matroid theory and chern–simons

Matroid theory and Chern–SimonsJ. A. Nieto and M. C. Marın Citation: Journal of Mathematical Physics 41, 7997 (2000); doi: 10.1063/1.1319518 View online: http://dx.doi.org/10.1063/1.1319518 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/41/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extension of Chern-Simons forms J. Math. Phys. 55, 062304 (2014); 10.1063/1.4882086 Chern-Simons matrix models and Stieltjes-Wigert polynomials J. Math. Phys. 48, 023507 (2007); 10.1063/1.2436734 On Chern–Simons (super) gravity, E 8 Yang–Mills and polyvector-valued gauge theories in Clifford spaces J. Math. Phys. 47, 112301 (2006); 10.1063/1.2363257 Uniqueness of the topological multivortex solution in the self-dual Chern–Simons theory J. Math. Phys. 46, 012305 (2005); 10.1063/1.1834694 Searching for a connection between matroid theory and string theory J. Math. Phys. 45, 285 (2004); 10.1063/1.1625416

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Matroid theory and Chern–SimonsJ. A. Nietoa) and M. C. MarınFacultad de Ciencias Fı´sico-Matema´ticas de la Universidad Auto´noma de Sinaloa,80010 Culiaca´n Sinaloa, Me´xico

~Received 1 June 2000; accepted for publication 16 August 2000!

It is shown that matroid theory may provide a natural mathematical framework fora duality symmetries not only for quantum Yang–Mills physics, but also forM-theory. Our discussion is focused in an action consisting purely of the Chern–Simons term, but in principle the main ideas can be applied beyond such an action.In our treatment the theorem due to Thistlethwaite, which gives a relationshipbetween the Tutte polynomial for graphs and Jones polynomial for alternatingknots and links, plays a central role. Before addressing this question we brieflymention some important aspects of matroid theory and we point out a connectionbetween the Fano matroid andD511 supergravity. Our approach also seems to berelated to loop solutions of quantum gravity based in an Ashtekar formalism.© 2000 American Institute of Physics.@S0022-2488~00!00712-X#

I. INTRODUCTION

In the last few years, duality has been a source of great interest to study nonperturbative, aswell as perturbative, dynamics of superstrings1 and supersymmetric Yang–Mills.2 In fact, dualityis the key physical concept that relates the five known superstring theories in 911 dimensions~i.e., nine space and one time!, Type I, Type IIA, Type IIB, Heterotic SO~32! and HeteroticE8

3E8 , which may now be understood as different manifestations of one underlying unique theorycalled M-theory.3–9 However, dualities are still a mystery and up to now a general understandingof how these dualities arises is missing. Nevertheless, just as the equivalence principle is a basicprinciple in general relativity, the recent importance of dualities in gauge field theories and stringtheories strongly suggest a duality principle as a basic principle in M-theory. In this sense, what itis needed is a mathematical framework to support such a duality principle.

M-theory is defined as a 1011 dimensional theory arising as the strong-coupling limit of typeIIA string theory. Essentially, M-theory is a nonpertubative theory and in addition to the fivesuperstring theories it describes supermembranes,10 5-branes,11 and D511 supergravity.12 Al-though the complete M-theory is unknown there are two main proposed routes to construct it. Oneis theN5(2,1) superstring theory13 and the other M~atrix!-theory.14 Martinec15 has suggested thatthese two scenarios may, in fact, be closely related. This scenario has been extended16 to includedualities involving compactifications on timelike circles as well as spacelike circles ones. Inparticular, it has been shown that T-duality on a timelike circle takes type IIA theory into a typeIIB* theory and type IIB* theory into a type IIA theory and that the strong-coupling limit of typeIIA * is a theory in 912 dimensional theory, denoted by M* .

