26 April 2001

Physics Letters B 505 (2001) 243–248www.elsevier.nl/locate/npe

Matrix model and Chern–Simons theory

J. KlusonDepartment of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotláˇrská 2, 611 37, Brno, Czech Republic

Received 21 December 2000; received in revised form 21 February 2001; accepted 26 February 2001Editor: M. Cvetic

Abstract

In this short note we would like to present a simple topological matrix model which has close relation with thenoncommutative Chern–Simons theory. 2001 Published by Elsevier Science B.V.

Keywords:Matrix models

1. Introduction

In recent years there was a great interest in the areaof the noncommutative theory and its relation to stringtheory. In particular, it was shown in the seminal paper[1] that the noncommutative theory can be naturallyembedded into the string theory. It was also shownin the recent paper [2] that there is a remarkableconnection between noncommutative gauge theoriesand matrix theory. For that reason it is natural toask whether we can push this correspondence further.In particular, we would like to ask whether othergauge theories, for example Chern–Simons theory, canbe also generalised to the case of noncommutativeones. It was recently shown [3] that this can bedone in a relatively straightforward way in the caseof Chern–Simons theory. It is then natural to askwhether, in analogy with [2], there is a relationbetween topological matrix models [4] and Chern–Simons noncommutative theory.

It was suggested in many papers [6,7,11] thatChern–Simons theory could play profound role in the

E-mail address:klu@physics.muni.cz (J. Kluson).

nonperturbative formulation of the string theory, M-theory. On the other hand, one of the most successful(up to date) formulation of M-theory is the matrixtheory [5], for review, see [12–16]. We can ask thequestion whether there could be some connectionbetween matrix models and Chern–Simons theory.This question has been addressed in interesting papers[6,7], where some very intriguing ideas have beensuggested.

Noncommutative Chern–Simons theory could alsoplay an important role in the description of the quan-tum hall effect in the framework of string theory [17,18]. All these works suggest plausible possibility todescribe some configurations in the physics of the con-dense systems in terms of D-branes, which can be verypromising area of research. On the other hand, someideas of physics of the condense systems could be use-ful in the nonperturbative formulation of the string the-ory. In summary, on all these examples we see that itis worth to study the basic questions regarding to thenoncommutativeChern–Simons theory and its relationto the matrix theory and consequently to the string the-ory.

In this Letter we will not address these excitingideas. We will rather ask the question whether some

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00337-9

244 J. Kluson / Physics Letters B 505 (2001) 243–248

from of the topological matrix model can lead tothe noncommutative Chern–Simons theory. Such amodel has been suggested in [8] and further elaboratedin [4]. We will show that the simple topologicalmatrix model [4] cannot lead (as far as we know)to the noncommutative Chern–Simons theory. Forthat reason we propose a simple modification of thismodel when we include additional term containingthe information about the background structure of thetheory. Without including this term in the action wewould not be able to obtain noncommutative versionof the Chern–Simons theory. It is remarkable fact thatthis term naturally arises in D-brane physics fromthe generalised Chern–Simons term in D-brane actionin the presence of the background Ramond–Ramondfields [19]. For that reason we believe that our proposalcould really be embedded in the string theory and alsocould have some relation with M-theory.

2. Brief review of Chern–Simons theory

In this section we would like to review the basicfacts about Chern–Simons actions and in particulartheir extensions to noncommutative manifolds. Wewill mainly follow [3].

The Chern–Simons action is the integral of the2n + 1 form C2n+1 over spacetime manifold1 whichsatisfies

(2.1)dC2n+1 = TrF n+1,

where the wedge operation∧ between formsF isunderstood. The action is defined as

(2.2)δS2n+1

δA= δ

δA

∫C2n+1 = (n + 1)F n,

with the conventions

A = Aµ dxµ,

F = dA − iA ∧ A

(2.3)= 1

2

(∂µAν − ∂νAµ − i[Aµ,Aν]

)dxµ ∧ dxν.

