an e_8 gauge theory of gravity in 8d, clifford spaces and grand unification

8/14/2019 An E_8 Gauge Theory of Gravity in 8D, Clifford Spaces and Grand Unification

1/22

An Exceptional E 8 Gauge Theory of Gravity in D = 8 , Clifford Spaces and

Grand Unication

Carlos CastroCenter for Theoretical Studies of Physical Systems

Clark Atlanta University, Atlanta, GA. 30314, [email protected]

September 2008

Abstract

A candidate action for an Exceptional E 8 gauge theory of gravity in 8 Dis constructed. It is obtained by recasting the E 8 group as the semi-directproduct of GL (8 , R ) with a deformed Weyl-Heisenberg group associatedwith canonical-conjugate pairs of vectorial and antisymmetric tensorialgenerators of rank two and three. Other actions are proposed, like thequartic E 8 group-invariant action in 8 D associated with the Chern-SimonsE 8 gauge theory dened on the 7-dim boundary of a 8 D bulk. To nalize,it is shown how the E 8 gauge theory of gravity can be embedded into amore general extended gravitational theory in Clifford spaces associatedwith the Cl (16) algebra and providing a solid geometrical program of a

grand-unication of gravity with Yang-Mills theories.

Keywords: C-space Gravity, Clifford Algebras, Grand Unication, Exceptionalalgebras, String Theory.

1 Introduction

Exceptional, Jordan, Division and Clifford algebras are deeply related and es-sential tools in many aspects in Physics [3], [8], [9]. Ever since the discovery[1] that 11 D supergravity, when dimensionally reduced to an n-dim torus ledto maximal supergravity theories with hidden exceptional symmetries E n forn 8, it has prompted intensive research to explain the higher dimensionalorigins of these hidden exceptional E n symmetries [2] . More recently, therehas been a lot of interest in the innite-dim hyperbolic Kac-Moody E 10 andnon-linearly realized E 11 algebras arising in the asymptotic chaotic oscillatorysolutions of Supergravity elds close to cosmological singularities [1], [2].

Grand-Unication models in 4 D based on the exceptional E 8 Lie algebrahave been known for sometime [7]. The supersymmetric E 8 model has more

1


2/22

recently been studied as a fermion family and grand unication model [6] un-der the assumption that there is a vacuum gluino condensate but this conden-sate is not accompanied by a dynamical generation of a mass gap in the pureE 8 gauge sector. Supersymmetric non-linear models of Kahler coset spaces

E 8SO (10) SU (3) U (1) ;

E 7SU (5) ;

E 6SO (10) U (1) are known to contain three generations

of quarks and leptons as (quasi) Nambu-Goldstone superfields [4] (and refer-ences therein). The coset model based on G = E 8 gives rise to 3 left-handedgenerations assigned to the 16 multiplet of SO (10), and 1 right-handed gen-eration assigned to the 16 multiplet of SO (10). The coset model based onG = E 7 gives rise to 3 generations of quarks and leptons assigned to the 5+ 10multiplets of SU (5), and a Higgsino (the fermionic partner of the scalar Higgs)in the 5 representation of SU (5).

A Chern-Simons E 8 Gauge theory of Gravity proposed in [15] is a uniedeld theory (at the Planck scale) of a Lanczos-Lovelock Gravitational theory

with a E 8 Generalized Yang-Mills (GYM) eld theory, and is dened in the 15 Dboundary of a 16D bulk space. The Exceptional E 8 Geometry of the Clifford(16) Superspace Grand-Unication of Conformal Gravity and Yang-Mills wasstudied by [16]. In particular, it was discussed how an E 8 Yang-Mills in 8D , aftera sequence of symmetry breaking processes E 8 E 7 E 6 SO (8, 2), leads toa Conformal gravitational theory in 8 D based on the conformal group SO (8, 2)in 8D . Upon performing a Kaluza-Klein-Batakis [19] compactication on CP 2 ,involving a nontrivial torsion , leads to a Conformal Gravity-Yang-Mills uniedtheory based on the Standard Model group SU (3)SU (2)U (1) in 4D. Batakis[19] has shown that, contrary to the standard lore that it is not possible to obtainthe Standard Model group from compactications of 8 D to 4D , the inclusion of a nontrivial torsion in the internal CP 2 = SU (3) /SU (2) U (1) space permitsto do so. Furthermore, it was shown [16] how a conformal (super) gravity and(super) Yang-Mills unied theory in any dimension can be embedded into a(super) Clifford-algebra-valued gauge eld theory by choosing the appropriateClifford group.

The action that denes a Chern-Simons E 8 gauge theory of ( Euclideanized )Gravity in 15-dim (the boundary of a 16 D space) was based on the octic E 8invariant constructed by [5] and is dened as [15]

S = M 16

< F F ...... F > E 8 =

M 16

(F M 1 F M 2...... F

M 8 ) M 1 M 2 M 3 ....M 8 = M 16 L

(15)CS (A , F ) (1.1)

The E 8 Lie-algebra valued 16-form < F 8 > is closed : d (< F M 1 T M 1 F

M 2

T M 2..... F M 8

T M 8 > ) = 0 and locally can always be written as an exactform in terms of an E 8-valued Chern-Simons 15-form as I 16 = dL(15)CS (A , F ).

For instance, when M16 = S 16 the 15-dim boundary integral (1.1) is evaluatedin the two coordinate patches of the equator S 15 = M16 of S 16 leading to theintegral of tr (g 1dg )15 (up to numerical factors ) when the gauge potential A iswritten locally as A = g 1dg and g belongs to the E 8 Lie-algebra. The integral

2


3/22

is characterized by the elements of the homotopy group 15 (E 8). S 16 can alsobe represented in terms of quaternionic and octonionic projectives spaces asHP 4 , OP 2 respectively.

In order to evaluate the operation < ....... > E 8 in the action (1.1) it involvesthe existence of an octic E 8 group invariant tensor M 1 M 2 ....M 8 that was recentlyconstructed by Cederwall and Palmkvist [5] using the Mathematica packageGAMMA based on the full machinery of the Fierz identities. The entire octicE 8 invariant contains powers of the SO (16) bivector X IJ and spinorial Y generators X 8 , X 6Y 2 , X 4Y 4 , X 2Y 6 , Y 8 . The corresponding number of terms is6, 11, 12, 5, 2 respectively giving a total of 36 terms for the octic E 8 invariantinvolving 36 numerical coefficients multiplying the corresponding powers of theE 8 generators. There is an extra term ( giving a total of 37 terms ) with anarbitrary constant multiplying the fourth power of the E 8 quadratic invariantI 2 = 12 tr [ (F IJ X J )2 + ( F Y )2 ].Thus, the E 8 invariant action has 37 terms containing : (i) the Lanczos-Lovelock (Euclideanized) Gravitational action associated with the 15-dim bound-ary M16 of the 16-dim manifold. ; (ii) 5 terms with the same structure as thePontryagin p4(F IJ ) 16-form associated with the SO (16) spin connection IJ and where the indices I, J run from 1 , 2, ...., 16; (iii) the fourth power of thestandard quadratic E 8 invariant [ I 2]4 ; (iv) plus 30 additional terms involvingpowers of the E 8-valued F IJ and F eld-strength (2-forms). The most salientfeature of the action (1.1) is that it furnishes a unication of gravity and E 8Yang-Mills theory in 16 D.