More recently, Khoury and Verlinde17 have shed some new light on the old idea of open/closed string duality.18 This duality is of special interest because it emphasizes the idea that closedstring dynamics~gravity! is dual to open string dynamics~gauge theory!. Two previous exampleson this direction are matrix theory,14 where gravity arises as an effect of open string quantumfluctuations and Maldacena’s conjecture19 according to which that anti-deSitter supergravity is insome sense dual to supersymmetric gauge theory.

a!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 12 DECEMBER 2000

79970022-2488/2000/41(12)/7997/9/$17.00 © 2000 American Institute of Physics


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Thus, just as the tensor theory makes mathematical sense of the postulate of relativity ‘‘thelaws of physics are the same for every observer,’’ we are pursuing the possibility that the math-ematical formalism necessary to make sense of a duality principle in M-theory is matroid theory.20

This theory is a generalization of matrices and graphs and, in contrast, to graphs in which dualitycan be defined only for planar graphs, it has the remarkable property that duality can be defined forevery matroid. Since M~atrix!-theory andN5(2,1) superstrings have had an important success ondescribing some essential features of M-theory a natural question is to see whether matroid theoryis related to these two approaches. As a first step in this direction we may attempt to see if matroidtheory is linked somehow toD511 supergravity which is a common feature of both formalisms.In fact, it has been shown21 that the Fano matroid and its dual are closely related to Englert’scompactification22 of D511 supergravity. This result is physically interesting because such arelation allows the connection between the fundamental Fano matroid or its dual23 and octonionswhich, at the same time, are one of the alternative division algebras.24 It is worth mentioning thatsome time ago Blencowe and Duff25 raised the question of whether the four forces of naturecorrespond to the four division algebras.

In this work, we make further progress on this program. Specifically, we find a route toincorporate matroid theory in quantum Yang–Mills in the context of Chern–Simons action. Ourmechanism is based on a theorem due to Thistlethwaite26 which connect the Jones polynomial foralternating knots with the Tutte polynomial for graphs. Since Witten27 showed that the Jonespolynomial can be understood in three dimensional terms through a Chern–Simons formalism itbecame evident that we have a bridge between graphs and Chern–Simons. In this context duality,which is the main subject in graphic matroids, can be associated to Chern–Simons in a math-ematical natural way. This connection may transfer important theorems from matroid theory tofundamental physics. For instance, the theorem due to Whitney20 that if M1 ,..,M p andM18 ,..,M p8are the components of the matroidsM and M 8, respectively, and ifMi8 is the dual ofMi ( i51, . . . ,p) thenM 8 is dual ofM and conversely, ifM andM 8 are dual matroids thenMi8 is dualof Mi may be applied to dual Chern–Simons partition functions. One of our aims in this work isto explain how this can be done.

The plan of this work is as follows. In Sec. II, we briefly review matroid theory and in Sec. IIIwe closely follow Ref. 21 to discuss a connection between matroid theory andD511 supergrav-ity. In Sec. IV, we study the relation between matroid theory and Witten’s partition function forknots. Finally, in Sec. V, we make some final comments.

II. A BRIEF REVIEW OF MATROID THEORY

In 1935, while working on abstract properties of linear dependence, Whitney20 introduced theconcept of a matroid. In the same year, Birkhoff28 established the connection between simplematroids~also known as combinatorial geometries29! and geometric lattices. In 1936, Mac Lane30

gave an interpretation of matroids in terms of projective geometry. And an important progress tothe subject was given in 1958 by Tutte23 who introduced the concept of homotopy for matroids. Atpresent, there is a large body of information about matroid theory. The reader interested in thesubject may consult the excellent books on matroid theory by Welsh,31 Lawler32 and Tutte.33 Wealso recommend the books of Wilson,34 Kung35 and Ribnikov.36

An interesting feature of matroid theory is that there are many different but equivalent waysof defining a matroid. In this respect, it seems appropriate to briefly review Whitney’s20 discoveryof the matroid concept. While working with linear graphs Whitney noticed that for certain matri-ces duality had a simple geometrical interpretation quite different than in the case of graphs.Further, he observed that any subset of columns of a matrix is either linearly independent orlinearly dependent and that the following two theorems must hold.

