spin transformations toa nontopological lattice gauge system one can expect topossess a partition function of the form

119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894

are holonomies around loops (that is aroundplaquettes but also around other surfaces made of severalplaquettes) and 119908119897

1119897119872

is a class function of its argumentsdescribing possible couplings between (Wilson) loops Acharacter expansion would introduce representation labelsnot only on the basic plaquettes but also on all of the othersurfaces encircled by loops appearing in (103) The resultingstructure is akin to a two-dimensional generalization of agraph Graphs would appear as effective descriptions forthe coarse graining of the Ising-like models discussed inSection 2 Such nonlocal ldquospin graphsrdquo would be general-izations of the spin nets introduced there Similarly graphshave been introduced in [27 28] to accommodate nonlo-cal couplings and to describe phase transitions between ageometric phase (ie a phase where for instance a spacetime dimension can be defined) and a nongeometric phaseWe leave the exploration of the ensuing structures for futurework

9 Outlook

In this work we discussed several concepts and tools whicharose in the spin foam loop quantum gravity and group fieldtheory approach to quantum gravity and applied these tofinite groups We encountered different classes of theoriesone is the well-known example of the Yang-Mills-like the-ories others are the topological BF-theories The latter arealso well known in condensed matter and quantum com-puting A third class of theories are the constrained modelsdiscussed in Section 53 which mimic the construction of thegravitational models These are only applicable for the non-Abelian groups as for theAbelian groups the invariantHilbertspace associated to the edges is always one dimensional andcannot be further restricted We plan to study these theoriesin more detail in the future in particular the symmetriesand the associated transfer or constraint operators Therelative simplicity of these models (compared with the fulltheory) allows for the prospect of a complete classificationof the choices for the edge projectors and hence of thedifferent constrained models For the study of the translationsymmetries in the non-Abelian models it might be fruitfulto employ a non-commutative Fourier transform [25 26207 244ndash246] This would define an alternative dual forthe non-Abelian models in which the edge projectors carrydelta function factors however defined on a noncommutativespace

We also mentioned the possibilities to obtain gravity-like models by breaking down the translation symmetries ofthe BF-theory This could be particularly promising in thecanonical formalism where one can be guided by the form of

the Hamiltonian constraints for gravity This strategy is alsoavailable for the Abelian groups In particular for Z2 it is inprinciple possible to parametrize all possible Hamiltoniansor partition functions and to study the associated symmetrycontent See also the universality result in [89] Hence theremight be a definitive answer whether gravity-like modelsexist that is 4D (Z2) lattice models with translation symme-tries based on the 4-cells (or the dual vertices)

Finally we hope that these models can be helpful in orderto develop techniques for coarse graining and renormaliza-tion of spin foam models and in group field theories Herethe connection to standard theories could be exploited andtechniques can be taken over from the known examples andwith adjustments being applied to quantum gravity modelsTo this end the finite group models could be an importantlink and provide a class of toy models on which ideas can betestedmore easily For instance the connection between (realspace) coarse graining of spin foams and renormalizationin group field theories which generate spin foams as theFeynman diagrams could be explored more explicitly thanin the (divergent) 119878119880(2)-based models

It will be particularly interesting to access the many-particle regime for instance using the Monte Carlo simula-tions which for the full models is yet out of reach In the idealcase it might be possible to explore the phase structure of theconstrained theories and to study the symmetry content ofthese different phases

Acknowledgments

Bianca Dittrich thanks Robert Raussendorf for discussionson topological models in quantum computing The authorsare thankful to Frank Hellmann for illuminating discussions

Endnotes

1 In some cases the resulting models might then bereformulated as ldquotilingrdquomodels on a fixed lattice [30ndash32]

2 The small-spin regime arises as the spin is just a labelfor representation of the group and for finite groupsthere is only a finite number of irreducible unitaryrepresentations

3 The models can be extended to oriented graphs We willhowever not be interested in the most general situationbut will assume that the lattice or more generally cellcomplex is sufficiently nice in order to construct the duallattices or complexes Later on we will also need to beable to identify higher-dimensional cells (2-cells and 3-cells) in the lattice

4 The reader may be more familiar with the form of themodel in which the variables take the values119892V isin minus1 1which corresponds to the matrix representation of Z2119892 rarr 119890

120587119894119892 These are two realizations of the samemodel and their partition functions may be related bythe following redefinition

119885minus11

120573= 119890

minus12057311988501

2120573 (104)


5 Note that the factors of 119902 disappear because

120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)

6 Note that in this particular example we are abusinglanguage twice as we do have neither a ldquospinrdquo nor aldquofoamrdquoThe term spin arises in the fullmodels fromusingthe representation labels (spin quantum numbers) of119878119880(2)The proper ldquofoamsrdquo arise for lattice gauge theoriesas a higher-dimensional generalization of the spin netsintroduced in this section

7 Here we assume that the lattice possesses trivial coho-mology otherwiseone may find the so-called topologi-cal models see [94]

8 This definition is taken over from [83] in which similarobjects known as branched surfaces were defined forspin foams We will discuss such objects directly in thefollowing section In general such discussions in thispaper are motivated by similar considerations for spinfoams-like structures appearing in lattice gauge theories[1ndash6 83 84]

9 The free energy is defined with respect to the partitionfunction as

119865 = log119885 (106)

In the perturbative expansion contributions to the freeenergy come from single spin nets whereas the partitionfunction contains contributions from arbitrary productsof spin nets embedded into the lattice

10 For the non-Abelian groups the branchings or spin foamedges carry intertwiners between the representationsassociated to the surfaces meeting at the edges

11 A Wilson loop is a contiguous loop of lattice edges12 The orientation of the edges is the one induced by the

orientation of the face and 119892119890 = 119892minus1

119890minus1 where 119890minus1 is the

edge with opposite orientation to 11989013 Indeed in the zero-temperature limit the partition

function for the Ising gaugemodels enforces all plaquetteholonomies to be trivial This coincides with the parti-tion function of the BF-theories discussed in Section 4

14 Here and and ⋆ are the exterior product and the Hodgedual operators on space time forms

15 As before the product on Z119902 is an addition modulo 119902

ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)

16 However one should note that not all divergences aredue to this gauge symmetry [105 106]

17 In 2119889 this symmetry degenerates into a global symmetrysince the Gauss constraints force all 119896119891 to be equal 119896119891 equiv

119896 However the contribution of every representationlabel 119896 to the partition function is the same

18 The gauge parameters are then the Lie algebra-valuedfields and for the Lie algebra of 119878119880(2) they trulycorrespond to translations

19 More specifically we consider only complexes which aresuch that the gluing maps used to successively buildup 120581 are injective This in particular means that theboundary of each face contains each edge at most onceWith certain choices for amplitudes this condition canbe relaxed slightly see for example [85ndash87]

20 This can be easily shown by proving that 119875119890 = 119875lowast

119890

resulting from the unitarity of all representations andthat (119875119890)

2= 119875119890 which uses the translation-invariance

and normalization of the Haar measure on 11986621 It is defined by 120588lowast(ℎ)119886119887 = 120588(ℎ

minus1)119887119886

22 Not that there is some ambiguity in the literature aboutwhat is actually called the 6119895-symbol which is a questionof the correct normalization We mean the normalized6119895-symbol here since we chose the intertwiners to benormalized in the first place The normalization whichusually results in nontrivial edge amplitudes can beabsorbed into the vertex amplitude Also there might beaccording to convention some sign factors assigned tothe edges 119890 which may not be absorbable into the vertexamplitudesAV

23 We are thankful to Frank Hellmann for this insight24 See [41 119ndash121] for a possibility to introduce a slicing

of triangulations into hypersurfaces and a transferoperator based on the so-called tent moves

25 All functions have finite norm since we consider finitegroups

26 In the limit of taking the discretization in time directionto the continuum we would obtain the same projectoras for finite time discretization Indeed the projectorsalready encode for finite time steps the ldquoHamiltonianrdquoconstraints which are the generators of time translations(time reparametrization symmetry)

27 Here ΓV119904

is a unitary operator so that the condition onthe physical states is in the form of a so-called stabilizerAlternatively the condition can be expressed by self-adjoint constraints119862V

119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904

(120574minus1)) for group

elements 120574 with 120574 = 120574minus1 Otherwise define 119862V

119904

(120574) =

ΓV119904

(120574) minus IdH Physical states are then annihilated by theconstraints

28 While we refer the reader to [18] for various definitionsit is perhaps not unwise to match up right here thedefining properties of a closed 119863-colored graphB withthose of a trace invariant (i)B has two types of vertexlabelled black and white that represent the two types oftensor 119879 and 119879 respectively (ii) Both types of vertex are119863-valent which matches the property that both types oftensor have 119863-indices (iii)B is bipartite meaning thatblack vertices are directly joined only to white verticesand vice versa This is in correspondence with the factthat indices are contracted in covariant-contravariantpairs (iv) Every edge ofB is colored by a single element


of 1 119863 such that the119863-edges emanating from anygiven vertex possess distinct colors This represents thefact that the indices index distinct vector spaces so thata covariant index in the 119894th position must be contractedwith some contravariant index in the 119894th position (v)Bis closed representing that every index is contracted

29 These couplings are in general exponentially suppressedwith respect to some nonlocality parameter like thedistance or size of the Wilson loops In this case onewould still speak of a local theory

References

[1] M P Reisenberger ldquoWorld sheet formulationsof gauge theoriesand gravityrdquo httparxivorgabsgr-qc9412035

[2] J Iwasaki ldquoADefinition of the Ponzano-Regge quantum gravitymodel in terms of surfacesrdquo Journal of Mathematical Physicsvol 36 no 11 p 6288 1995

[3] M P Reisenberger and C Rovelli ldquoSum over surfaces form ofloop quantum gravityrdquo Physical Review D vol 56 no 6 pp3490ndash3508 1997

[4] M Reisenberger and C Rovelli ldquoSpin foams as Feynmandiagramsrdquo httparxivorgabsgr-qc0002083