It is the purpose of this work to explore further the Exceptional theories of Gravity based on gauging the E 8 group in 8D and how to embed these theoriesinto generalized theories of gravity in C -spaces (Clifford spaces) providing asolid geometrical program of Grand-Unication of Gravity with the other forces

in Nature. Recent approaches to the E 8 group based on Clifford algebras canbe found in [11], [29].

2 An E 8 Gauge Theory of Gravity in D = 8

We will base our construction of the E 8 Gauge Theory of Gravity in D = 8on Borns deformed reciprocal complex gravitational theory and Noncommuta-tive Gravity in 4 D [18] which was constructed as a local gauge theory of thedeformed Quaplectic group U (1, 3) s H(4) advanced by [10], and that wasgiven [18] by the semi-direct product of U (1, 3) with the deformed (noncom-mutative) Weyl-Heisenberg group corresponding to noncommutative generators[Z a , Z b] = 0. To achieve our goal we need to show why the E 8 group can berecast as the semi-direct product of GL(8, R) with a deformed Weyl-Heisenberggroup involving canonical-conjugate pairs of vectorial and antisymmetric tenso-rial generators.

The commutation relations of E 8 can be expressed in terms of the 120 SO (16)bivector generators X [IJ ] and the 128 SO (16) chiral spinorial generators Y as[12] (and references therein)

3


4/22

[X IJ , X KL ] = 4 ( IK X LJ IL X KJ + JK X IL JL X IK ).[X IJ , Y ] =

12

[IJ ] Y ; [Y , Y ] =

14

[IJ ] X IJ . (2.1a)

where X IJ = X JI . It is required to choose a representation of the gammamatrices such that [IJ ] =

[IJ ] since [Y , Y ] is antisymmetric under .The Jacobi identities among the triplet [ Y , [Y , Y ]] + cyclic permutation are

IJ IJ Y

+ cyclic permutation among (,, ) = 0 . (2.2a)

the above Jacobi identity can be shown to be satised by contracting two of thespinorial indices ( , ) in (2.2a) after multiplying (2.2a) by KL and

K 1 K 2 ....K 6 ,

respectively, giving

IJ KL

IJ +

IJ

KL

IJ +

IJ

KL

IJ = 0 . (2.2b)

and

IJ K 1 K 2 ....K 6

IJ +

IJ

K 1 K 2 ....K 6

IJ +

IJ

K 1 K 2 ....K 6

IJ = 0 . (2.2c)

Eqs-(2.2b, 2.2c) are zero (which implies that eq-(2.2a) is also zero) due to thevery special properties of the chiral representation of the Clifford gamma ma-trices in 16D and after decomposing the 12 (128127) = 8128 dimensional spaceof antisymmetric [ ] matrices into a space involving 120 antisymmetric IJ and 8008 I 1 I 2 ....I 6 matrices in their chiral spinorial indices [21] .

The E 8 algebra as a sub-algebra of Cl (8)Cl (8) is consistent with theSL (8, R ) 7-grading decomposition of E 8(8) (with 128 noncompact and 120 com-pact generators) as shown by [12]. Such SL (8, R) 7-grading is based on thediagonal part [ SO (8) SO (8)]diag SO (16) described in full detail by [12] andcan be deduced from the Cl (8)Cl (8) 7-grading decomposition of E 8 providedby Larsson [11] as follows,

[ (1) (1)

(1) ]1 (2) + 1 (1)[

(2)

(2)

(2) ] +

(1)

(2) . (2.3)

these tensor products of elements of the two factor Cl (8) algebras, described bythe subscripts (1) , (2), furnishes the 7 grading of E 8(8)

8 + 28 + 56 + 64 + 56 + 28 + 8 = 248 . (2.4)8 corresponds to the 8 D vector ; 28 is the 8D bivector ; 56 is the 8Dtri-vector , and 64 = 8 8 corresponds to the tensor product

(1)

(2) .

In essence one can rewrite the E 8 algebra in terms of 8 + 8 vectors Z a , Z a (a = 1 , 2,...8); 28 + 28 bivectors Z [ab ], Z [ab ]; 56 + 56 tri-vectors E [abc ], E [abc ],

4


5/22

and the SL (8, R ) generators E ba which are expressed in terms of a 8 8 = 64-component tensor Y ab that can be decomposed into a symmetric part Y (ab )with 36 independent components, and an anti-symmetric part Y [ab ] with 28independent components. Its trace Y cc = N yields an element N of the Cartansubalgebra such that the degrees 3, 2, 1, 0, 3, 2, 1 of the 7-grading of E 8(8)can be read from [12]

We begin by following very closely [12] by writing the full E 8 commutatorsin the SL (8, R ) basis of [13], after decomposing the SO (16) representations intorepresentations of the subgroup SO (8) SO (8) SO (8) diag SO (16). Theindices corresponding to the 8 v , 8 s and 8 c representations of SO (8), respectively,will be denoted by a, and . After a triality rotation the SO (8) vector andspinor representations decompose as [12]

16 8 s 8 c . (2.5)

128 s (8 s 8 c )(8 v 8 v ) = 8 v 56 v 128 35 v . (2.6a)

128 c (8 v 8 s )(8 c 8 v ) = 8 s 56 s 8 c 56 c . (2.6b)respectively. We thus have I = ( , ) and A = ( ,ab), and the E 8 generatorsdecompose as

X [IJ ] (X [ ], X [ ], X ); Y A (Y , Y ab ). (2.7)Next we regroup these generators as follows. The 63 generators

E ba =

18(

ab X

[ ]+

ab X

[ ]) + Y

(ab )

18

ab

Y cc

. (2.8)

for 1 a, b 8 span an SL (8, R ) subalgebra of E 8 . The generator given bythe trace N = Y cc extends this subalgebra to GL(8, R). ab , abc , .. are signedsums of antisymmetrized products of gammas. The remainder of the E 8 Liealgebra then decomposes into the following representations of SL (8, R ):

Z a =14

a ( X + Y ). (2.9a)

Z [ab ] = Z ab =18

ab X [ ] ab X [ ] + Y [ab ]. (2.9b)