The extension of this action to the case of the noncom-mutative background is straightforward [3]. The easi-est way to see this is in terms of the operator formalism

1 In this article we will consider the Euclidean spacetimes only.

of the noncommutative geometry [9]. In this short arti-cle we will not discuss the operator formalism in moredetails since it is well know from the literature. (See[9] and reference therein.) The definition of the non-commutative Chern–Simons action is very straightfor-ward in the operator formalism as was shown in [3],where the whole approach can be found. We will seethe emergence of the noncommutative Chern–Simonsaction in the operator formalism in the next sectionwhere this action naturally arises from the modifiedtopological matrix model [4].

3. Matrix model of Chern–Simons theory

It was argued in [4,8] that we can formulate thetopological matrix model which has many propertiesas the Chern–Simons theory [8]. The action for thismodel was proposed in the form

(3.1)S = εµ1···µD TrXµ1 · · ·XµD .

It is easy to see that this model can be defined in theodd dimensions only:

S = εµ1···µD TrXµD Xµ1 · · ·XµD−1

= (−1)D−1εµDµ1···µD−1 TrXµD Xµ1 · · ·XµD−1

(3.2)= (−1)D−1S,

so we haveD − 1 = 2n ⇒ D = 2n + 1. The equationsof motion obtained from (3.1) are

δS

δXµ= εµµ1···µD Xµ1 · · ·XµD = 0,

(3.3)µ = 0, . . . ,2n.

It was argued in [4] that there are solutions corre-sponding to D-branes. However, it is difficult to seewhether these solutions corresponding to some phys-ical objects since we do not know how to study thefluctuations around these solutions. For example, forD = 3 we obtain from the equation of motion forµ = 0,1,2

(3.4)[X1,X2]= 0,

[X0,X1]= 0,

[X2,X0]= 0.

We see that the only possible solutions correspond toseparate objects where the matricesX are diagonalor solution X1 = 0 = X2 with any X0. We do notknow any physical meaning of the second solution.For that reason we propose the modification of the

J. Kluson / Physics Letters B 505 (2001) 243–248 245

topological matrix model which, as we will see, hasa close relation with the noncommutative Chern–Simons theory [3]. We propose the action in the form

S = (2π)nεµ1···µD Tr

((−1)n/2D + 1

2DXµ1 · · ·XµD

(3.5)

+ (−1)(n−1)/2 D + 1

4(D − 2)θµ1µ2Xµ3 · · ·XµD

),

whereD = 2n + 1 and the numerical factors(−1)n/2,

(−1)(n−1)/2 arise from the requirement of the real-ity of the action. The other factors(D + 1)/(2D),

(D + 1)/(4(D − 2)) were introduced to have a con-tact with the work [3]. In the previous expression (3.5)we have also introduced the matrixθµν which char-acterises given configuration. The equations of motionhave a form

(−1)n/2(n + 1)εµµ1···µ2nXµ1Xµ2 · · ·Xµ2n

+ (−1)(n−1)/2n + 1

2εµµ1µ2···µ2n

(3.6)× θµ1µ2Xµ3 · · ·Xµ2n = 0.

We would like to find solution corresponding to thenoncommutative Chern–Simons action. From the factthat we have odd number of dimensions we see thatone dimension should correspond to the commutativeone. In order to obtain Chern–Simons action in thenoncommutative spacetime we will follow [10] andcompactify the commutative directionX0. For thatreason we write any matrix as

XµIJ = (

Xµij

)mn

,

(3.7)I = m × M + i, J = n × M + j,

whereXij is M × M matrix with M → ∞ and alsom,n go from −N/2 to N/2 and we again take thelimit N → ∞. In other words, the previous expressioncorresponds to the direct product of the matrices

X = A ⊗ B ⇒ Xxy = Aij Bkl,

(3.8)x = i × M + k, y = j × M + l,

with M × M matrix B. We impose the followingconstraints on the various matrices [10](Xi

ij

)mn

= (Xi

ij

)m−1,n−1, a = 1, . . . ,2n,(

X0ij

)mn

= (X0

ij

)m−1,n−1, m = n,

(3.9)(X0

ij

)nn

= 2πRδij + (X0

ij

)n−1,n−1,

where R is a radius of compact dimension. Theseconstraints (3.9) can be solved as [10](Xi

ij

)mn

= (Xi

ij

)0,m−n

= (Xi

ij

)m−n

,

(3.10)(X0

ij

)mn

= 2πRmδmn ⊗ δij + (X0

ij

)m−n

.