[5] M P Reisenberger and C Rovelli ldquoSpace-time as a Feynmandiagram the connection formulationrdquo Classical and QuantumGravity vol 18 no 1 pp 121ndash140 2001

[6] J C Baez ldquoSpin foam modelsrdquo Classical and Quantum Gravityvol 15 no 7 p 1827 1998

[7] A Perez ldquoSpin foammodels for quantum gravityrdquo Classical andQuantum Gravity vol 20 no 6 p R43 2003

[8] A Perez ldquoIntroduction to loop quantum gravity and spinfoamsrdquo httparxivorgabsgrqc0409061

[9] R Loll ldquoDiscrete approaches to quantum gravity in fourdimensionsrdquo Living Reviews in Relativity vol 1 p 13 1998

[10] F J Wegner ldquoDuality in generalized Ising models and phasetransitions without local order parametersrdquo Journal of Mathe-matical Physics vol 12 no 10 pp 2259ndash2272 1971

[11] C Rovelli Quantum Gravity Cambridge University PressCambridge UK 2004

[12] T Thiemann Modern Canonical Quantum General RelativityCambridge University Press Cambridge UK 2005

[13] J Ambjorn ldquoQuantum gravity represented as dynamical trian-gulationsrdquoClassical andQuantumGravity vol 12 no 9 p 20791995

[14] J Ambjorn D Boulatov J L Nielsen J Rolf and Y WatabikildquoThe spectral dimension of 2Dquantumgravityrdquo Journal ofHighEnergy Physics vol 02 p 010 1998

[15] J Ambjorn B Durhuus and T Jonsson Quantum GeometryA Statistical Field Theory Approach Cambridge Monographsin Mathematical Physics Cambridge University Press Cam-bridge UK 1997

[16] J Ambjorn J Jurkiewicz and R Loll ldquoThe universe fromscratchrdquo Contemporary Physics vol 47 no 2 pp 103ndash117 2006

[17] J Ambjorn A Gorlich J Jurkiewicz R Loll J Gizbert-Studnicki and T Trzesniewski ldquoThe semiclassical limit ofcausal dynamical triangulationsrdquoNuclear Physics B vol 849 no1 pp 144ndash165 2011

[18] R Gurau and J P Ryan ldquoColored tensor models-a reviewrdquoSIGMA vol 8 article 020 78 pages 2012

[19] D Oriti ldquoTheGroup field theory approach to quantum gravityrdquoin Approaches to Quantum Gravity D Oriti Ed pp 310ndash331Cambridge University Press Cambridge UK 2007

[20] D Oriti ldquoGroup field theory as the microscopic descriptionof the quantum spacetime fluid a new perspective on thecontinuum in quantum gravityrdquo PoS QG p 030 2007

[21] L Freidel ldquoGroup field theory an Overviewrdquo InternationalJournal of Theoretical Physics vol 44 no 10 pp 1769ndash17832005

[22] T Krajewski J Magnen V Rivasseau A Tanasa and P VitaleldquoQuantum corrections in the group field theory formulation ofthe Engle-Pereira-Rovelli-Livine and Freidel-Krasnov modelsrdquoPhysical Review D vol 82 no 12 Article ID 124069 2010

[23] J B Geloun R Gurau and V Rivasseau ldquoEPRLFK group fieldtheoryrdquo EPL vol 92 no 6 Article ID 60008 2010

[24] E R Livine andDOriti ldquoCoupling of spacetime atoms and spinfoam renormalisation from group field theoryrdquo Journal of HighEnergy Physics vol 0702 p 092 2007

[25] A Baratin and D Oriti ldquoQuantum simplicial geometry in thegroup field theory formalism reconsidering the Barrett-Cranemodelrdquo New Journal of Physics vol 13 Article ID 125011 2011

[26] A Baratin and D Oriti ldquoGroup field theory and simplicialgravity path integrals a model for Holst-Plebanski gravityrdquoPhysical Review D vol 85 no 4 Article ID 044003 2012

[27] T Konopka F Markopoulou and L Smolin ldquoQuantumgraphityrdquo httparxivorgabshep-th0611197

[28] T Konopka F Markopoulou and S Severini ldquoQuantumgraphity a model of emergent localityrdquo Physical Review D vol77 no 10 Article ID 104029 2008

[29] D Benedetti and J Henson ldquoImposing causality on a matrixmodelrdquo Physics Letters B vol 678 no 2 pp 222ndash226 2009

[30] B Dittrich and R Loll ldquoA hexagon model for 3D Lorentzianquantum cosmologyrdquo Physical Review D vol 66 no 8 ArticleID 084016 2002

[31] B Dittrich and R Loll ldquoCounting a black hole in Lorentzianproduct triangulationsrdquoClassical andQuantumGravity vol 23no 11 Article ID 3849 pp 3849ndash3878 2006

[32] D Benedetti R Loll and F Zamponi ldquo(2+1)-dimensionalquantum gravity as the continuum limit of causal dynamicaltriangulationsrdquo Physical Review D vol 76 no 10 Article ID104022 2007

[33] P Hasenfratz and F Niedermayer ldquoPerfect lattice action forasymptotically free theoriesrdquo Nuclear Physics B vol 414 no 3pp 785ndash814 1994

[34] P Hasenfratz ldquoProspects for perfect actionsrdquoNuclear Physics Bvol 63 no 1ndash3 pp 53ndash58 1998

[35] W Bietenholz ldquoPerfect scalars on the latticerdquo InternationalJournal of Modern Physics A vol 15 no 21 p 3341 2000

[36] B Bahr and B Dittrich ldquoImproved and perfect actions indiscrete gravityrdquo Physical Review D vol 80 no 12 Article ID124030 2009

[37] B Bahr and B Dittrich ldquoBreaking and restoring of diffeomor-phism symmetry in discrete gravityrdquo in Proceedings of the 25thMax Born Symposium on the Planck Scale pp 10ndash17 July 2009

[38] B Bahr B Dittrich and S Steinhaus ldquoPerfect discretization ofreparametrization invariant path integralsrdquo Physical Review Dvol 83 no 10 Article ID 105026 2011

[39] B Dittrich ldquoDiffeomorphism symmetry in quantum gravitymodelsrdquo httparxivorgabs08103594


[40] B Bahr and B Dittrich ldquo(Broken) Gauge symmetries andconstraints in Regge calculusrdquo Classical and Quantum Gravityvol 26 no 22 Article ID 225011 2009

[41] B Dittrich and P A Hoehn ldquoFrom covariant to canonical for-mulations of discrete gravityrdquo Classical and Quantum Gravityvol 27 no 15 Article ID 155001 2010

[42] M Rocek and R M Williams ldquoQuantum Regge CalculusrdquoPhysics Letters B vol 104 no 1 pp 31ndash37 1981

[43] M Rocek and R M Williams ldquoThe Quantization Of ReggeCalculusrdquo Zeitschrift fur Physik C vol 21 no 4 pp 371ndash3811984

[44] P A Morse ldquoApproximate diffeomorphism invariance in nearat simplicial geometriesrdquo Classical and QuantumGravity vol 9no 11 p 2489 1992

[45] H W Hamber and R M Williams ldquoGauge invariance insimplicial gravityrdquo Nuclear Physics B vol 487 no 1-2 pp 345ndash408 1997

[46] B Dittrich L Freidel and S Speziale ldquoLinearized dynamicsfrom the 4-simplex Regge actionrdquo Physical Review D vol 76no 10 Article ID 104020 2007

[47] T Regge ldquoGeneral relativity without coordinatesrdquo Il NuovoCimento vol 19 no 3 pp 558ndash571 1961

[48] T Regge and R M Williams ldquoDiscrete structures in gravityrdquoJournal of Mathematical Physics vol 41 no 6 p 3964 2000

[49] L Freidel and D Louapre ldquoDiffeomorphisms and spin foammodelsrdquo Nuclear Physics B vol 662 no 1-2 pp 279ndash298 2003

[50] G T Horowitz ldquoExactly soluble diffeomorphism invarianttheoriesrdquoCommunications inMathematical Physics vol 125 no3 pp 417ndash437 1989

[51] R Dijkgraaf and E Witten ldquoTopological gauge theories andgroup cohomologyrdquo Communications in Mathematical Physicsvol 129 no 2 pp 393ndash429 1990

[52] M de Wild Propitius and F A Bais ldquoDiscrete gauge theoriesrdquoin Banff 1994 Particles and Fields pp 353ndash439 1995

[53] M Mackaay ldquoFinite groups spherical 2-categories and 4-manifold invariantsrdquo Advances in Mathematics vol 153 no 2pp 353ndash390 2000

[54] A Y Kitaev ldquoFault-tolerant quantum computation by anyonsrdquoAnnals of Physics vol 303 no 1 pp 2ndash30 2003

[55] M A Levin and X-G Wen ldquoString-net condensation aphysical mechanism for topological phasesrdquo Physical Review Bvol 71 no 4 Article ID 045110 2005

[56] J F Plebanski ldquoOn the separation of Einsteinian substructuresrdquoJournal of Mathematical Physics vol 18 no 12 pp 2511ndash25201976

[57] J W Barrett and L Crane ldquoRelativistic spin networks andquantum gravityrdquo Journal of Mathematical Physics vol 39 no6 Article ID 3296 1998

[58] J Engle R Pereira and C Rovelli ldquoThe loop-quantum-gravityvertex-amplituderdquo Physical Review Letters vol 99 no 16Article ID 161301 2007

[59] J Engle E Livine R Pereira and C Rovelli ldquoLQG vertex withfinite Immirzi parameterrdquo Nuclear Physics B vol 799 no 1-2pp 136ndash149 2008

[60] E R Livine and S Speziale ldquoA new spinfoam vertex forquantum gravityrdquo Physical Review D vol 76 no 8 Article ID084028 2007

[61] L Freidel and K Krasnov ldquoA new spin foam model for 4dgravityrdquo Classical and Quantum Gravity vol 25 no 12 ArticleID 125018 2008

[62] S Alexandrov ldquoSimplicity and closure constraints in spin foammodels of gravityrdquo Physical Review D vol 78 no 4 Article ID044033 2008