E [abc ] = E abc = 14

abc ( X Y ). (2.9c)

and

Z a = 14

a (X Y ). (2.10a)

Z [ab ] = Z ab = 18

ab X [ ] ab X [ ] + Y [ab ]. (2.10b)

5


6/22

E [abc ] = E abc = 14

abc ( X + Y ). (2.10c)

It is important to emphasize that Z a = ab Z b, Z ab = ac db Z cd , ...... and forthese reasons one could use the more convenient notation for the generators

Z a (Z a , Z a ); Z ab (Z ab , Z ab ); Z abc (E abc , E abc ). (2.11)which permits to view these doublets of generators (2.11) as pairs of canonicallyconjugate variables, and which in turn, allows us to view their commutationrelations as a dening a generalized deformed Weyl-Heisenberg algebra withnoncommuting coordinates and momenta as shown next. One may now denethe pairs of complex generators to be used later

V a =1

2(

Z a+

i

Z a ), V

a =1

2(

Z a+ + i

Z a ). (2.12a)

V ab =12 (Z

ab+ i Z ab ), V ab =

12 (Z

ab+ + i Z ab ). (2.12b)

V abc =12 (Z

ab+ i Z ab ), V abc =

12 (Z

abc+ + i Z abc ). (2.12c)

The remaining GL (8, R) = Sl (8, R) U (1) generators areE ab = E (ab ) + E [ab ]. (2.13)

The Cartan subalgebra is spanned by the diagonal elements E 11 , .......,E 77and N , or, equivalently, by Y 11 , .......,Y 88 . The elements E b

afor a < b (or

a > b ) together with the elements for a < b < c generate the Borel subalgebraof E 8 associated with the positive (negative) roots of E 8 . Furthermore, thesegenerators are graded w.r.t. the number of times the root 8 (correspondingto the element N in the Cartan subalgebra) appears, such that for any basisgenerator X we have [N, X ] = deg (X ) X .The degree can be read off from

[N, Z a ] = 3Z a , [N, Z a ] = 3 Z a , [N, Z ab ] = 2Z ab ; [N, Z ab ] = 2Z ab[N, E abc ] = E abc , [N, E abc ] = E abc ; [N, E ba ] = 0. (2.14)

The remaining commutation relations dening the generalized deformed

Weyl-Heisenberg algebra involving pairs of canonical conjugate generators are

[Z a , Z b] = 0; [Z a , Z b] = 0; [Z a , Z b] = E ba 38

ba N. (2.15)

This last commutator between the pairs of conjugate Z a , Z b generators (likephase space coordinates) yields the deformed Weyl-Heisenberg algebra. The

6


7/22

latter algebra is deformed due to the presence of the E ba generator in the r.h.s of (2.15) and also because the

N trace generator does not commute with Z a , Z a as

seen in (2.14). Similarly, one has the deformed Weyl-Heisenberg algebra amongthe pairs of conjugate Z ab , Z ab antisymmetric rank-two tensorial generators (liketensorial phase space coordinates)

[Z ab , Z cd ] = 0; [Z ab , Z cd ] = 0; [Z ab , Z cd ] = 4[c[a E

d ]b] +

12

cdab N ; (2.16)

The commutators among the pairs of conjugate and noncommuting E abc , E abc

antisymmetric rank-three generators (like noncommuting tensorial phase spacecoordinates) are

[E abc , E def ] =

1

32abcdefgh Z

gh= 0 [E

abc, E

def ] =

1

32 abcdefghZ gh = 0

(2.17)

[E abc , E def ] = 18

[ab[de E f ]c ]

34

abcdef N. (2.18)

The other commutators among the generalized antisymmetric tensorial genera-tors are

[Z ab , Z c ] = 0; [Z ab , Z c ] = E abc ; [Z ab , Z c ] = E abc ; [Z ab , Z c ] = 0 . (2.19)

[E abc , Z d ] = 0; [E abc , Z d ] = 3d[a Z bc ]; [E abc , Z de ] =

6[abde Z

c]; [E abc , Z de ] = 0 .(2.20)

[E abc , Z d ] = 3[ad Z

bc ]; [E abc , Z d ] = 0; [E abc , Z de ] = 0; [E abc , Z de ] = 6de[ab Z c].(2.21)

The homogeneous commutators among the GL (8, R) generators and those be-longing to the deformed Weyl-Heisenberg algebra are

[E ba , Z c ] = ca Z b +

18

ba Z c ; [E ba , Z c ] =

bc Z a

18

ba Z c . (2.22)

[E ba , Z cd ] = 2

b[c Z d ]a

14

ba Z cd ; [E

ba , Z

cd] = 2

[ca Z

d ]b+

14

ba Z

cd.

[E ba , E cde ] = 3[ca E de ]b +

38

ba E cde ; [E a b, E cde ] = 3b[c E de ]a

38

ba E cde .(2.23)

7


8/22

Finally, the commutators among the GL (8, R) generators are

[E ba , E dc ] = bc E da da E bc . (2.24)The elements {Z a , Z ab }(or equivalently {Z a , Z ab }) span the maximal 36-dimensional abelian nilpotent subalgebra of E 8 [12], [13]. Finally, the generators

are normalized according to the values of the traces given by

T r (NN ) = 6 0 8; T r (Z a Z b) = 6 0 ab , T r (Z ab Z cd ) = 6 0 2! abcdT r (E abc E def ) = 6 0 3! def abc , T r (E ba E dc ) = 6 0 da bc

152

ba dc . (2.25)

with all other traces vanishing.Using the redenitions of the generators in eqs-(2.11, 2.12) allows to write

the E 8 Hermitian gauge connection associated with the E 8 generators as

A = E a V a + E a V a + E ab V ab + E ab V ab +E abc V abc + E

abc V abc + i (

ab ) E (ab ) + [ab ] E [ab ] (2.26)

where one may set the length scale L = 1, scale that is attached to the viel-beins to match the ( length ) 1 dimensions of the connection in (2.26). TheGL (8, R) components of the E 8 (Hermitian) gauge connection are the (real-valued symmetric) (ab ) shear and (real-valued antisymmetric) [ab ] rotationalparts of the GL (8, R) anti-Hermitian gauge connection i ((ab ) i

[ab ] ) such

that the GL (8, R) Lie-algebra-valued connection i ab E ab is Hermitian becausethe GL (8, R) generators E (ab ) , E [ab ], and the remaining ones appearing in the E 8commutators of eqs-(2.14-2.24), are all chosen to be anti-Hermitian (there areno i factors in the r.h.s of the latter commutators). The (generalized) vielbeinselds are E a , E ab , E abc plus their complex conjugates. These (generalized) viel-beins elds involving antisymmetric tensorial tangent space indices also appearin generalized gravity in Clifford spaces (C-spaces) where one has polyvector-valued coordinates in the base space and in the tangent space such that thegeneralized vielbeins are represented by square and rectangular matrices [22].The trace part N is included in the symmetric shear-like generator E (ab ) of Gl(8, R ). The rotational part corresponds to E [ab ].The E 8 (Hermitian) eld strength (in natural units h = c = 1) is

F = i [ D , D ] = ( AA AA + i f ABC AB AC ) LA . (2.27a)where the indices A = 1 , 2, 3, ...... 248 are spanned by the 248 generators LA of E 8

V a , V a , V ab , V ab , V abc , V abc , E (ab ) , E [ab ]. (2.27b)giving a total of 8 + 8 + 28 + 28 + 56 + 56 + 36 + 28 = 248, respectively.