~a! Any subset of an independent set is independent.~b! If Np andNp11 are independent sets ofp andp11 columns, respectively, thenNp together

with some column ofNp11 forms an independent set ofp11 columns.

7998 J. Math. Phys., Vol. 41, No. 12, December 2000 J. A. Nieto and M. C. Marin


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Moreover, he discovered that if these two statements are taken as axioms then there areexamples that do not represent any matrix and graph. Thus, he concluded that a system satisfying~a! and~b! should be a new one and therefore deserve a new name: He called this kind of systema matroid.

The definition of a matroid in terms of independent sets has been refined and is now expressedas follows: A matroidM is a pair (E,I), whereE is a nonempty finite set, andI is a nonemptycollection of subsets ofE ~called independent sets! satisfying the following properties:

~I i! any subset of an independent set is independent;~I ii ! if I andJ are independent sets withI #J, then there is an elemente contained inJ but

not in I , such thatI ø$e% is independent.A base is defined to be any maximal independent set. By repeatedly using the property~I ii !

it is straightforward to show that any two bases have the same number of elements. A subset ofEis said to be dependent if it is not independent. A minimal dependent set is called a circuit.Contrary to the bases not all circuits of a matroid have the same number of elements.

An alternative definition of a matroid in terms of bases is as follows.A matroidM is a pair (E,B), whereE is a nonempty finite set andB is a nonempty collection

of subsets ofE ~called bases! satisfying the following properties:~B i! no base properly contains another base;~B ii ! if B1 andB2 are bases and ifb is any element ofB1 , then there is an elementg of B2

with the property that (B1-$b%)ø$g% is also a base.A matroid can also be defined in terms of circuits:A matroid M is a pair (E,C), whereE is a nonempty finite set, andC is a collection of

nonempty subsets ofE ~called circuits! satisfying the following properties:~C i! no circuit properly contains another circuit;~C ii ! if C1 andC2 are two distinct circuits each containing an elementc, then there exists a

circuit in C1øC2 which does not containc.If we start with any of the three definitions the other two follows as a theorems. For example,

it is possible to prove that~I! implies ~B! and ~C!. In other words, these three definitions areequivalent. There are other definitions also equivalent to these three, but for the purpose of thiswork it is not necessary to consider them.

Notice that even from the initial structure of a matroid theory we find relations such asindependent–dependent and base-circuit which suggests duality. The dual ofM , denoted byM* ,is defined as a pair (E,B* ), whereB* is a nonempty collection of subsets ofE formed with thecomplements of the bases ofM . An immediate consequence of this definition is that every matroidhas a dual and this dual is unique. It also follows that the double-dualM** is equal toM .Moreover, ifA is a subset ofE, then the size of the largest independent set contained inA is calledthe rank ofA and is denoted byr(A). If M5M11M2 andr(M )5r(M1)1r(M2) we shall saythat M is separable. Any maximal nonseparable part ofM is a component ofM . An importanttheorem of Whitney20 is that if M1 ,..,M p andM18 ,..,M p8 are the components of the matroidsMand M 8, respectively, and ifMi8 is the dual ofMi ( i 51, . . . ,p), then M 8 is a dual of M .Conversely, letM and M 8 be dual matroids, and letM1 ,..,M p be components ofM . LetM18 ,..,M p8 be the corresponding submatroids ofM 8. ThenM18 ,..,M p8 are the components ofM 8,andMi8 is a dual ofMi .