[63] S Alexandrov ldquoThe new vertices and canonical quantizationrdquoPhysical Review D vol 82 no 2 Article ID 024024 2010

[64] B Dittrich and J P Ryan ldquoSimplicity in simplicial phase spacerdquoPhysical Review D vol 82 Article ID 064026 2010

[65] ROeckl ldquoGeneralized lattice gauge theory spin foams and statesum invariantsrdquo Journal of Geometry and Physics vol 46 no 3-4 pp 308ndash354 2003

[66] R OecklDiscrete GaugeTheory From Lattices to Tqft ImperialCollege London UK 2005

[67] J W Barrett R J Dowdall W J Fairbairn H Gomes andF Hellmann ldquoAsymptotic analysis of the EPRL four-simplexamplituderdquo Journal of Mathematical Physics vol 50 no 11Article ID 112504 2009

[68] J W Barrett R J Dowdall W J Fairbairn H Gomes FHellmann and R Pereira ldquoAsymptotics of 4d spin foammodelsrdquoGeneral Relativity andGravitation vol 43 p 2421 2011

[69] F Conrady and L Freidel ldquoOn the semiclassical limit of 4Dspin foam modelsrdquo Physical Review D vol 78 no 10 ArticleID 104023 2008

[70] S Liberati F Girelli and L Sindoni ldquoAnalogue models foremergent gravityrdquo httparxivorgabs09093834

[71] Z C Gu and X G Wen ldquoEmergence of helicity plusmn2 modes(gravitons) from qubit modelsrdquo Nuclear Physics B vol 863 no1 pp 90ndash129 2012

[72] A Hamma and F Markopoulou ldquoBackground independentcondensed matter models for quantum gravityrdquo New Journal ofPhysics vol 13 Article ID 095006 2011

[73] T Fleming M Gross and R Renken ldquoIsing-Link quantumgravityrdquo Physical Review D vol 50 no 12 pp 7363ndash7371 1994

[74] W Beirl H Markum and J Riedler ldquoTwo-dimensional latticegravity as a spin systemrdquo International Journal ofModern PhysicsC vol 5 p 359 1994

[75] E Bittner A Hauke H Markum J Riedler C Holm andW Janke ldquoZ(2)-Regge versus standard Regge calculus in twodimensionsrdquo Physical Review D vol 59 no 12 Article ID124018 1999

[76] E Bittner W Janke and H Markum ldquoIsing spins coupled to afour-dimensional discrete Regge skeletonrdquo Physical Review Dvol 66 no 2 Article ID 024008 2002

[77] H A Kramers and G H Wannier ldquoStatistics of the two-dimensional ferromagnet Part irdquo Physical Review vol 60 no3 pp 252ndash262 1941

[78] R Savit ldquoDuality in field theory and statistical systemsrdquoReviewsof Modern Physics vol 52 no 2 pp 453ndash487 1980

[79] W Fulton and J Harris RepresentAtion Theory A First CourseSpringer New York NY USA 1991

[80] F Girelli and H Pfeiffer ldquoHigher gauge theory differentialversus integral formulationrdquo Journal of Mathematical Physicsvol 45 no 10 Article ID 3949 2004

[81] F Girelli H Pfeiffer and E M Popescu ldquoTopological highergauge theory from BF to BFCG theoryrdquo Journal of Mathemati-cal Physics vol 49 no 3 Article ID 032503 2008

[82] J Oitmaa C Hamer and W Zheng Series Expansion Methodsfor Strongly Interacting Lattice Models Cambridge UniversityPress Cambridge UK 2006

[83] F Conrady ldquoGeometric spin foams Yang-Mills theory andbackground-independent modelsrdquo httparxivorgabsgr-qc0504059


[84] J Drouffe and J Zuber ldquoStrong coupling and mean fieldmethods in lattice gauge theoriesrdquo Physics Reports vol 102 no1-2 pp 1ndash119 1983

[85] M Bojowald and A Perez ldquoSpin foam quantization andanomaliesrdquoGeneral Relativity andGravitation vol 42 no 4 pp877ndash907 2010

[86] B Bahr ldquoOn knottings in the physical Hilbert space of LQG asgiven by the EPRL modelrdquo Classical and Quantum Gravity vol28 no 4 Article ID 045002 2011

[87] B Bahr F Hellmann W Kaminski M Kisielowski and JLewandowski ldquoOperator spin foam modelsrdquo Classical andQuantum Gravity vol 28 no 10 Article ID 105003 2011

[88] C Rovelli and M Smerlak ldquoIn quantum gravity summing isrefiningrdquo Classical and Quantum Gravity vol 29 Article ID055004 2012

[89] G De Las Cuevas W Dur H J Briegel and M A Martin-Delgado ldquoMapping all classical spin models to a lattice gaugetheoryrdquo New Journal of Physics vol 12 Article ID 043014 2010

[90] J B Kogut ldquoAn introduction to lattice gauge theory and spinsystemsrdquo Reviews of Modern Physics vol 51 no 4 pp 659ndash7131979

[91] R Oeckl and H Pfeiffer ldquoThe dual of pure non-Abelian latticegauge theory as a spin foammodelrdquo Nuclear Physics B vol 598no 1-2 pp 400ndash426 2001

[92] F Conrady ldquoAnalytic derivation of dual gluons and monopolesfrom SU(2) lattice Yang-Mills theory I BFYang-Mills represen-tationrdquo httparxivorgabshep-th0610236

[93] F Conrady ldquoAnalytic derivation of dual gluons and monopolesfrom SU(2) lattice Yang-Mills theory II Spin foam representa-tionrdquo httparxivorgabshep-th0610237

[94] M Rakowski ldquoTopological modes in dual lattice modelsrdquoPhysical Review D vol 52 no 1 pp 354ndash357 1995

[95] I Montvay and G Munster Quantum Fields on a LatticeCambridge University Press Cambridge UK 1994

[96] M Martellini and M Zeni ldquoFeynman rules and beta-functionfor the BF Yang-Mills theoryrdquo Physics Letters B vol 401 no 1-2pp 62ndash68 1997

[97] A S Cattaneo P Cotta-Ramusino F Fucito et al ldquoFour-dimensional Yang-Mills theory as a deformation of topologicalBF theoryrdquo Communications in Mathematical Physics vol 197no 3 pp 571ndash621 1998

[98] L Smolin and A Starodubtsev ldquoGeneral relativity with a topo-logical phase an action principlerdquo httparxivorgabshep-th0311163

[99] A Starodubtsev Topological methods in quantum gravity [PhDthesis] University of Waterloo 2005

[100] D Birmingham and M Rakowski ldquoState sum models and sim-plicial cohomologyrdquo Communications in Mathematical Physicsvol 173 no 1 pp 135ndash154 1995

[101] D Birmingham and M Rakowski ldquoOn Dijkgraaf-Witten typeinvariantsrdquo Letters in Mathematical Physics vol 37 no 4 pp363ndash374 1996

[102] SWChungM Fukuma andAD Shapere ldquoStructure of topo-logical lattice field theories in three-dimensionsrdquo InternationalJournal of Modern Physics A vol 9 no 8 p 1305 1994

[103] M Asano and S Higuchi ldquoOn three-dimensional topologicalfield theories constructed from D120596(G) for finite groupsrdquo Mod-ern Physics Letters A vol 9 no 25 p 2359 1994

[104] J C Baez ldquoAn introduction to spin foam models of BF theoryand quantum gravityrdquo in Geometry and Quantum Physics vol543 of Lecture Notes in Physics pp 25ndash93 2000

[105] V Bonzom and M Smerlak ldquoBubble divergences from cellularcohomologyrdquo Letters in Mathematical Physics vol 93 no 3 pp295ndash305 2010

[106] V Bonzom and M Smerlak ldquoBubble divergences from twistedcohomologyrdquo Communications in Mathematical Physics vol312 no 2 pp 399ndash426 2012

[107] B Dittrich and J P Ryan ldquoPhase space descriptions forsimplicial 4d geometriesrdquo Classical and Quantum Gravity vol28 no 6 Article ID 065006 2011

[108] L Freidel and K Krasnov ldquoSpin foam models and the classicalaction principlerdquo Advances in Theoretical and MathematicalPhysics vol 2 pp 1183ndash1247 1999

[109] H Pfeiffer ldquoDual variables and a connection picture for theEuclidean Barrett-Crane modelrdquo Classical and Quantum Grav-ity vol 19 no 6 pp 1109ndash1137 2002

[110] P Cvitanovic Group Theory Birdtracks Liersquos and ExceptionalGroups Princeton University Pres New Jersey NJ USA 2008

[111] J Kogut and L Susskind ldquoHamiltonian formulation ofWilsonrsquoslattice gauge theoriesrdquo Physical Review D vol 11 no 2 pp 395ndash408 1975

[112] J Smit Introduction to Quantum Fields on a lattice A RobustMate vol 15 of Cambridge Lecture Notes in Physics CambridgeUniversity Press Cambridge UK 2002

[113] J J Halliwell and J B Hartle ldquoWave functions constructed froman invariant sum over histories satisfy constraintsrdquo PhysicalReview D vol 43 no 4 pp 1170ndash1194 1991

[114] C Rovelli ldquoThe projector on physical states in loop quantumgravityrdquo Physical Review D vol 59 no 10 Article ID 1040151999

[115] K Noui and A Perez ldquoThree-dimensional loop quantum grav-ity physical scalar product and spin-foam modelsrdquo Classicaland Quantum Gravity vol 22 no 9 pp 1739ndash1761 2005

[116] T Thiemann ldquoAnomaly-free formulation of non-perturbativefour-dimensional Lorentzian quantum gravityrdquo Physics LettersB vol 380 no 3-4 pp 257ndash264 1996

[117] T Thiemann ldquoQuantum spin dynamics (QSD)rdquo Classical andQuantum Gravity vol 15 no 4 p 839 1998

[118] T Thiemann ldquoQSD III quantum constraint algebra and phys-ical scalar product in quantum general relativityrdquo Classical andQuantum Gravity vol 15 no 5 pp 1207ndash1247 1998