8


9/22

The structure constants are determined by the commutators eqs-(2.14-2.24)in terms of the redenitions of the generators in eqs-(2.11, 2.12). It is theGL (8, R) eld strength sector of the E 8 eld strength the one which is associatedwith the Hermitian GL (8, R )-valued curvature two form

R = ( i R (ab ) E (ab ) + R [ab ] E [ab ] ) dx dx =i ( R (ab ) i R [ab ] ) ( E (ab ) + E [ab ] ) dxdx (2.28)

and whose components are given by

R [ab ] = [ab ] [ab ] +

[ac ][

[cb ] ]

(ac )[ (cb ) ] +

1L2

E a[ E b ] +

1L2

E a[ E b ] + ........ (2.29)

R (ab ) = (ab ) (ab ) + (ac )[ [cb ] ] + (bc )[ [ca ] ] +1

L2E a[ E

b ] +

1L2

E b[ E a ] + ............. (2.30)

A summation over the repeated c indices is implied and [ ] denotes the anti-symmetrization of indices with weight one. One may set the length scale L = 1(necessary to match dimensions).

The components of the (generalized) torsion two-form correspond to the eldstrength associated with the (generalized) vielbeins

F a = E a E a +

[ac ][ E

c ] + ....... (2.31)

F ab = E

ab E

ab +

[ac ][ E

cb ] + ........ (2.32)

F abc = E abc E abc +

[ad ][ E

dbc ] + ...... (2.33)

plus their complex conjugates F a , F ab , F abc .The complex Hermitian metric with symmetric g( ) and antisymmetric g[ ]

components (which could play the role of a symplectic structure) associated withthe Exceptional E 8 Geometry is dened in terms of the (generalized) complexvielbeins

E a =12 ( e

a + if

a ); E

a =

12 ( e

a if a ). (2.34)

E ab =12 ( e

ab + if

ab ); E

ab =

12 ( eab if ab ). (2.35)

E abc =12 ( e

abc + if

abc ); E

abc =

12 ( e

abc if abc ). (2.36)

9


10/22

as

g g( ) + i g[ ] E a E b ab + E ab E cd abcd + E abc E def abcdef .(2.37)such that ( g ) = g (g )

= g and where the generalized (tangentspace) area and volume metrics are given as

abcd = ac bd bc ad . (2.38a)abcdef = ad be cf permutations of a,b, c indices (2.38b)

The complex-valued Hermitian curvature tensor is dened

R = ( R (ab ) i R [ab ] ) ( E a E b + E a E b ). (2.39a)

R = ( R (ab ) i R [ab ] ) ( E a E b + E a E b ). (2.39b)

where

E a = ab E b , E a = ab E b , E

a E

b =

ba , E

a E

b =

ba . (2.39c)

E ab = E cd abcd , E abc = E def abcdef , E

ab E

cd =

cdab , E

abc E

def =

def abc .

(2.39d)The contraction of spacetime indices of the Hermitian curvature tensor with

the complex Hermitian metric g yields two different complex valued HermitianRicci tensors 1 given by

R = g g R =

R

= R ( ) + i R [ ]; (R ) = R (2.40)

and

S = g g R = S ( ) + i S [ ]; (S ) = S (2.41)

due to the fact that

g g = but g g = . (2.42)

because g = g . The position of the indices is crucial.A further contraction yields the generalized (real-valued) Ricci scalars

R = g R = ( g( ) i g[ ]) ( R ( ) + i R [ ] ) =R = g( ) R ( ) + B R [ ]. (2.43)

1 There is a third Ricci tensor Q [ ] = R

related to the curl of the nonmetricity

Weyl vector Q [24]

10


11/22

S = g S = ( g( )

i g[ ]) ( S ( ) + i S [ ] ) =

S = g( ) S ( ) + B S [ ]. (2.44)The antisymmetric part of the metric g[ ] B can be identied with aKalb-Ramond eld. The rst term g( ) R ( ) corresponds to the usual scalarcurvature of the ordinary Riemannian geometry. The presence of the extraterms B R [ ] and B S [ ] due to the anti-symmetric components of thecomplex metric, the two different types of Ricci tensors and the presence of generalized vielbeins with antisymmetric tensorial tangent space indices in thedenition of the complex Hermitian metric (2.37, 2.38) are one of the hallmarksof this Exceptional complex gravity based on E 8 . We should notice that theinverse complex metric is

g( ) + ig [ ] = [ g( ) + ig[ ] ] 1 = ( g( ) ) 1 + ( ig[ ]) 1 . (2.45)

so g( ) is now a complicated expression of both g and g [ ] = B . The sameoccurs with g[ ] = B . Rigorously we should have used a different notationfor the inverse metric g( ) + iB [ ], but for notational simplicity we chose todrop the tilde symbol.

The generalized real-valued torsion tensor T is dened in terms of thecomplex-valued torsion T , T quantities as

T T + T =F a E

a + F

a E

a + F

ab E

ab + F

ab E

ab + F

abc E

abc + F

abc E

abc . (2.46a)

The real-valued torsion vector is

T = T = T + T (2.46b)where the complex valued torsion tensors are

T = T g = F a E a + F

ab E ab + F

abc E abc . (2.46c)

T = T g = F a E a + F

ab E ab + F

abc E abc . (2.46d)

the complex-valued torsion vectors are

T = T g = T , T = T g

= T . (2.46e)

The inverse vielbeins are dened by

E a E b =

ba , E

ab E

cd =

cdab , E

abc E

def =

def abc , ....... (2.47a)

E a E a =

, E

ab E

ab =

, E

abc E

abc =

, .... (2.47b)

and one has the following relationships

g E a = E a , g E a = E a , g E

ab = E ab , g E

ab = E ab , ..........