III. MATROID THEORY AND SUPERGRAVITY

Among the most important matroids are the binary and regular matroids. A matroid is binaryif it is representable over the integers modulo two. Let us clarify this definition. An importantproblem in matroid theory is to see which matroids can be mapped in some set of vectors in avector space over a given field. When such a map exists we are speaking of a coordinatization~orrepresentation! of the matroid over the field. Let GF(q) denote a finite field of orderq. Thus, wecan express the definition of a binary matroid as follows: A matroid which has a coordinatizationover GF~2! is called binary. Furthermore, a matroid which has a coordinatization over every field

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is called regular. It turns out that regular matroids play a fundamental role in matroid theory,among other things, because they play a similar role as planar graphs in graph theory.34 It isknown that a graph is planar if and only if it contains no subgraph homeomorphic toK5 or K3,3.The analog of this theorem for matroids was provided by Tutte.23 In fact, Tutte showed that amatroid is regular if and only if is binary and includes no Fano matroid or the dual of this. In orderto understand this theorem it is necessary to define the Fano matroid. We shall show that the Fanomatroid may be connected with octonions which, in turn, are related to the Englert’s compactifi-cation ofD511 supergravity.

A Fano matroidF is the matroid defined on the setE5$1,2,3,4,5.6.7% whose bases are allthose subsets ofE with three elements exceptf 15$1,2,4%, f 25$2,3,5%, f 35$3,4,6%, f 4

5$4,5,7%, f 55$5,6,1%, f 65$6,7,2% and f 75$7,1,3%. The circuits of the Fano matroid are pre-cisely these subsets and its complements. It follows that these circuits define the dualF* of theFano matroid.

Let us write the setE in the formE5$e1 ,e2 ,e3,e4,e5 ,e6 ,e7%. Thus, the subsets used to definethe Fano matroid now becomef 15$e1 ,e2 ,e4%, f 25$e2 ,e3 ,e5%, f 35$e3 ,e4 ,e6%, f 4

5$e4 ,e5 ,e7%, f 55$e5 ,e6 ,e1%, f 65$e6 ,e7 ,e2% and f 75$e7 ,e1 ,e3%. The central idea is to iden-tify the quantitiesei , where i 51, . . . ,7, with the octonionic imaginary units. Specifically, wewrite an octonionq in the formq5q0e01q1e11q2e21q3e31q4e41q5e51q6e61q7e7 , whereq0 andqi are real numbers. Here,e0 denotes the identity. The product of two octonions can beobtained with the rule

eiej52d i j 1c i jk ek , ~1!

whered i j is the Kronecker delta andc i jk5c i jl d lk is the fully antisymmetric structure constants,

with i , j ,k51, . . . ,7. Bytaking the fact thatc i jk equals 1 for one of the seven combinationsf i wemay derive all the values ofc i jk .

The octonion~Cayley! algebra is not associative, but alternative. This means that the basicassociator of any three imaginary units is

~ei ,ej ,ek!5~eiej !ek2ei~ejek!5w i jkmem , ~2!

wherew i jkl is a fully antisymmetric four index tensor. It turns out thatw i jkl andc i jk are related bythe expression

w i jkl 5~1/3!!e i jklmnrcmnr , ~3!

wheree i jklmnr is the completely antisymmetric Levi-Civita tensor, withe12 . . . 751. It is interest-ing to note that given the numerical valuesf i for the indices ofcmnr and using~3! we get the otherseven subsets ofE with four elements of the dual Fano matroidF* . For instance, if we takef 1

then we havec124 and ~3! givesw3567 which leads to the circuit subset$3,5,6,7%.We would like now to relate the above structure to the Englert’s octonionic solution22 of

eleven dimensional supergravity. First, let us introduce the metric

gab5d i j hai hb

j , ~4!

wherehai 5ha

i (xc) is a sieben-bein, witha,b,c51, . . . ,7.Here,xc are a coordinates patch of thegeometrical seven sphereS7. The quantitiesc i jk can now be related to theS7 torsion in the form

Tabc5R021c i jkha

i hbj hc

k , ~5!

whereR0 is theS7 radius. While the quantitiesw i jkl can be identified with the four index gaugefield Fabcd through the formula

Fabcd5R021w i jkl ha

i hbj hc

khdl . ~6!