We then immediately obtain([X0,Xi

]ij

)mp

= 2πRmδmn

(Xi

ij

)np

− (Xi

ij

)mn

2πRnδnp

+ (X0

ik

)mn

(Xi

kj

)np

− (Xi

ik

)mn

(X0

kj

)np

(3.11)

= 2πR(m − p)(Xi

ij

)m−p

+ ([X0,Xi

]ij

)m−p

,

where(Xµij )0m = (X

µij )m. We see that the commutator

X0 with anyXi has a form of the covariant derivative[10] where the first term correspond to the ordinaryderivative−i∂0 with respect to the dual coordinatex0which is identified asx0 ∼ x0 + 2π/R. The secondterm is the commutator of the gauge fieldX0 = A0with any matrix. We could then proceed as in [10]and rewrite the action in the form of the dual theorydefined on the dual torus with the radiusR = 1/R, butfor simplicity we will use the original variables. Usingthis result we will writeX0 = KC0 as

(3.12)(C0,ij )mn = p0,mn ⊗ 1M×M + (A0,ij )mn,

where the acting ofp0 on various matrices is definedin (3.11) and where the numerical factorK will bedetermined for letter convenience.

For illustration of the main idea, let us considermatrix model defined inD = 2n + 1 = 3 dimensions.Let us consider the matrixθµν in the form

(3.13)θµν = 1mn ⊗(0 0 0

0 0 θ1N×N

0 −θ1N×N 0

),

where 1N×N is a unit matrix withN going to infinity.Then the equations of motion, which arise from (3.5),are

iε012X1X2 + iε021X

2X1 + 12

(ε012θ

12 + ε021θ21)= 0,

iε102X0X2 + iε120X

2X0 = 0,

(3.14)iε201X0X1 + iε210X

1X0 = 0,

The second and the third equation gives the condition[X0,Xi ] = 0 which leads to the solutionA0 = 0 and[p0,Xi ] = 0. These equations, together with the first

246 J. Kluson / Physics Letters B 505 (2001) 243–248

one, can be solved as

(3.15)Xi = δmn ⊗ xijk,

[xi, xj

]= iθ ij .

Thanks to the presence of the unit matrixδmn, Xi

commutes withp0 ⊗ 1M×M and so is the solution ofthe equation of motion (3.14). Following [2], we canstudy the fluctuations around this solution with usingthe ansatz

X0 = ω12C0 = ω12(p0 ⊗ 1M×M + (A0,ij )mn

),

Xi = θ ij Cj ,

Ci = 1N×N ⊗ pi + (Ai,ij )mn,

(3.16)pi = ωij xj , i = 1,2, ωij = (θ−1)

ij,

It is easy to see that this configuration corresponds tothe noncommutative Chern–Simons action inD = 3dimensions [3]. More precisely, let us introduce formalparameters

dxµ, µ = 0, . . . ,2,

dxµ ∧ dxν = −dxν ∧ dxµ,

(3.17)εµ1...µD = dxµ1 ∧ · · · ∧ dxµD

and the matrix valued one form

(3.18)C = Cµ dxµ = d + A,

whereCµ is given in (3.16). Then the action describ-ing the fluctuations around the classical solution (3.15)has a form

(3.19)S = 2π√

detθ Tr(−2i

3 C ∧ C ∧ C + 2ω ∧ C).

We rewrite this action in the form which has a closercontact with the commutative Chern–Simons theory.Firstly we prove the cyclic symmetry of the trace ofthe forms

Tr(A1 ∧ · · · ∧ AD

)= Tr

(A1

µ1A2

µ2· · ·AD

µD

)dxµ1 ∧ · · · ∧ dxµD

= Tr(A2

µ2· · ·AD

µDA1

µ1

)dxµ2 ∧ dxµD ∧ dxµ1

(3.20)= Tr(A2 ∧ · · · ∧ AD ∧ A1),

where we have used the fact thatD is odd number sothat dxµ1 commutes with even numbers ofdx. Thenthe expression (3.19) is equal to

S = −2π√

detθ

(3.21)

× Tr(iA ∧ (d ∧ A + A ∧ d) + i2

3 A ∧ A ∧ A),

where we have used

d ∧ d = pµpν dxµ dxν

= 1

2[pµ,pν]dxµ ∧ dxν = −iω,

(3.22)

ω = 1

2ωµν dxµ ∧ dxν, ωij = (

θ−1)ij

, ω0i = 0.