(2.47c)

11


12/22


13/22

For these reasons, the E 8 gauge theory of gravity in 8 D constructed here is veryappealing. We will discuss in the next section how the inherent E 8 Geometrypresent in the action (2.48) can be seen as a particular case of a more generalClifford space gravity associated with the Clifford algebra Cl (16) in 16D [16],[22].

The standard E 8 Yang-Mills action in 8 D associated with the eld strengthsin eq-(2.27) and involving the ordinary real symmetric metric g = ea eb ab =g( ) of the manifold M8 is

I Y M =1

4g2 d8x |det g | Trace ( F F ). (2.49c)The trace operation is performed in the 248-dim adjoint representation andto evaluate the action (2.49) it is more convenient to use the normalizationcondition of the 248 original anti-Hermitian generators given by eq-(2.25) in

terms of the trace operation. The Yang-Mills coupling g2

has dimensions of (length )4 in 8D .A topological invariant action based on the quartic E 8-invariant action in

8D is given by

I (4) = M 8

< F F F F > E 8 . (2.50)

the < ...... > operation involves the existence of a quartic E 8 group invarianttensor that allows us to contract group indices and which contains powers of the 120 SO (16) bivectors X IJ and the chiral spinorial Y generators. A di-mensionless factor in front of the integral can be included. The action (2.50) islocally a total derivative and since the E 8 Lie-algebra valued 8-form < F 4 > isclosed : d (< F M 1 T M 1 F

M 2 T M 2 ..... F M 4 T M 4 > ) = 0, the action locally

can always be written as an exact form in terms of an E 8-valued Chern-Simons7-form as I 8 = dL

(7)CS (A , F ), exactly as we did in the 16 D case [15].

A generalized Yang-Mills (GYM) action in 8 D involving quartic powers of the eld strength is

I GY M = M 8

< (F F ) (F F ) > . (2.51)

where the < ...... > operation requires again the use of the quartic E 8 group-invariant tensor in order to contract group indices like the Killing Lie groupinvariant metric in ordinary quadratic Yang-Mills actions. The Hodge star dualoperation is dened in terms of the ordinary real symmetric metric g of themanifold M8 . Scalar and spinorial matter elds (minimally coupled to thegauge/geometric elds of the E 8 gauge theory of gravity) can be added to theaction (2.48) and their equations of motion can be found, the Noether symmetrycurrents (associated with conservations laws) can be constructed, etc .... like inthe metric affine theories of gravity [23].

13


14/22

3 Affine Theories of Extended Gravity in Clifford-

SpacesWe begin this nal section by showing how to embed the E 8 gauge theory of gravity and E 8 Yang-Mills theories into more general actions associated with thegauging of the Cl (16) algebra in 16 D . Let us start by constructing the actionsassociated to the most general Clifford Cl (16) gauge eld theory by writing theCl (16)-valued gauge eld

A = AA A = A 1 + Aa a + Aa 1 a 2 a 1 a 2 + Aa 1 a 2 a 3 a 1 a 2 a 3 + ......... +Aa 1 a 2 ....a 16 a 1 a 2 .......a 16 . (3.1)

the Cl (16)-algebra-valued eld strength ( omitting numerical coefficients at-tached to the s ) is

F A A = [ A ] 1 + [ [ Aa ] + Ab2[ Ab1 a ] b1 b2 + ..... ] a +[ [ Aab ] + A

a[ A

b ] Aa 1 a[ Ab1 b ] a 1 b1 Aa 1 a 2 a[ Ab1 b2 b ] a 1 b1 a 2 b2 + ..... ] ab +

[ [ Aabc ] + Aa 1 a[ A

b1 bc ] a 1 b1 + ...... ] abc + [ [ A

abcd ] Aa 1 a[ Ab1 bcd ] a 1 b1 + ...... ] abcd

+[ [ Aa 1 a 2 ....a 5 b1 b2 .....b 5 ] + Aa 1 a 2 ...a 5[ A

b1 b2 ....b 5 ] + ...... ] a 1 a 2 ....a 5 b1 b2 .....b 5 + ....

(3.2)and is obtained from the evaluation of the commutators of the Clifford-algebragenerators. The most general formulae for all commutators and anti-commutatorsof , 1 2 , ..... , with the appropriate numerical coefficients, can be found in[20], in general for pq = odd one has

[ b1 b2 .....b p , a 1 a 2 ......a q ] = 2 a 1 a 2 ......a qb1 b2 .....b p

2 p!q!2!( p2)!(q 2)!

[a 1 a 2[b1 b2 a 3 ....a q ]b3 .....b p ] +

2 p!q!4!( p4)!(q 4)!

[a 1 ....a 4[b1 ....b 4 a 5 ....a q ]b5 .....b p ] ......

(3.3a)for pq = even one has

[ b1 b2 .....b p , a 1 a 2 ......a q ] =

(1) p 12 p!q!1!( p 1)!(q 1)!

[a 1[b1 a 2 a 3 ....a q ]b2 b3 .....b p ]

(1) p 12 p!q!3!( p

3)!(q

3)!

[a 1 ....a 3[b1 ....b 3 a 4 ....a q ]b4 .....b p ] + ...... (3.3b)

The anti-commutators of the gammas can also be found in [20], and one has thereciprocal situation as eqs-(3.3), one has instead that for pq = even

{ b1 b2 .....b p , a 1 a 2 ......a q } = 2 a 1 a 2 ......a qb1 b2 .....b p

14


15/22

2 p!q!2!( p

2)!(q

2)!

[a 1 a 2[b1 b2 a 3 ....a q ]b3 .....b p ] +

2 p!q!4!( p

4)!(q

4)!

[a 1 ....a 4[b1 ....b 4 a 5 ....a q ]b5 .....b p ] ......

(3.4a)for pq = odd one has

{ b1 b2 .....b p , a 1 a 2 ......a q } = (1) p 12 p!q!

1!( p1)!(q 1)![a 1[b1

a 2 a 3 ....a q ]b2 b3 .....b p ]

(1) p 12 p!q!3!( p 3)!(q 3)!

[a 1 ....a 3[b1 ....b 3 a 4 ....a q ]b4 .....b p ] + ...... (3.4b)

Therefore, one of the most salient features of this work is that the octic E 8-invariant actions can be embedded into a more general octic Cl (16)-invariantaction involving a large number of terms. The octic Cl (16)-invariant action in16D is of the form

S = d16 x < F A 1 1 1 F A 2 2 2 ....... F A 8 8 8 A 1 A 2 ...... A 8 > 1 1 2 2 ..... 8 8 .(3.5)

where < ....... > denotes the scalar part of the Clifford geometric product asso-ciated with the products of the Cl (16) algebra generators. For instance

< a b > = ab , < a 1 a 2 b1 b2 > = a 1 b1 a 2 b2 a 1 b2 a 2 b1< a 1 a 2 a 3 > = 0 , < a 1 a 2 a 3 b1 b2 b3 > = a 1 b1 a 2 b2 a 3 b3 ......< a 1 a 2 a 3 a 4 > = a 1 a 2 a 3 a 4 a 1 a 3 a 2 a 4 + a 2 a 3 a 1 a 4 , etc ...... (3.6)

The integrand of the 16-dim action (3.5) is locally a total derivative and uponintegration yields the Chern-Simons Cl (16) gauge theory of gravity in 15 D andwhich is an extension of the action for the Chern-Simons E 8 gauge theory of gravity in 15 D described by eq-(1.1) [15].