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Furthermore, it is possible to prove that the Englert’s 7-dimensional covariant equations are solvedwith the identificationFabcd5lT[abc ,d] , wherel is a constant. Therefore,lTabc5Aabc is thefully antisymmetric gauge field which is a fundamental object in 2-brane theory.6

It is important to mention that in the Englert’s solution ofD511 supergravity the torsionsatisfies the Cartan–Schouten equations,

TacdTbcd56R022gab , ~7!

TeadTdb fTf ce53R022Tabc. . ~8!

But as Gursey and Tze37 noted, these equations are mere septad-dressed, i.e., covariant forms ofthe algebraic identities,

c iklc jkl56d i j , ~9!

c l imcm jncnkl53c i jk , ~10!

respectively. It is worth mentioning that the Englert solution realizes the Riemannian curvature-less but torsion-full Cartan-geometries of absolute parallelism onS7.

So, we have shown that the Fano matroid is closely related to octonions which at the sametime are an essential part of the Englert’s solution of absolute parallelism onS7 of D511 super-gravity. The Fano matroid and its dual are the only minimal binary irregular matroids. We knowfrom the Hurwitz theorem~see Ref. 24! that octonions are one of the alternative division algebras~the others are the real numbers, the complex numbers and the quaternions!. While among the onlyparallelizable spheres we findS7 @the other are the spheresS1 andS3 ~Ref. 38!#. This distinctiveand fundamental role played by the Fano matroid, octonions andS7 in such different areas inmathematics as combinatorial geometry, algebra and topology, respectively, lead us to believe thatthe relationship between these three concepts must have a deep significance not only in mathemat-ics, but also in physics. Of course, it is known that the parallelizability ofS1, S3 andS7 has to dowith the existence of the complex numbers, the quaternions and the octonions, respectively~seeRef. 39!. It is also known that using an algebraic topology called K-theory40 we find that the onlydimensions for division algebras structures on Euclidean spaces are 1, 2, 4 and 8. We can add tothese remarkable results another fundamental concept in matroid theory; the Fano matroid.

IV. MATROID THEORY AND CHERN–SIMONS

Before going into detail, it turns out to be convenient to slightly modify the notation of theprevious section. In this section, we shall assume that the Greek indicesa,b, . . .,etc. run from 0 to3, the indicesi , j , . . . ,etc. run from 0 to 2 and finally the indicesa,b, . . . ,etc. take values in therank of a compact Lie GroupG. Further, we shall denote a compact oriented four manifold asM4.

Consider the second Chern class action,

S5k

16p EM4

emnabFmna Fab

b gab , ~11!

with the curvature given by

Faba 5]aAb

a2]bAaa1Cbc

a AabAb

c . ~12!

Heregab is the Killling–Cartan metric andCbca are the completely antisymmetric structure con-

stants associated with the compact simple Lie groupG. The action~11! is a total derivative andleads to the Chern–Simons action,

SCS5k

4p EM3H e i jk S Ai

a~] jAkb2]kAj

b!gab12

3CabcAi

aAjbAk

cD J , ~13!



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whereM35]M4 is a compact oriented three dimensional manifold. In a differential form notation~13! can be rewritten as follows:

SCS5k

2p EM3

TrS A`dA12

3A`A`AD , ~14!

whereA5AiaTa dxi , with Ta the generators of the Lie algebra ofG.

Given a link L with r components and irreducible representationr r of G, one for eachcomponent of the link, Witten27 defines the partition function,

Z~L,k!5E DA exp~ iSCS!)r 51

n

W~Lr ,r r !, ~15!

whereW(Ci ,r i) is the Wilson line,

W~Lr ,r r !5TrrrP expS E

Lr

AiaTa dxi D . ~16!