[119] R Sorkin ldquoTime evolution problem in Regge Calculusrdquo Physi-cal Review D vol 12 no 2 pp 385ndash396 1975

[120] R Sorkin ldquoErratum time-evolution problem in Regge calcu-lusrdquo Physical Review D vol 23 no 2 p 565 1981

[121] JW Barrett M GalassiW AMiller R D Sorkin P A Tuckeyand R MWilliams ldquoA Paralellizable implicit evolution schemefor Regge calculusrdquo International Journal of Theoretical Physicsvol 36 no 4 pp 815ndash835 1997

[122] B Bahr B Dittrich and S He ldquoCoarse-graining free theorieswith gauge symmetries the linearized caserdquo New Journal ofPhysics vol 13 Article ID 045009 2011

[123] T Thiemann ldquoThe Phoenix project master constraint pro-gramme for loop quantum gravityrdquo Classical and QuantumGravity vol 23 no 7 Article ID 2211 2006

[124] B Dittrich and T Thiemann ldquoTesting the master constraintprogramme for loop quantum gravity I General frameworkrdquoClassical and Quantum Gravity vol 23 no 4 pp 1025ndash10652006

[125] B Dittrich and T Thiemann ldquoTesting the master constraintprogramme for loop quantum gravity II finite dimensional


systemsrdquo Classical and Quantum Gravity vol 23 no 4 ArticleID 1067 2006

[126] DMarolf ldquoGroup averaging and refined algebraic quantizationwhere are we nowrdquo httparxivorgabsgrqc0011112

[127] P G Bergmann ldquoObservables in general relativityrdquo Reviews ofModern Physics vol 33 no 4 pp 510ndash514 1961

[128] C Rovelli ldquoWhat is observable in classical and quantumgravityrdquo Classical and Quantum Gravity vol 8 no 2 p 2971991

[129] C Rovelli ldquoPartial observablesrdquo Physical Review D vol 65Article ID 124013 2002

[130] B Dittrich ldquoPartial and complete observables for Hamiltonianconstrained systemsrdquoGeneral Relativity andGravitation vol 39no 11 pp 1891ndash1927 2007

[131] B Dittrich ldquoPartial and complete observables for canonicalgeneral relativityrdquo Classical and Quantum Gravity vol 23 no22 Article ID 6155 2006

[132] C G Torre ldquoGravitational observables and local symmetriesrdquoPhysical Review D vol 48 no 6 pp 2373ndash2376 1993

[133] S Elitzur ldquoImpossibility of spontaneously breaking local sym-metriesrdquo Physical Review D vol 12 no 12 pp 3978ndash3982 1975

[134] L Freidel and D Louapre ldquoPonzano-Regge model revisited IGauge fixing observables and interacting spinning particlesrdquoClassical and Quantum Gravity vol 21 no 24 Article ID 56852004

[135] K Noui and A Perez ldquoThree dimensional loop quantumgravity coupling to point particlesrdquo Classical and QuantumGravity vol 22 no 21 Article ID 4489 2005

[136] K Noui ldquoThree dimensional loop quantum gravity particlesand the quantum doublerdquo Journal of Mathematical Physics vol47 no 10 Article ID 102501 2006

[137] A Perez ldquoOn the regularization ambiguities in loop quantumgravityrdquo Physical Review D vol 73 Article ID 044007 18 pages2006

[138] J C Baez andA Perez ldquoQuantization of strings and branes cou-pled to BF theoryrdquo Advances in Theoretical and MathematicalPhysics vol 11 Article ID 060508 pp 451ndash469 2007

[139] W J Fairbairn and A Perez ldquoExtended matter coupled to BFtheoryrdquo Physical Review D vol 78 no 2 Article ID 0240132008

[140] W J Fairbairn ldquoOn gravitational defects particles and stringsrdquoJournal of High Energy Physics vol 2008 no 9 p 126 2008

[141] H Bombin andM A Martin-Delgado ldquoFamily of non-AbelianKitaev models on a lattice topological condensation and con-finementrdquo Physical Review B vol 78 no 11 Article ID 1154212008

[142] Z Kadar A Marzuoli and M Rasetti ldquoBraiding and entangle-ment in spin networks a combinatorical approach to topologi-cal phasesrdquo International Journal of Quantum Information vol7 p 195 2009

[143] Z Kadar AMarzuoli andM Rasetti ldquoMicroscopic descriptionof 2d topological phases duality and 3D state sumsrdquo Advancesin Mathematical Physics vol 2010 Article ID 671039 18 pages2010

[144] L Freidel J Zapata and M Tylutki Observables in the Hamil-tonian formulation of the Ponzano-Regge model [MS thesis]Uniwersytet Jagiellonski Krakow Poland 2008

[145] H Waelbroeck and J A Zapata ldquoTranslation symmetry in 2+1Regge calculusrdquo Classical and Quantum Gravity vol 10 no 9Article ID 1923 1993

[146] H Waelbroeck and J A Zapata ldquo2 +1 covariant lattice theoryand trsquoHooftrsquos formulationrdquo Classical and Quantum Gravity vol13 p 1761 1996

[147] V Bonzom ldquoSpin foam models and the Wheeler-DeWittequation for the quantum 4-simplexrdquo Physical Review D vol84 Article ID 024009 13 pages 2011

[148] A Ashtekar ldquoNew variables for classical and quantum gravityrdquoPhysical Review Letters vol 57 no 18 pp 2244ndash2247 1986

[149] A Ashtekar New Perspepectives in Canonical Gravity Mono-graphs and Textbooks in Physical Science Bibliopolis NapoliItaly 1988

[150] A Ashtekar Lectures on Non-Perturbative Canonical GravityWorld Scientific Singapore 1991

[151] M Campiglia C Di Bartolo R Gambini and J PullinldquoUniform discretizations a new approach for the quantizationof totally constrained systemsrdquo Physical Review D vol 74 no12 Article ID 124012 2006

[152] E Alesci K Noui and F Sardelli ldquoSpin-foam models and thephysical scalar productrdquo Physical Review D vol 78 no 10Article ID 104009 2008

[153] E Alesci and C Rovelli ldquoA regularization of the hamiltonianconstraint compatible with the spinfoam dynamicsrdquo PhysicalReview D vol 82 no 4 Article ID 044007 2010

[154] P Di Francesco P Ginsparg and J Zinn-Justin ldquo2D gravity andrandom matricesrdquo Physics Report vol 254 no 1-2 pp 1ndash1331995

[155] F David ldquoSimplicial quantum gravity and random latticesrdquohttparxivorgabshep-th9303127

[156] F David ldquoPlanar diagrams two-dimensional lattice gravity andsurface modelsrdquo Nuclear Physics B vol 257 pp 45ndash58 1985

[157] V A Kazakov ldquoBilocal regularization of models of randomsurfacesrdquo Physics Letters B vol 150 no 4 pp 282ndash284 1985

[158] D J Gross and A A Migdal ldquoNonperturbative two-dimensional quantum gravityrdquo Physical Review Lettersvol 64 no 2 pp 127ndash130 1990

[159] D J Gross and A A Migdal ldquoA nonperturbative treatment oftwo-dimensional quantum gravityrdquo Nuclear Physics B vol 340no 2-3 pp 333ndash365 1990

[160] V A Kazakov ldquoIsing model on a dynamical planar randomlattice exact solutionrdquo Physics Letters A vol 119 no 3 pp 140ndash144 1986

[161] DV Boulatov andVAKazakov ldquoThe isingmodel on a randomplanar lattice the structure of the phase transition and the exactcritical exponentsrdquo Physics Letters B vol 186 no 3-4 pp 379ndash384 1987

[162] V A Kazakov and A A Migdal ldquoInduced QCD at large NrdquoNuclear Physics B vol 397 no 1-2 Article ID 920601 pp 214ndash238 1993

[163] P H Ginsparg and G W Moore ldquoLectures on 2-D gravity and2-D stringrdquo httparxivorgabshep-th9304011

[164] P Zinn-Justin and J B Zuber ldquoMatrix integrals and the count-ing of tangles and linksrdquo httparxivorgabsmath-ph9904019

[165] J L Jacobsen and P Zinn-Justin ldquoA Transfer matrix approachto the enumeration of knotsrdquo httparxivorgabsmath-ph0102015

[166] P Zinn-Justin ldquoThe General O(n) quartic matrix model and itsapplication to counting tangles and linksrdquo Communications inMathematical Physics vol 238 pp 287ndash304 2003


[167] P Zinn-Justin and J Zuber ldquoMatrix integrals and the generationand counting of virtual tangles and linksrdquo Journal of KnotTheoryand its Ramifications vol 13 no 3 pp 325ndash355 2004

[168] M L Mehta RandomMatrices Academic Press New York NYUSA 1991

[169] G T Hooft ldquoA planar diagram theory for strong interactionsrdquoNuclear Physics B vol 72 no 3 pp 461ndash473 1974

[170] G T Hooft ldquoA two-dimensional model for mesonsrdquo NuclearPhysics B vol 75 no 3 pp 461ndash470 1974

[171] E Brezin C Itzykson G Parisi and J B Zuber ldquoPlanardiagramsrdquoCommunications inMathematical Physics vol 59 no1 pp 35ndash51 1978

[172] M L Mehta ldquoA method of integration over matrix variablesrdquoCommunications inMathematical Physics vol 79 no 3 pp 327ndash340 1981

[173] C Itzykson and J-B Zuber ldquoThe planar approximation IIrdquoJournal of Mathematical Physics vol 21 no 3 pp 411ndash421 1979

[174] D Bessis C Itzykson and J B Zuber ldquoQuantum field theorytechniques in graphical enumerationrdquo Advances in AppliedMathematics vol 1 no 2 pp 109ndash157 1980

[175] P Di Francesco and C Itzykson ldquoA Generating function forfatgraphsrdquo Annales de lrsquoInstitut Henri Poincare (A) PhysiqueTheorique vol 59 pp 117ndash140 1993