A Cl (16)-invariant Yang-Mills action is

S Y M [Cl (16)] =1

4g2 d16 x g < F A F B A B > scalar g g . (3.7)where < A B > = GAB 1 denotes the scalar part of the Clifford geometricproduct of the gammas . There are a total of 2 16 = 65536 terms in

F A F B GAB = F F + F

a F

a + F

a 1 a 2 F

a 1 a 2 + .......... +

F a 1 a 2 .......a 16

F a 1 a 2 ......a 16

. (3.8)

where the indices run as a = 1 , 2, ..... 16. The Clifford algebra Cl (16) = Cl (8)Cl (8) has the graded structure ( scalars, bivectors, trivectors,....., pseudoscalar) given by

1 16 120 560 1820 4368 8008 11440 12870

15


16/22

11440 8008 4368 1820 560 120 16 1. (3.9)

consistent with the dimension of the Cl (16) algebra 216

= 256 256 = 65536.The anomaly-free group of the Heterotic string E 8 E 8 Cl (16)Cl (16) =Cl (32), and whose bivector generators can be identied with the SO (32) algebragenerators that is consistent with the fact that SO (32) is the anomaly-free groupof the open superstring.

Let us extend the above actions to the more general case involving the C -space (Clifford space) associated with the Cl (16) and Cl (8) algebras. In par-ticular , we will focus on the latter where the 2 8 = 256 components of theCl (8)-space polyvector X can be expanded as

X = 1 + x + x 1 2 1 2 + ........ x 1 2 3 ....... 8 1 2 .... 8 . (3.10)

In order to match dimensions in the expansion (3.10) one requires to introduce

powers of a length scale [22] which we could set equal to the Planck scale and setit to unity. In Clifford Phase Spaces [30] one needs two length scales parameters,a lower and an upper scale.

The novel affine theories of gravity in the C -space associated with the Cl (8)algebra involves gauging the semi-direct product of the Cl (8) group with thepolyvector-valued translation group T in 28 = 256 dimensions. An extendedtheory of gravity in C -spaces was presented by [22] based on generalizations of the Poincare group ( SO (D 1, 1) s T D ) to Clifford spaces of dimension 2 Dcorresponding to polyvector-valued rotations, boosts and translations. The ordi-nary affine group GL(D, R ) s T D is a further extension of the Poincare groupby including the shear transformations in additional to rotations. Therefore,the affine theories of extended gravity in C -spaces proposed here is a furthergeneralization of the C -space gravity results in [22].

The Cl (8) polyvector-valued gauge connection can be decomposed into sym-metric and antisymmetric pieces as

(AB )M {A , B }+ [AB ]M [A , B ] = (

(AB )M d

C AB +

[AB ]M f

C AB ) C C M C .(3.11)

where the structure constants f C AB and dC AB are given by eqs-(3.3, 3.4). Addingthe polyvector-valued translations P A allows us to construct the affine connec-tion in the Cl (8)-space as follows

AM = AM A + E AM P A . (3.12)where A are the Cl (8) generators corresponding to the generalized Lorentz andshear transformations in C -space, and P A are the polyvector-valued translationgenerators. The polyvector-valued gauge connection AM has 256 256 compo-nents, since the base space index M and tangent space index A span 2 8 = 256degrees of freedom. The C -space vielbein E AM which gauges the polyvector-valued translations has also 256 256 components. For instance we can see whynow there are square and rectangular matrices of the form

16


17/22

E AM = E a , E a1 a 2 , ..........,E a 1 a 2 ....a 8 , E a 1 2 , ............, E a 1 2 ........ 8 ,

E a 1 a 2 1 2 , E a 1 a 2 1 2 3 , ......, E

a 1 a 2 1 2 ........ 8 , ...... (3.13)

we must also include the (pseudo)scalar-(pseudo) scalar components E, E a 1 a 2 .....a 8 1 2 ..... 8as well.

The generalized curvature and torsion two-forms in C -space associated withthe Cl (8) gauge connection and vielbein one-forms

A AM dX M , E A E AM dX M . (3.14)are

RAMN dX

M

dX N

= RA

= d A

+ f ABC

B

C . (3.15)

T AMN dX M dX N = T A = d E A + gABC B E C . (3.16)To illustrate why RAMN is a true generalized curvature in C -space despite thefact that it has 3 polyvector-valued indices it suffices to select A among the 28bivector components [ a1a2] of the tangent Cl (8)-space and M, N to be thevectorial components of the base Cl (8)-space manifold. Hence, R [a 1 a 2 ] hasthe correct number of indices corresponding to the SO (7, 1)-valued curvaturetwo-form R [a 1 a 2 ] dx dx

. Care must be taken when working with Cl (8) orCl (7, 1), Cl (1, 7) algebras since they are not isomorphic.

Notice that the expressions in eqs-(3.15, 3.16) are just the polyvector valuedextensions of the usual Poincare algebra involving the commutators [ M , M ]and [M , P ], when the Lorentz algebra generators are realized in terms of Clifford bivectors as M [ , ] = 2 . In Clifford affine spaces associatedwith Cl (8) s T the commutators involving the polyvector generators are

[A , B ] = f C AB C , [A , P B ] = gC AB P C , [P A , P B ] = 0. (3.17)that permits us to evaluate each one of the (very large number of ) components

RAMN , T AMN in eqs-(3.15, 3.16). Further contractions of the curvature and torsionwith the inverse vielbeins give

RM = RAMN E N A ; T M = T AMN E N A ; RMNP = RAMN E AP , T MN P = T AMN E AP ,(3.18)where the C -space metric is dened in terms of the vielbein and the 256-dim

tangent space metric AB as

GMN E AM E BN AB ; AB = GMN E M A E N B ; E BN = AB E AN . (3.19a)

17


18/22

Two important remarks are in order. Firstly, note that there is in general anscalar component in

RM when the polyvector-valued index M corresponds to

the scalar element of the Clifford algebra as described by the rst term in theexpansion of the polyvector X in eq-(3.10). Secondly, instead of working withthe expressions for the curvature RAMN given in terms of the Clifford connec-tion C M by eqs-(3.15-3.18), one could work alternatively with the expressionsR