Here the symbolP means the path-ordering along the knotsLr .If we chooseM35S3, G5SU(2) andr r5C2 for all link components then the Witten’s

partition function~15! reproduces the Jones polynomial,

Z~L,k!5VL~ t !, ~17!

where

t5e2p i /k. ~18!

HereVL(t) denotes the Jones polynomial satisfying the skein relation:

t21VL12tVL2

5S At21

AtD VL0

, ~19!

where L1 , L2 and L0 are the standard notations for overcrossing, undercrossing and zerocrossing.

Now, let us pause about the relation between the knots and the Chern–Simons term and let usdiscuss the Tutte polynomial. To each graphG, we associate a polynomialTG(x,x21) with theproperty that ifG is composed solely of isthmus and loops thenTG(x,x21)5xIx2 l , whereI is thenumber of isthmuses andl is the number of loops. The polynomialTG satisfies the skein relation,

TG5TG 81TG 9 , ~20!

whereG 8 andG 9 are obtained by delating and contracting, respectively, an edge that is neither aloop nor an isthmus ofG.

There is a theorem due to Thistlethwaite26 which assures that ifL is an alternating link andG(L) the corresponding planar graph then the Jones polynomialVL(t) is equal to the TuttepolynomialTG(2t,2t21) up to a sign and factor power oft. Specifically, we have

VL~ t !5~2t3/4!w(L)t2(r 2n)/4TG~2t,2t21!, ~21!

wherew(L) is the writhe andr andn are the rank and the nullity ofG, respectively. HereVL(t)is the Jones polynomial of alternating linkL.

On the other hand, a theorem of Tutte allows us to computeTG(2t,2t21) from the maximaltrees ofG. In fact, Tutte proved that ifB denotes the maximal trees in a graphG, i (B) denotes the



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number of internally active edges inG ande(B) the number the externally active edges inG ~withrespect to a given maximal treeBeB! then the Tutte polynomial is given by the formula

TG~2t,2t21!5( xi (B)x2e(B), ~22!

where the sum is over all elements ofB.First, note thatB is the collection of bases ofG. If we now remember our definition of matroid

M in terms of bases discussed in Sec. II we note the Tutte polynomialTG(2t,2t21) computedaccording to~22! uses the concept of a graphic matroidM (G) defined as the pair (E,B), whereEis the set of edges ofG. In fact, the elements ofB satisfy the two properties:

~B i! no base properly contains another base;~B ii ! if B1 andB2 are bases and ifb is any element ofB1 , then there is and elementg of B2

with the property that (B12$b%)ø$g% is also a base;which identifies aM (G) as a matroid. With this remarkable connection between the Tutte poly-nomial and a matroid we have found in fact a connection between the partition functionZ(L,k)given in ~15! and matroid theory. This is because according to~21! the Tutte polynomialTG(2t,2t21) are related to the Jones polynomialVL(t) which at the same time according to~17!are related to the partition functionZ(L,k). Specifically, forM35S3, G5SU(2), r r5C2 for allalternating link components ofL, we have the relation

Z~L,k!5VL~ t !5~2t3/4!w(L)t2(r 2n)/4TG~2t,2t21!. ~23!

Thus, the matroid (E,B) used to computeTG(2t,2t21) can be associated not only toVL(t), butalso toZ(L,k).

Now that we have at hand this slightly but an important connection between matroid theoryand Chern–Simons theory we are able to transfer information from matroid theory to Chern–Simons and conversely from Chern–Simons to matroid theory. Let us discuss two examples forthe former possibility.