[176] V A KazakovM Staudacher and TWynter ldquoCharacter expan-sion methods for matrix models of dually weighted graphsrdquoCommunications in Mathematical Physics vol 177 no 2 pp451ndash468 1996

[177] J Ambjoslashrn D V Boulatov A Krzywicki and S VarstedldquoThe vacuum in three-dimensional simplicial quantum gravityrdquoPhysics Letters B vol 276 no 4 pp 432ndash436 1992

[178] R Gurau ldquoColored group field theoryrdquo Communications inMathematical Physics vol 304 pp 69ndash93 2011

[179] J Ben Geloun R Gurau and V Rivasseau ldquoEPRLFK groupfield theoryrdquoEurophysics Letters vol 92 no 6 Article ID 600082010

[180] R Gurau ldquoLost in translation topological singularities in groupfield theoryrdquo Classical and Quantum Gravity vol 27 no 23Article ID 235023 2010

[181] M Smerlak ldquoComment on lsquoLost in translation topologicalsingularities in group field theoryrsquordquo Classical and QuantumGravity vol 28 no 17 Article ID 178001 2011

[182] R Gurau ldquoReply to comment on lsquoLost in translation topolog-ical singularities in group field theoryrsquordquo Classical and QuantumGravity vol 28 no 17 Article ID 178002 2011

[183] L Freidel R Gurau andDOriti ldquoGroup field theory renormal-ization in the 3D case power counting of divergencesrdquo PhysicalReview D vol 80 no 4 Article ID 044007 2009

[184] J Magnen K Noui V Rivasseau and M Smerlak ldquoScalingbehavior of three-dimensional group field theoryrdquoClassical andQuantum Gravity vol 26 no 18 Article ID 185012 2009

[185] J B Geloun T Krajewski J Magnen and V RivasseauldquoLinearized group field theory and power-counting theoremsrdquoClassical andQuantumGravity vol 27 no 15 Article ID 1550122010

[186] R Gurau ldquoThe 1N Expansion of Colored Tensor ModelsrdquoAnnales Henri Poincare vol 12 no 5 pp 829ndash847 2011

[187] R Gurau and V Rivasseau ldquoThe 1N expansion of coloredtensor models in arbitrary dimensionrdquo EPL vol 95 no 5Article ID 50004 2011

[188] R Gurau ldquoThe Complete 1N Expansion of Colored TensorModels in Arbitrary Dimensionrdquo Annales Henri Poincare vol13 no 3 pp 399ndash423 2012

[189] V Bonzom ldquoNew 1N expansions in random tensor modelsrdquoJournal of High Energy Physics vol 1306 p 062 2013

[190] V Bonzom R Gurau A Riello and V Rivasseau ldquoCriticalbehavior of colored tensor models in the large N limitrdquo NuclearPhysics B vol 853 no 1 pp 174ndash195 2011


[192] V Bonzom R Gurau and V Rivasseau ldquoThe Ising model onrandom lattices in arbitrary dimensionsrdquo Physics Letters B vol711 no 1 pp 88ndash96 2012

[193] V Bonzom ldquoMulticritical tensor models and hard dimers onspherical random latticesrdquo Physics Letters A vol 377 no 7 pp501ndash506 2013

[194] V Bonzom and H Erbin ldquoCoupling of hard dimers to dynami-cal lattices via random tensorsrdquo Journal of Statistical Mechanicsvol 1209 Article ID P09009 2012

[195] D Benedetti and R Gurau ldquoPhase transition in dually weightedcolored tensor modelsrdquo Nuclear Physics B vol 855 no 2 pp420ndash437 2012

[196] J Ambjoslashrn B Durhuus and T Jonsson ldquoSumming over allgenera for dgt 1 a toy modelrdquo Physics Letters B vol 244 no 3-4pp 403ndash412 1990

[197] P Bialas and Z Burda ldquoPhase transition in uctuating branchedgeometryrdquo Physics Letters B vol 384 no 1ndash4 pp 75ndash80 1996

[198] R Gurau and J P Ryan ldquoMelons are branched polymersrdquohttparxivorgabs13024386

[199] R Gurau ldquoUniversality for random tensorsrdquohttparxivorgabs 11110519

[200] R Gurau ldquoA generalization of the Virasoro algebra to arbitrarydimensionsrdquoNuclear Physics B vol 852 no 3 pp 592ndash614 2011

[201] R Gurau ldquoThe Schwinger Dyson equations and the algebraof constraints of random tensor models at all ordersrdquo NuclearPhysics B vol 865 no 1 pp 133ndash147 2012

[202] V Bonzom ldquoRevisiting random tensormodels at large N via theSchwinger-Dyson equationsrdquo Journal of High Energy Physicsvol 1303 p 160 2013

[203] R De Pietri andC Petronio ldquoFeynman diagrams of generalizedmatrixmodels and the associatedmanifolds in dimension fourrdquoJournal of Mathematical Physics vol 41 no 10 pp 6671ndash66882000

[204] D V Boulatov ldquoA Model of three-dimensional lattice gravityrdquoModern Physics Letters A vol 7 no 18 p 1629 1992

[205] H Ooguri ldquoTopological lattice models in four-dimensionsrdquoModern Physics Letters A vol 7 no 30 pp 2799ndash2810 1992

[206] R De Pietri L Freidel K Krasnov and C Rovelli ldquoBarrett-Crane model from a Boulatov-Ooguri field theory over ahomogeneous spacerdquoNuclear Physics B vol 574 no 3 pp 785ndash806 2000

[207] A Baratin F Girelli and D Oriti ldquoDiffeomorphisms in groupfield theoriesrdquo Physical Review D vol 83 no 10 Article ID104051 2011

[208] J Ben Geloun and V Bonzom ldquoRadiative corrections in theBoulatov-Ooguri tensor model the 2-point functionrdquo Interna-tional Journal ofTheoretical Physics vol 50 no 9 pp 2819ndash28412011


[209] J B Geloun and V Rivasseau ldquoA renormalizable 4-dimensionaltensor field theoryrdquo Communications in Mathematical Physicsvol 318 no 1 pp 69ndash109 2013

[210] J B Geloun and D O Samary ldquo3D tensor field theory renor-malization and one-loop 120573-functionsrdquo httparxivorgabs12010176

[211] J B Geloun ldquoTwo and four-loop 120573-functions of rank 4renormalizable tensor field theoriesrdquo Classical and QuantumGravity vol 29 Article ID 235011 2012

[212] J B Geloun and V Rivasseau ldquoAddendum to lsquoA renormal-izable 4-dimensional tensor field theoryrsquordquo httparxivorgabs12094606

[213] J B Geloun ldquoAsymptotic freedom of rank 4 tensor group fieldtheoryrdquo httparxivorgabs12105490

[214] J B Geloun and E R Livine ldquoSome classes of renormalizabletensor modelsrdquo httparxivorgabs12070416

[215] S Carrozza and D Oriti ldquoBounding bubbles the vertex rep-resentation of 3d group field theory and the suppression ofpseudomanifoldsrdquo Physical Review D vol 85 no 4 Article ID044004 2012

[216] S Carrozza and D Oriti ldquoBubbles and jackets new scalingbounds in topological group field theoriesrdquo Journal of HighEnergy Physics vol 1206 p 092 2012

[217] S Carrozza D Oriti and V Rivasseau ldquoRenormalization oftensorial group field theories Abelian U(1) models in fourdimensionsrdquo httparxivorgabs12076734

[218] S Carrozza D Oriti and V Rivasseau ldquoRenormalization ofan SU(2) tensorial group field theory in three dimensionsrdquohttparxivorgabs13036772

[219] D Oriti and L Sindoni ldquoToward classical geometrodynamicsfrom the group field theory hydrodynamicsrdquo New Journal ofPhysics vol 13 Article ID 025006 2011

[220] L Freidel D Oriti and J Ryan ldquoA group field the-ory for 3d quantum gravity coupled to a scalar fieldrdquohttparxivorgabsgr-qc0506067

[221] D Oriti and J Ryan ldquoGroup field theory formulation of 3-D quantum gravity coupled to matter fieldsrdquo Classical andQuantum Gravity vol 23 pp 6543ndash6576 2006

[222] R J Dowdall ldquoWilson loops geometric operators and fermionsin 3d group field theoryrdquo httparxivorgabs09112391

[223] F Girelli and E R Livine ldquoA deformed Poincare invariance forgroup field theoriesrdquoClassical andQuantumGravity vol 27 no24 Article ID 245018 2010

[224] M Dupuis F Girelli and E R Livine ldquoSpinors group fieldtheory and Voros star product first contactrdquo Physical ReviewD vol 86 Article ID 105034 2012

[225] W J Fairbairn and E R Livine ldquo3D spinfoam quantum gravitymatter as a phase of the group field theoryrdquo Classical andQuantum Gravity vol 24 no 20 pp 5277ndash5297 2007

[226] F Girelli E R Livine and D Oriti ldquo4D deformed specialrelativity from group field theoriesrdquo Physical Review D vol 81no 2 Article ID 024015 2010

[227] E R Livine D Oriti and J P Ryan ldquoEffective Hamiltonianconstraint from group field theoryrdquo Classical and QuantumGravity vol 28 no 24 Article ID 245010 2011

[228] J B Geloun ldquoClassical group field theoryrdquo Journal ofMathemat-ical Physics Article ID 022901 p 53 2012

[229] G Calcagni S Gielen and D Oriti ldquoGroup field cosmology acosmological field theory of quantum geometryrdquo Classical andQuantum Gravity vol 29 no 10 Article ID 105005 2012

[230] V Bonzom R Gurau and V Rivasseau ldquoRandom tensormodels in the large N limit uncoloring the colored tensormodelsrdquo Physical Review D vol 85 no 8 Article ID 0840372012

[231] V A Kazakov ldquoThe appearance of matter fields from quantumfluctuations of 2D-gravityrdquo Modern Physics Letters A vol 4 p2125 1989

[232] V A Kazakov ldquoExactly solvable Potts models bond- and tree-like percolation on dynamical (random) planar latticerdquoNuclearPhysics B vol 4 pp 93ndash97 1988