(AB )MN , R

[AB ]MN given in terms of

(AB )M ,

[AB ]M which follow explicitly from eq-

(3.11). In the latter case one can construct a C -space Ricci curvature and Ricciscalar using the standard contractions

RMP = RABMN E N B E AP ; R = RABMN E N B E M A . (3.19b)We prefer at the moment to work with eqs-(3.18) where the tangent spacepolyvector-valued indices are A,B,C,... and span a 2 8 = 256 dim space. Thebase space polyvector-valued indices are M,N,P,Q,.. . and also span a 2 8 = 256dim space. The inverse vielbein E M A is dened as E M A E BM = BA and E M A E AN = M N .In the traditional description of C -spaces [22] there is one component of theC -space metric GMN = Gscalar scalar = corresponding the scalar element of the Clifford algebra that must be included as well. Such scalar component is adilaton-like Jordan-Brans-Dicke scalar eld. In [31] we were able to show howWeyl-geometry solves the riddle of the cosmological constant within the contextof a Robertson-Friedmann-Lemaitre-Walker cosmology by coupling the Weylscalar curvature to the Jordan-Brans-Dicke scalar eld with a self-interactingpotential V () and kinetic terms ( D )(D ). Upon eliminating the Weylgauge eld of dilations A from its algebraic (non-propagating) equations of motion, and xing the Weyl gauge scalings, by setting the scalar eld to aconstant o such that 2o = 1 / 16G N , where GN is the present day observedNewtonian constant, we were able to prove that V (

o) = 3 H 2

o/ 8G and which

was precisely equal to the observed vacuum energy density of the order of 10 122 M 4Planck . H o is the present value of the Hubble scale. One must alsoinclude the pseudo-scalar elements of GMN as well when both indices M, N are[12 ..... 8] corresponding to the top grade part of the Cl (8) polyvector. Thiscomponent of the C -space metric corresponds to another scalar eld.

The affine theory of extended gravity in Cl (8)-space admits an action

S =1

22 M 256 [d(256) X ] |det G MN | L. (3.20)whose Lagrangian density is

[ a1 RM RM + a2 T M T M + a3 RMN P RMNP + a4 T MN P T MN P ]. (3.21)where the 256-dim measure of integration is dened by

[d(256) X ] = d dx dx 1 2 dx 1 2 3 ....... dx 1 2 3 ....... 8 . (3.22)

in terms of the 256 components of the polyvector X

18


19/22

A generalized Einstein-Hilbert gravity action based on gauging the general-ized Poincare group in C -spaces was given by [22] where in very special casesthe C -space scalar curvature Radmits an expansion in terms of sums of pow-ers of the ordinary scalar curvature R , Riemann curvature R and RicciR tensor of the underlying Riemannian spacetime manifold. The exteriorproducts of the (Clifford-algebra-valued) spin-connection and vielbein one-formsin Clifford-spaces can also be constructed in Clifford-Superspaces by includingboth orthogonal and symplectic Clifford algebras and generalizing the Cliffordsuper-differential exterior calculus in ordinary superspace [33], to the full edgedClifford-Superspace outlined in [16]. Clifford-Superspace is far richer than or-dinary superspace and Clifford-Supergravity involving polyvector-valued exten-sions of Poincare and (Anti) de Sitter supergravity [27] is far richer than ordinarysupergravity. Fermionic matter and scalar-eld actions can be constructed inC -spaces in terms of Dirac-Barut-Hestenes spinors as in [22], [32].

To nalize we write down the most general extension of the Cl (2n) Chern-Simons gravitational action in D = 2 n in the case when one replaces the ordinaryD = 2 n-dim space with a Cl (2n)-space of dimensions 2 D = 2 2n associated withpolyvector-valued X coordinates instead of ordinary vectors x . The action isof the form

I = M 2

2 n< F F F......... F > . (3.23)

where the summands are of the form

< F I 1 I 2 1 2 F I 3 I 4 3 4 ..... F

I 2 2 n 1 I 2 2 n 2 2 n 1 2 2 n I 1 I 2 I 3 I 4 ...... >

1 2 .......... 2 2 n 1 2 2 n . (3.25)

< F I 1 I 2 I 3 I 4 1 2 ... 4 ........ F I 2 2 n 3 ......I 2 2 n 2 2 n 3 ........ 2 2 n I 1 I 2 I 3 I 4 I 5 I 6 I 7 I 8 .......... >

1 2 .......... 2 2 n .(3.26)

etc ....... and where the brackets < ...... > denotes taking the scalar parts of the Clifford geometric product of the gamma factors inside the bracket. Theproperties of the most general action (3.23) warrants further investigation.

Acknowledgements

We thank Frank (Tony) Smith and Matej Pavsic for discussions and M.Bowers for assistance.

References

[1] E. Cremmer, B. Julia and J. Scherk, Phys. Letts B 76 (1978 ) 409;

[2] S. de Buyl, M. Henneaux and L. Paulot, Extended E 8 invariance of 11-dim Supergravity, hep-th/0512292; P. West, Class. Quan. Grav 18 (2001)4443; V. A. Belinksy, I. M Khalatnikov and E. M. Lifshitz, Adv. Phys. 19(1970) 525.

19


20/22

[3] I. R. Porteous, Clifford algebras and Classical Groups (Cambridge Univ.Press, 1995).

[4] K. Itoh, T. Kugo and H. Kunimoto, Progress of Theoretical Physics 75 ,no. 2 (1986) 386.

[5] M. Cederwall and J. Palmkvist, The octic E 8 invariant, hep-th/0702024.

[6] S. Adler, Further thoughts on Supersymmetric E 8 as family and grandunication theory, hep-ph/0401212.

[7] I. Bars and M. Gunaydin, Phys. Rev. Lett 45 (1980) 859; N. Baaklini,Phys. Lett 91 B (1980) 376; S. Konshtein and E. Fradkin, Pisma Zh.Eksp. Teor. Fiz 42 (1980) 575; M. Koca, Phys. Lett 107 B (1981) 73; R.Slansky, Phys. Reports 79 (1981) 1.

[8] C. H Tze and F. Gursey, On the role of Divison, Jordan and Related Al-gebras in Particle Physics (World Scientic, Singapore 1996); S. Okubo,Introduction to Octonion and other Nonassociative Algebras in Physics(Cambridge Univ. Press, 2005); R. Schafer, An introduction to Nonas-sociative Algebras (Academic Press, New York 1966); G. Dixon, Division Algebras, Octonions, Quaternions, Complex Numbers, and the AlgebraicDesign of Physics ( Kluwer, Dordrecht, 1994); G. Dixon, J. Math. Phys45 no 10 (2004) 3678; P. Ramond, Exceptional Groups and Physics, hep-th/0301050; J. Baez, Bull. Amer. Math. Soc 39 no. 2 (2002) 145.