First of all, it is known that in matroid theory the concept of duality is of fundamentalimportance. For example, there is a remarkable theorem that assures that every matroid has a dual.So, the question arises about what are the implications of this theorem in Chern–Simons formal-ism. In order to address this question let us first make a change of notationTG(2t,2t21)→TM (G)(t) and Z(L,k)→ZM (G)(k). The idea of this notation is to emphasize the connectionbetween matroid theory, Tutte polynomial and Chern–Simons partition function. Consider theplanar dual graphG* of G. In matroid theory we haveM (G* )5M* (G). Therefore, the dualityproperty of the Tutte polynomial,

TG~2t,2t21!5TG* ~2t21,2t !, ~24!

can be expressed as

TM (G)~ t !5TM* (G)~ t21!, ~25!

and consequently from~23! we discover that for the partition functionZM (G)(k) we should havethe duality property

ZM (G)~k!5ZM* (G)~2k!. ~26!

This duality symmetry for the partition functionZM (G)(k) is not really new, but is already knownin the literature as mirror image symmetry~see, for instance, Ref. 41, and references quoted there!.However, what seems to be new is the way we had derived it.

As a second example let us first mention another theorem of Whitney:20 If M1 ,..,M p andM18 ,..,M p8 are the components of the matroidsM andM 8, respectively, and ifMi8 is the dual ofMi ( i 51, . . . ,p), thenM 8 is a dual ofM . Conversely, letM andM 8 be dual matroids, and let



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M1 ,..,M p be components ofM . Let M18 ,..,M p8 be the corresponding submatroids ofM 8. ThenM18 ,..,M p8 are the components ofM 8, andMi8 is a dual ofMi . Thus, according to~26! we findthat

ZMi (Gi )~k!5ZM

i8(Gi )~2k!, ~27!

if and only if

ZM (G)~k!5ZM8(G)~2k!, ~28!

whereGi are the components ofG.

V. COMMENTS

Motivated by a possible duality principle in M-theory we have started to bring informationfrom matroid theory to fundamental physics. We now have two good examples which indicate thatthis task makes sense. In the first example, we have found enough evidence for a connectionbetween the Fano matroid and supergravity inD511. While in the second example, we havefound a relation between the graphic matroid and the Witten’s partition function for Chern–Simons. This relation is of special importance because it leads us to a duality symmetry in thepartition functionZM (G)(k). In fact, if there is a duality principle in M-theory we should expect toa have a duality symmetry in the corresponding partition function associated with M-theory.

In this work, we have concentrated on the original connection between Chern–Simons actionand knots theory. But it is well known that the Chern–Simons formalism and knots connection hasa number of extensions.41 It will be interesting to study such extensions from the point of view ofmatroid theory. It is also known that Chern–Simons formalism is closely related to conformal fieldtheory and this in turn is closely related to string theory. So, it seems that the present work mayeventually lead to a connection between matroid theory and string theory. In order to achieve thisgoal we need to study the relation between matroids and Chern–Simons using signed graphs.42

This is because general knots and links~not only alternating! are in a one to one correspondencewith signed planar graphs. This in turn is straightforward connected with Kauffmannpolynomials43 which at the same time leads to the Jones polynomials. But, signed graphs leads tosigned matroids. So, one of our future goals will be to find a connection between signed matroidsand string theory. Moreover, matrix Chern–Simons theory44 has a straightforward relation withthe Matrix-model and noncommutative geometry.45 So, a natural extension of the present workwill be to study the relation between matroid theory and matrix Chern–Simons theory.

An important duality in M-theory is the strong/weak couplingS-duality46 which provides uswith one of the most important techniques to study nonperturbative aspects of gauge field theoryand string theory. For further work it may also be important to find the relation between the dualitysymmetry forZM (G)(k) given in ~27! andS-duality.

Besides the possible connection between M~atroid!-theory and M-theory there is anotherinteresting physical application of the present work. This has to do with loop solutions of quantumgravity based on Ashtekar formalism. It is known that the Witten’s partition function provides asolution of the Ashtekar constraints.47 So, the duality symmetries~27! also applies to such solu-tions. In other words, it seems that we have also found a connection between matroid theory andloop solutions of quantum canonical gravity.

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matroid theory and chern–simons

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