[234] V Bonzom andM Smerlak ldquoBubble Divergences from TwistedCohomologyrdquo Communications in Mathematical Physics pp 1ndash28 2012

[235] V Bonzom and M Smerlak ldquoBubble divergences sorting outtopology from cell structurerdquo Annales Henri Poincare vol 13no 1 pp 185ndash208 2012

[236] F Markopoulou ldquoAn algebraic approach to coarse grainingrdquohttparxivorgabshep-th0006199

[237] F Markopoulou ldquoCoarse graining in spin foam modelsrdquo Clas-sical and Quantum Gravity vol 20 no 5 pp 777ndash799 2003

[238] R Oeckl ldquoRenormalization of discrete models without back-groundrdquo Nuclear Physics B vol 657 no 1ndash3 pp 107ndash138 2003

[239] M Levin and C P Nave ldquoTensor renormalization groupapproach to two-dimensional classical lattice modelsrdquo PhysicalReview Letters vol 99 no 12 Article ID 120601 2007

[240] Z-C Gu M Levin and X-G Wen ldquoTensor-entanglementrenormalization group approach as a unified method forsymmetry breaking and topological phase transitionsrdquo PhysicalReview B vol 78 no 20 Article ID 205116 2008

[241] L Tagliacozzo and G Vidal ldquoEntanglement renormalizationand gauge symmetryrdquo Physical Review B vol 83 no 11 ArticleID 115127 2011

[242] B Dittrich F C Eckert and M Martin-Benito ldquoCoarse grain-ing methods for spin net and spin foammodelsrdquoNew Journal ofPhysics vol 14 Article ID 35008 2012

[243] B Dittrich ldquoFrom the discrete to the continuous towards acylindrically consistent dynamicsrdquo New Journal of Physics vol14 Article ID 123004 2012

[244] L Freidel and E R Livine ldquoEffective 3D quantum gravityand effective noncommutative quantum field theoryrdquo PhysicalReview Letters vol 96 no 22 Article ID 221301 2006

[245] L Freidel and S Majid ldquoNoncommutative harmonic analysissampling theory and the Duflo map in 2+1 quantum gravityrdquoClassical and Quantum Gravity vol 25 no 4 Article ID045006 2008

[246] A Baratin B Dittrich D Oriti and J Tambornino ldquoNon-commutative flux representation for loop quantum gravityrdquoClassical and QuantumGravity vol 28 no 17 Article ID 1750112011

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


FluidsJournal of

Atomic and Molecular Physics

Journal of



Advances in Condensed Matter Physics

OpticsInternational Journal of



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ThermodynamicsJournal of


function Such vertex translation symmetries are realized in3D gravity where general relativity is a topological theory[49] This means that the theory has no local degrees offreedom only a topology-dependent finite number of globalones The first-order formalism of 3D gravity coincides witha 3D version of the BF-theory [50] The BF-theory is a gaugetheory and can be formulated with a gauge group given bya Lie group The 3D gravity is obtained by choosing 119878119880(2)However one can also choose finite groups [51ndash53] In thiscase one obtains models used in quantum computing [54] aswell as models describing topological phases of condensed-matter systems so called string-net models [55] Moreoverthe BF-systems can be seen as Yang-Mills-like systems fora special choice of the coupling parameter (correspondingto zero temperature) Hence in 3D we have Yang-Mills-liketheories with the usual (lattice) gauge symmetry as well asthe topological BF-theories with the additional translationsymmetries

The BF-theory (in any dimension) is a topological theorybecause it has a large gauge group of symmetries whichreduces the physical degrees of freedom to a finite numberIt is characterized not only by the usual (lattice) gaugesymmetry determined by the gauge group but also by theso-called translational symmetries based on the 3-cells (iecubes) of the lattice and parametrized by the Lie algebraelements (if we work with the Lie groups) For 3D gravitythese symmetries can be interpreted as being associatedto the vertices of the dual lattice (for instance given bya triangulation) and are hence termed vertex translationsHowever for 4D the symmetries are still associated to the 3-cells of the lattice and hence to the edges of the dual lattice ortriangulation As in general there aremore edges than verticesin a triangulation we obtainmuchmore symmetries than thevertex translations onemight look for in 4D gravity To obtainvertex translations of the dual lattice in 4D one would ratherneed a symmetry based on the 4-cells of the lattice

In 4D gravity is not a topological theoryThere is howevera formulation due to Plebanski [56] that starts with the BF-theory and imposes the so-called simplicity constraints on thevariablesThese break the symmetries of the BF-theory downto a subgroup that can be interpreted as diffeomorphismsymmetry (in the continuum) This Plebanski formulation isalso the one used in many spin foam models Here one ofthe main problems is to implement the simplicity constraints[57ndash64] into the BF-partition function The status of thesymmetries is quite unclear in these models however thereare indications [40] that in the discrete the reduction of theBF-translation symmetries does not leave a sufficiently largegroup to be interpretable as diffeomorphisms Neverthelessif these models are related to gravity then there is a pos-sibility that diffeomorphism symmetry as the fundamentalsymmetry of general relativity arises as a symmetry forlarge scales In the abovementioned method of regainingsymmetries by coarse graining one can then expect thatthis diffeomorphism symmetry arises as vertex translationsymmetry (on the vertices of the dual lattice)

Hence we want to emphasize that in 4D spin foammodels are candidates for a new class of lattice models inaddition to Yang-Mills-like systems and the BF-theory This

new class would be characterized by a symmetry group inbetween those for the Yang-Mills theory (usual lattice gaugesymmetry) and BF-symmetry (usual lattice gauge symmetryand translation symmetry associated to the 3-cells) Here wewould look for a symmetry given by the usual lattice gaugesymmetry and translation symmetries associated to the 4-cells

As we will show in this work the 4D spin foam modelswhich as gravity models are based on 119878119874(4) 119878119874(3 1) or119878119880(2) can be easily generalized to finite groups (or moregenerally tensor categories [65 66]) Although the immediateinterpretation as gravity models is lost one can neverthelessask whether translation symmetries are realized and if nothow these could be implemented This question is mucheasier to answer for finite groups than in the full gravitycase One can therefore see these models as a test bed forthe full theory This also applies for renormalization andcoarse graining techniques which need to be developed toaccess the large-scale limit of spin foams With finite groupmodels it might be in particular possible to access themany-particle (that is many simplices or building blocks inthe triangulation) and small-spin (corresponding to smallgeometrical size of the building blocks) regime2 This is incontrast to the few-particle and large-spin (semiclassical)regime [67ndash69] which is accessible so far

In the emergent gravity approaches [70ndash72] one attemptsto construct models which do not necessarily start fromgravitational or even geometrical variables but neverthelessshow features typical of gravity The models proposed herecan also be seen as candidates for an emergent gravityscenario A related proposal is the truncated Regge modelsin [73ndash76] in which the continuous length variables of theRegge calculus are replaced by values in Z119902 with 119902 = 2 forldquothe Ising quantum gravityrdquo

Apart fromproviding awealth of testmodels for quantumgravity researchers another intention of this work is to givean introduction to some of the spin foam concepts and ideasto researchers outside quantum gravity Indeed spin foamsarise as graphical tools in high-temperature expansions oflattice theories or more generally in the construction ofdual models We will review these ideas in Section 2 wherewe will discuss the Ising-like systems and introduce spinnets as a graphical tool for the high-temperature expansionNext in Section 3 we will discuss lattice gauge theories withthe ldquoAbelianrdquo finite groups Here spin foams arise in thehigh-temperature expansion The zero-temperature limit ofthese theories gives topological BF-theories which we willdiscuss inmore detail in Section 4 In Section 5 we detail spinfoammodels with the non-Abelian finite groups Following astrategy from quantum gravity we will also discuss the so-called constrained models which arise from the BF-modelsby implementing simplicity constraints here in the formof edge projectors into the partition function This will ingeneral change the topological character of the BF-models tonontopological ones

We will then discuss in Section 6 a canonical descriptionfor lattice gauge theories and show that for the BF-theoriesthe transfer operators are given by projectors onto the so-called physical Hilbert space These are known as stabilizer







119885 =1

2♯Vsum

119892V

prod

119890

exp (120573119892119904(119890)119892minus1

119905(119890)) (1)


120573sum119890119892119904(119890)119892

minus1





119885 =1

119902♯Vsum

119892V

prod

119890

119908119890 (119892119904(119890)119892minus1

119905(119890)) (2)


119908 (119892) =

119902minus1

sum

119896=0

119908 (119896) 120594119896 (119892)

119908 (119896) =1

119902

119902minus1

sum

119892=0

119908 (119892) 120594119896(119892)

(3)


minus1) = [120594119896(119892)]

minus1= 120594

119896(119892)


119885 =1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890) 120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890))(prod

119890

120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119896119890

(prod

119890

119908119890 (119896119890))prod

V(sum

119892V

prod

119890supV

120594119896119890

(119892119900(V119890)V ))

(4)


119890supV120594119896119890



119885 = sum

119896119890

(prod

119890

119908119890 (119896119890))prod

V120575(119902)

(sum

119890supV

119900 (V 119890) 119896119890) (5)











119885 = 119902♯Vsum

119896

(prod

119890

119908119890 (119896)) = 119902♯Vsum

119896

119908(119896)♯119890 (6)




119908 (0) = 1198901205732 cosh120573

119908 (1) = 1198901205732 cosh120573 tanh120573

(7)






prod

119890119896119890= 0

119908119890 (119896119890) (8)













minus1

119905(119890)and the




119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)




119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)


minus1) =



119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)


119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)



(119902)(sum





119885 = sum

119896

(119908119896)♯119891 (13)











119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics









119885 =1

2♯Vsum

119892V

prod

119890

exp (120573119892119904(119890)119892minus1

119905(119890)) (1)


120573sum119890119892119904(119890)119892

minus1





119885 =1

119902♯Vsum

119892V

prod

119890

119908119890 (119892119904(119890)119892minus1

119905(119890)) (2)