[9] P. Jordan, J von Neumann and E. Wigner, Ann. Math 35 ( 1934 ) 2964;K. MacCrimmon, A Taste of Jordan Algebras (Springer Verlag, New York2003); H. Freudenthal, Nederl. Akad. Wetensch. Proc. Ser 57 A (1954 )

218; J. Tits, Nederl. Akad. Wetensch. Proc. Ser 65 A (1962 ) 530; T.Springer, Nederl. Akad. Wetensch. Proc. Ser 65 A (1962 ) 259.

[10] S. Low: Jour. Phys A Math. Gen 35 , 5711 (2002). J. Math. Phys. 38 ,2197 (1997). S. Low, Hamilton Relativity Group for Noninertial States InQuantum Mechanics [arXiv : math-ph/0710.3599]. S. Low, ReciprocallyRelativity of Noninertial Frames : Quantum Mechanics [arXiv : math-ph/0606015]. J. Govaerts, P. Jarvis, S. Morgan and S. Low, WorldlineQuantization of a Reciprocally Invariant System arXiv : 0706.3736. P.Jarvis and S. Morgan, Constraint Quantization of a worldline system in-variant under reciprocal Relativity, II arXiv.org : 0806.4794. P. Jarvisand S. Morgan, Born reciprocity and the granularity of spacetime [arXiv: math-ph/0508041]. R. Delbourgo and D. Lashmar, Born reciprocity and

the 1 /r potential hep-th/0709.0776.[11] F.D. Smith Jr, The Physics of E 8 and Cl (16) = Cl (8) Cl (8) www.tony5m17h.net/E8physicsbook.pdf (Carterville, Georgia, June 2008,

367 pages).

20


21/22

[12] K. Koepsell, H. Nicolai and H. Samtleben, An Exceptional Geometry of d = 11 Supergravity hep-th/0006034.

[13] E. Cremmer, B. Julia, H. Lu, and C. N. Pope, Nucl. Phys. B523 (1998)73.

[14] A. Malcev. Izv. Akad. Nauk SSR, Ser. Mat. 9 (1945) 291.

[15] C. Castro, A Chern-Simons E 8 Gauge theory of Gravity in D = 15,Grand-Unication and Generalized Gravity in Clifford Spaces IJGMMP4 , No. 8 (2007) 1239;

[16] C. Castro, The Exceptional E 8 Geometry of Clifford (16) Superspace andConformal Gravity Yang-Mills Grand Unication submitted to the IJG-MMP journal, June 2008.

[17] C. Castro, Int. J. Mod. Phys A 23 , (2008) 1487.[18] C. Castro, On Borns Deformed Reciprocal Complex Gravitational Theory

and Noncommutative Gravity Phys Letts B 668 (2008) 442.

[19] N. Batakis, Class and Quantum Gravity 3 (1986) L 99.

[20] K. Becker, M. Becker and J. Schwarz, String Theory and M Theory (Cam-bride University Press, 2007, pages 543-545).

[21] B. Green, J. Schwarz and E. Witten, Superstring Theory vol I, Appendix6A (Cambridge University Press 1987).

[22] C. Castro and M. Pavsic, Progress in Physics vol 1 , (2005) 31. Phys. LettsB 559 (2003) 74; Int. J. Theor. Phys 42 (2003) 1693.

[23] F. Hehl, J. McCrea, E. Mielke and Y. Neeman, Phys. Reports 258 (1995)1-171.

[24] J. Moffat, J. Math. Phys 36 , no. 10 (1995) 5897. J. Moffat and D. Boal,Phys. Rev D 11 , 1375 (1975). J. Moffat, Noncommutative Quantum Grav-ity arXiv : hep-th/0007181. N. Poplawski, On Nonsymmetric purelyaffine Gravity gr-qc/0610132, T. Damour, S. Deser and J. McCarthy,Phys. Rev D 47 , 1541 (1993). T. Jansen and T. Prokopec, Problems andHopes in Nonsymmetric Gravity arXiv : gr-qc/0611005. C. Mantz andT. Prokopec, Resolving Curvature Singularities in Holomorphic GravityarXiv : 0805.4721.

[25] A. Chamseddine, Comm. Math. Phys218

, 283 (2001). Gravity in Com-plex Hermitian Spacetime arXiv : hep-th/0610099.

[26] A. Einstein, Ann. Math 46 , 578 (1945). A. Einstein and E. Strauss, Ann.Math 47 , 731 (1946).

21


22/22

[27] J. Cornwell, Group Theory in Physics, vol. 3 , Appendix L, pp. 475-502,(Academic Press, London, 1989).

[28] J. A. Nieto Towards a Background Independent Quantum Gravity inEight Dimensions arXiv : 0704.2769 (hep-th). J. A. Nieto, Class. Quant.Grav. 22, 947 (2005). Towards an Ashtekar formalism in eight dimensionsarXiv; hep-th/0410260. J. A. Nieto and J. Socorro, Is nonsymmetric grav-ity related to string theory?, arXiv: hep-th/9610160.

[29] M. Pavsic, A Novel view on the Physical Origin of E 8 arXiv : 0806.4365(hep-th). M.Pavsic, The Landscape of Theoretical Physics: A Global View,From Point Particles to the Brane World and Beyond, in Search of a Uni- fying Principle (Kluwer Academic Publishers, Dordrecht-Boston-London,2001).

[30] C. Castro, Foundations of Physics 35 , no. 6 (2005 ) 971; Progress in Physics 1 (2005) 20.

[31] C. Castro, Foundations of Physics 37 , no. 3 (2007) 366; Mod. Phys. LettA 21 , no. 35 (2006) 2685-2701. Does Weyl Geometry solves the Riddleof Dark Energy Quantization Astrophysics, Brownian Motion and Su-persymmetry pp. 88-96 (eds F. Smarandache and V. Christianato, Math.Tiger, Chennai, India 2007).

[32] C. Castro, On Generalized Yang-Mills and Extensions of the StandardModel in Clifford (Tensorial ) Spaces, Annals of Physics 321 no.4 (2006)813; Mod. Phys Letters A 19 (2004) 14.

[33] H. De Bie and F. Sommen, A Clifford analysis to Superspace arXiv

: 0707.2859; Hermite and Gegenbauer polynomials in superspace usingClifford analysis arXiv : 0707. 2863.

[34] C.Y Lee, Class. Quan Grav 9 (1992) 2001. C. Y. Lee and Y. Neeman,Phys. Letts B 242 (1990) 59.

22

an e_8 gauge theory of gravity in 8d, clifford spaces and grand unification

Documents