119908 (119892) =

119902minus1

sum

119896=0

119908 (119896) 120594119896 (119892)

119908 (119896) =1

119902

119902minus1

sum

119892=0

119908 (119892) 120594119896(119892)

(3)


minus1) = [120594119896(119892)]

minus1= 120594

119896(119892)


119885 =1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890) 120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119892V

sum

119896119890

(prod

119890

119908119890 (119896119890))(prod

119890

120594119896119890

(119892119904(119890)119892minus1

119905(119890)))

=1

119902♯Vsum

119896119890

(prod

119890

119908119890 (119896119890))prod

V(sum

119892V

prod

119890supV

120594119896119890

(119892119900(V119890)V ))

(4)


119890supV120594119896119890



119885 = sum

119896119890

(prod

119890

119908119890 (119896119890))prod

V120575(119902)

(sum

119890supV

119900 (V 119890) 119896119890) (5)











119885 = 119902♯Vsum

119896

(prod

119890

119908119890 (119896)) = 119902♯Vsum

119896

119908(119896)♯119890 (6)




119908 (0) = 1198901205732 cosh120573

119908 (1) = 1198901205732 cosh120573 tanh120573

(7)






prod

119890119896119890= 0

119908119890 (119896119890) (8)













minus1

119905(119890)and the




119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)




119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)


minus1) =



119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)


119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)



(119902)(sum





119885 = sum

119896

(119908119896)♯119891 (13)











119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics










119885 = 119902♯Vsum

119896

(prod

119890

119908119890 (119896)) = 119902♯Vsum

119896

119908(119896)♯119890 (6)




119908 (0) = 1198901205732 cosh120573

119908 (1) = 1198901205732 cosh120573 tanh120573

(7)






prod

119890119896119890= 0

119908119890 (119896119890) (8)













minus1

119905(119890)and the




119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)




119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)


minus1) =



119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)


119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)



(119902)(sum





119885 = sum

119896

(119908119896)♯119891 (13)











119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics














minus1

119905(119890)and the




119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)




119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)


minus1) =



119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)


119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)



(119902)(sum





119885 = sum

119896

(119908119896)♯119891 (13)











119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878 = 120572sum

119891

Re (tr (119880 (ℎ119891))) (9)




119885 =1

119902♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891) (10)


minus1) =



119885 =1

119902♯119890sum

119892119890

prod

119891

sum

119896119891

119908119891 (119896119891) 120594119896119891

(ℎ119891) (11)


119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896119891

(ℎ119891))

=1

119902♯119890sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

(sum

119892119890

prod

119891sup119890

120594119896119891

(119892119900(119891119890)

119890))

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

119890

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

= sum

119896119891

(prod

119891

119908119891 (119896119891))prod

Vprod

119890supV

120575(119902)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

(12)



(119902)(sum





119885 = sum

119896

(119908119896)♯119891 (13)











119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119885 =1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) 120594119896 (ℎ119891))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

119908119891 (119896119891) exp(2120587119894

119902119896119891 sdot ℎ119891 (119892119890)))

=1

119902♯119890sum

119892119890

sum

119896119891

(prod

119891

exp(2120587119894119902

119896119891 sdot ℎ119891 (119892119890) + ln119908119891 (119896119891)))

(14)




119885 = (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)

times (prod

119891

(120575(2)

(119896119891) + (1 + 120572) 120575(2)

(119896119891 minus 1)))

= (1198901205732 cosh120573)

♯119891

sum

119896119891

prod

119890

120575(2)

(sum

119891sup119890

119900 (119891 119890) 119896119891)


1198911

120575(2)

(1198961198911

minus 1) + 1205722

times sum

1198911lt1198912

120575(2)

(1198961198911

minus 1) 120575(2)

(1198961198912


times sum


♯119891

120575(2)

(1198961198911


(119896119891♯119891

minus 1))

(15)


(2)(119896119891 minus 1) = 1 + 120572 120575

(2)(119896119891 minus 1) Thus





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890)) (16)





119885 = ♯[

[


prod

119890sub119891

119892119890 = id forall119891]

]

times1

119902♯119890 (17)



119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119885 =1

119902♯119890sum

119892119890

prod

119891

120575(119902)

(ℎ119891 (119892119890))

=1

119902♯119890+♯119891sum

119892119890

sum

119896119891

exp(2120587119894

119902sum

119891

119896119891 sdot ℎ119891 (119892119890))

(18)


sum119890sub119891



119885 =1

119902♯119891sum

119896119891

prod

V(prod

119890supV

120575(119902)

( sum

119891sup119890

119900 (119891 119890) 119896119891)) (19)


119885 = ♯[

[


sum

119891sup119890

119900 (119891 119890) 119896119891 equiv 0 (mod 119902) forall119890]

]

times1

119902♯119891

(20)




minus1




1198961015840

119891equiv 119896119891 + sum

119888sup119891

119900 (119888 119891) 119896119888 (mod 119902) (21)




1015840










119891 (119892) = sum

120588

radicdim 120588119891120588119886119887 120588(119892)119886119887 (22)



119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119891120588119886119887 =1

|119866|sum

119892

radicdim 120588119891 (119892) 120588(119892)119886119887 (23)


119866 rarr C by

⟨1198911 | 1198912⟩ =1

|119866|sum

119892isin119866

1198911 (119892)1198912 (119892) = sum

120588119886119887

(1198911)120588119886119887(1198912)120588119886119887

(24)


⟨120588119894119895 | 120588119896119897⟩ =120575120588120588120575119894119896120575119895119897

dim 120588 (25)


120594120588 (119892) = tr (120588 (119892)) (26)


120575119866 (119892) = sum

120588

dim 120588120594120588 (119892) (27)

which satisfies

1

|119866|sum

119892

120575119866 (119892) 119891 (119892) = 119891 (1) (28)


ℎ119891 =

997888997888rarr

prod

119890isin120597119891

119892119900(119891119890)

119890 (29)




119885 =1

|119866|♯119890sum

119892119890

prod

119891isin119865

119908119891 (ℎ119891) (30)

f

e1

e2

e3

e4

e5


5 The curvature is

given by ℎ119891= 119892

minus1

1198905

1198921198904

119892minus1

1198903

1198921198902

1198921198901

(note the ordering)


119885 =1

|119866|♯119890sum

119892119890

sum

120588119891

prod

119891

(119908119891)120588120588119891(119892

plusmn1

1198901

)11989411198942

times 120588119891(119892plusmn1

1198902

)11989421198943


119890119899

)1198941198991198941

(31)




=1

|119866|sum

119892isin119866

1205881198911

(119892)11989411198951

1205881198912

(119892)11989421198952

sdot sdot sdot 120588119891119899

(119892)119894119899119895119899

(32)


(119875119890120595)11989411198942sdotsdotsdot119894119899


12059511989511198952sdotsdotsdot119895119899

(33)


H119890 = 1198811205881

otimes 1198811205882


(34)




1198990




119875119890 =

119898

sum

119896=1

10038161003816100381610038161003816120580(119896)

119890⟩ ⟨120580

(119896)

119890

10038161003816100381610038161003816 (35)


f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




f

e1

e2


2to 119891


1198902 the factor 119881

120588119891occurs

Moreover |1198941198901⟩ and ⟨119894





minus1

119890in (32) which


119890labels a



lowast

120588 in 120580119890

1

and 1205801198902




119885 = sum

120588119891120580119890

prod

119891

(119908119891)120588prod

VAV (120588119891 120580119890) (36)



dim 120588119891 we get

Z = sum

120588119891120580119890

prod

119891

dim 120588119891prod

VAV (37)


22


119878119884119872 (ℎ) =120572

2(120594120588 (ℎ) + 120594120588 (ℎ

minus1)) = 120572Re (120594120588 (ℎ)) (38)




+

119891 120588

minus



119891and 120588minus

119891are


H119890 = 119881120588+1198911


(39)


119891










119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119891=



A(119861119862)

V (120588plusmn

119891 120580119861119862) = sum

120580119890

10038161003816100381610038161003816A(119861119865)

V (120588119891 120580119890)10038161003816100381610038161003816

2

(40)






120588 ((12)) = (1 0

0 minus1) 120588 ((123)) = (

cos 21205873

minussin 21205873

sin 21205873

cos 21205873

)

(41)




(42)




119891= 120588

minus

119891 which


rarr 119881+

120588119891

otimes 119881minus

120588119891








119885 = trH119873 (43)


119890119904

H119890119904






120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120588(119892119890)1198861198871003816100381610038161003816119892⟩ = 120588(119892119890)119886119887

1003816100381610038161003816119892⟩ (44)


⟨1198921015840| 119892⟩ = prod

119890119904

120575(119866)

(1198921015840

119890119904

119892119890119904

) (45)


119892120575(119866)


relation

IdH =1

|119866|♯119890119904

sum

119892119890119904

1003816100381610038161003816119892⟩ ⟨1198921003816100381610038161003816 (46)


119885 = trH119873=

1

|119866|♯119890119904

sum

119892119890119904 119899

119873

prod

119899=0

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ (47)


119885 =1

|119866|♯119890sum

119892119890

prod

119891

119908119891 (ℎ119891)

=1

|119866|♯119890

119873

prod

119899=0

prod

(119891119904)119899+1

11990812

119891119904

(ℎ119891119904

)

times prod

(119891119905)119899

119908119891119905

(ℎ119891119905

) prod

(119891119904)119899

11990812

119891119904

(ℎ119891119904

)

(48)





⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ = prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899+1

)1

|119866|♯119890119905

timessum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)prod

119891119904

11990812

119891119904

(ℎ(119891119904)119899

)

(49)

t1

t2

t3

t4



= (50)

with

= prod

119891119904

11990812

119891119904

(ℎ119891119904

)

⟨119892119899+11003816100381610038161003816

1003816100381610038161003816119892119899⟩ =1

|119866|♯119890119905

sum

119892119890119905

prod

(119891119905)

119908119891119905

(ℎ(119891119905)119899

)

(51)


119908119891119905

(119892(119890119904)119899

119892(119890119905)119899

119892minus1

(119890119904)119899+1

119892minus1

(1198901015840119905)119899

) (52)



Γ (120574)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816120574minus1

⊳ 119892⟩ (53)

where (120574 ⊳ 119892)119890 = 120574119904(119890)119892119890120574minus1



⟨1198921003816100381610038161003816 Γ (120574)

1003816100381610038161003816120595⟩ = ⟨120574 ⊳ 119892 | 120595⟩ = 120595 (120574 ⊳ 119892) (54)



119905


= 1198660 = 0119866 (55)

where

119866 =1

|119866|♯V119904

sum

120574

Γ (120574)

⟨119892119899+11003816100381610038161003816 0

1003816100381610038161003816119892119899⟩ = prod

119891119905

119908119891119905

(119892(119890119904)119899

119892minus1

(119890119904)119899+1

)

(56)




♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






♯V119904


= 1198660119866 (57)





prod

119890

radicdim 120588119890120588119890(119892119890)119886119890119887119890

= ⟨119892 | 120588 119886 119887⟩ (58)


1015840 119887

1015840⟩ =



(ℎ)1003816100381610038161003816119892⟩ =

10038161003816100381610038161003816ℎminus1119892⟩ (ℎ)

1003816100381610038161003816119892⟩ =1003816100381610038161003816119892ℎ⟩ (59)


⟨1198921003816100381610038161003816 (ℎ)

1003816100381610038161003816120588 119886 119887⟩ = ⟨ℎ119892 | 120588 119886 119887⟩ = radicdim 120588120588(ℎ119892)119886119887

= radicdim 120588120588(ℎ)119886119888120588(119892)119888119887

= 120588(ℎ)119886119888 ⟨119892 | 120588 119888 119887⟩

(60)


(ℎ)1003816100381610038161003816120588 119886 119887⟩ = 120588(ℎ)119886119888

1003816100381610038161003816120588 119888 119887⟩

1003816100381610038161003816120588 119886 119887⟩ =

1003816100381610038161003816120588 119886 119888⟩ 120588(ℎminus1)119888119887

(61)


0 =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) =1

|119866|sum

ℎ

119908119891119905

(ℎ) (ℎ) (62)



0

1003816100381610038161003816120588 119886 119887⟩ =1

|119866|sum

ℎ

119908119891119905

120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

=1

|119866|sum

ℎ

sum

1205881015840

11988812058810158401205941205881015840 (ℎ) 120588(ℎ)1198861198881003816100381610038161003816120588 119888 119887⟩

= sum

120588

1

dim 1205881198881205881003816100381610038161003816120588 119886 119887⟩

(63)


1

|119866|sum

ℎ

1205941205881015840 (ℎ) 120588(ℎ)119886119888 =1

dim 1205881205751205881205881015840120575119886119888 (64)


119888120588 =1

|119866|sum

ℎ

119908119891119905

(ℎ) 120594120588(ℎ) (65)





Γ (120574) 120595 (ℎ119890) = 120595 (120574119904(119890)ℎ119890120574minus1

119905(119890)) (66)


120595 (ℎ119890) = sum

120588119890119886119890119887119890

120588119890119886119890119887119890

1003816100381610038161003816120588119890 119886119890 119887119890⟩ (67)


119890119886119890119887119890

namely

120588119890119886119890119887119890

= prod

V120580(V)1198861198901

119887119890119899

(68)



120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120580(V)

isin Inv(⨂

V=119904(119890)

119881120588119890

otimes ⨂

V=119905(119890)

119881lowast

120588119890

) (69)


lowast

120588



= 1198660119866 (70)




= prod

119891119904

(120575119866 (ℎ119891119904

))12

(71)




0 = |119866|♯119890119904IdH (72)


= |119866|♯119890119904+♯119891119904 119866119865 = |119866|

♯119890119904+♯119891119904 119865119866 (73)


= 119866 119873

119865= 119865 the


119885 = trH119873= |119866|

119873(♯119890119904+♯119891119904)trH119866119865

= |119866|119873(♯119890119904+♯119891119904) dim (Hphys)

(74)








(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics




(a) (b)





⟨1205951 | 1205952⟩phys = ⟨12059511003816100381610038161003816

10038161003816100381610038161205952⟩ (75)







119904

(120574)



ΓV119904

(120574)1003816100381610038161003816119892⟩ = prod

119890119904 119904(119890119904)=V119904

119890119904

(120574) prod

1198901015840119904 119905(1198901015840119904)=V119904

1198901015840119904

(120574)1003816100381610038161003816119892⟩ (76)


ΓV119904

(120574)1003816100381610038161003816120595⟩ =

1003816100381610038161003816120595⟩ (77)



ΓV119904

(120574)1003816100381610038161003816120588 119886 119887⟩ = prod

119890 119904(119890)=V119904

120588119890(120574)119886119890119888119890

prod

1198901015840 119905(1198901015840)=V119904

1205881198901015840(120574minus1)11988911989010158401198871198901015840

times1003816100381610038161003816120588 (119888119890 1198861198901015840 11988611989010158401015840) (119887119890 1198891198901015840 11988711989010158401015840)⟩

(78)




120588V119904

= ⨂

119890 119904(119890)=V119904

120588119890 otimes ⨂

1198901015840 119905(1198901015840)=V119904

120588lowast

1198901015840 (79)





sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







sum

119890 119904(119890)=V119904

119896119890 = sum

1198901015840 119905(1198901015840)=V119904

1198961198901015840 (80)



119865120588

119891119904119886119887

1003816100381610038161003816120595⟩ = 1205751198861198871003816100381610038161003816120595⟩ (81)


119865120588

119891119904119886119887

= (

prod

119890sub119891119904

120588 (119892119900(119891119904119890)

119890))

119886119887

(82)





















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics















1 119892



119894




1

(119879 119879) = 119879119892 120575119892119892 119879119892





119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics







119878 (119879 119879)

= trB1

(119879 119879) +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB (119879 119879)

(83)

where Γ(119863)119896



119864 (119905B) =1

119873119863log(int119889119879119889119879119890


119878(119879119879)) (84)


119860G =(minus1)

|120588|

SYM (G)(prod

120588

119905B(120588)

)119873minus(2(119863minus1))120596(G)

(85)

where

120596 (G) =(119863 minus 1)

2[119863 +

119863 (119863 minus 1)

4

1003816100381610038161003816VG1003816100381610038161003816 minus

1003816100381610038161003816FG1003816100381610038161003816]

(86)





max that is we


1

2

3

ℬ1



0 00

0

0

1

11

1

12

2

2

2 23

3

3

33






SYM (G) 119860G = 119890120581119863minus2

N119863minus2

minus120581119863N119863 (87)


120581119863minus2 = log119873

120581119863 =1

2(1


(88)



119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878EDT = Λsum

120590119863

vol (120590119863) minus1

16120587119866Newton

times sum

120590119863minus2

vol (120590119863minus2) 120575 (120590119863minus2)



2arccos( 1

119863)N119863)

(89)


119896119896)radic(119896 + 1)2119896




log119892

=119863

16120587119866Newtonvol (120590119863minus2)


minus 2Λvol (120590119863)

= minus2119886119863Λ

(90)






119864LO (119905B) = sum

G120596(G)=0

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

(91)



119864LO (119892) sim (1 minus119892

119892119888

)

2minus120574

(92)

where

119892119888 = minus1

48 120574 = minus

1

2 for 119863 = 2

119892119888 =119863119863

(119863 + 1)119863+1

120574 =1

2 for 119863 ge 3

(93)



⟨Volume⟩ = 119886119863⟨N119863⟩

= 2119886119863119892120597

120597119892log119864LO (119892)

sim 119886119863(1 minus

119892

119892119888

)

minus1

=Λ

log119892(1 minus

119892

119892119888

)

minus1

(94)


119892 997888rarr 119892119888 119886 997888rarr 0 (95)

while keeping

Λ 119877 = minusΛ

log119892(1 minus

119892

119892119888

) fixed (96)





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





119878 (1198791 119879

2 119879

1

1198792

)

= 119862119894119895trB1

(119879119894 119879

119895

) +

2

sum

119894=1

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873(2(119863minus2))120596(B)trB

times (119879119894 119879

119894

)

(97)

where

119862119894119895 = (1 minus119888

minus119888 1) (98)




119878 (119879 119879) = 119879119892119870119892119892119879119892 +

infin

sum

119896=2

sum

BisinΓ(119863)

119896

119905B

119873120572(B)trB (119879 119879) (99)


119870119892119892 = sum

ℎisinZ119873

119863

prod

119894=0

120575119892119894ℎ119892minus1

119894

(100)


119864 (119905B) = sum

G

(minus1)|120588|

SYM (G)prod

120588

119905B(120588)

[

[

prod

119890isinEG

sum

ℎ119890

prod

119891isinFG

120575119867119891

]

]

(101)

where

119867119891 = prod

119890isin120597119891

ℎ119900(119890119891)

119890 (102)














119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






119885 = sum

119892

infin

prod

119872=1

prod

1198971119897119872

1199081198971119897119872

(ℎ1198971

ℎ119897119872

) (103)

where ℎ119897119894


1119897119872


9 Outlook






Acknowledgments


Endnotes






119885minus11

120573= 119890

minus12057311988501

2120573 (104)



120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





120575(119902)

(119896) =1

119902

119902minus1

sum

119892=0

120594119896(119892) =

1

119902

119902minus1

sum

119892=0

120594119896 (119892) (105)





119865 = log119885 (106)










ℎ119891 (119892119890) = sum

119890sub119891

119900 (119891 119890) 119892119890 (107)







119890




minus1)119887119886






27 Here ΓV119904


119904

(120574) = 119894(ΓV119904

(120574)minusΓV119904



119904

(120574) =

ΓV119904






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics






References


































































































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics





























































































































































































































FluidsJournal of


Journal of










Superconductivity




GravityJournal of








Journal of






Volume 2014


PhotonicsJournal of


Journal of

Biophysics



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