Now it is easy to see that (3.19) is a correct actionfor the fluctuation fieldsA. The equations of motionarising from (3.19) are

(3.23)−2i(d + A) ∧ (d + A) + 2ω = 0.

Looking at (3.22) it is easy to see that the configurationA = 0 is a solution of equation of motion as it shouldbe for the fluctuating field. With using

d ∧ A + A ∧ d = [pµ,Aν]dxµ ∧ dxν

(3.24)= i∂µAν dxµ ∧ dxν = id · A,

we obtain the derivatived· that is an analogue ofthe exterior derivative in the ordinary commutativegeometry. In this case the action has a form

(3.25)S = 2π√

detθ Tr(A ∧ d · A − 2i

3 A ∧ A ∧ A),

which is the standard Chern–Simons action in threedimensions.

We observe that this action differs from the actiongiven in [3] since there is no the termω ∧ A in ouraction. This is a consequence of the presence of thesecond term in (3.5) that is needed for the emergenceof noncommutative structure in the Chern–Simonsaction. On the other hand, from the fact that similarmatrix structure arises in the study of quantum halleffect in D-brane physics [18] we believe that ourproposal of topological action could have relation tothe string theory and M-theory. As usual, this actioncan be rewritten using in terms of the integral overspacetime with ordinary multiplication replaced withstar product [9].

Generalisation to the higher dimensions is straight-forward. The equations of motion (3.6) give

iεµνXµXν + 1

2εµνθµν = 0⇒ [

Xµ,Xν]= iθµν,

(3.26)µ, ν = 1, . . . ,2n.

J. Kluson / Physics Letters B 505 (2001) 243–248 247

We restrict ourselves to the case ofθ of the maximalrank. For simplicity, we considerθ in the form

(3.27)θµν =

0 . . . . . . . . . . . . 00 0 θ1 0 . . . 00 −θ1 0 . . . . . . 0

. . . . . . . . . . . . . . . . . .

0 . . . . . . . . . 0 θn

0 . . . . . . 0 −θn 0

.

As in 3-dimensional case we introduce the matrixω

defined as follows

(3.28)

ωij = (θ−1)

ij, i, j = 1, . . . ,2n, ωi0 = ω0i = 0.

and we defineC = Cµ dxµ = d + A,µ = 0, . . . ,2n,where the dimensionx0 is compactified as above. Andfinally variousCµ are defined as

Xi = θ ij Cj ,

(3.29)X0 =(

n∏i=1

ωi

)C0, ωi = −θ−1

i ,

with asCµ same as in (3.16). Then the action has aform

S2n+1 = (2π)n√

detθ Tr

((−1)n(−1)n/2 n + 1

2n + 1C2n+1

(3.30)

+ (−1)n−1(−1)(n−1)/2 n + 1

2n − 1ω ∧ C2n−1

).

In order to obtain more detailed description of theaction we will follow [3]. Since δ

δC= δ

δA, we can

write2

δS2n+1

δC= (2π)n

√detθ

((−1)n(−1)n/2(n + 1)C2n

+ (−1)n−1(−1)(n−1)/2(n + 1)ω ∧ C2n−2)

= (2π)n√

detθ((n + 1)(F − ω)n

(3.31)+ (n + 1)ω ∧ (F − ω)n−1),where we have used the fact thatC2 = −iω +i(−idA − iAd − iA2) = −iω + iF . Sinceω andF

are both two forms andω is a pure number from thepoint of view of the noncommutative geometry weimmediately see thatF andω commute so that we can

2 We will write Cn instead ofC ∧ · · · ∧ C.

write

(3.32)(F − ω)n =n∑

k=0

(n

k

)(−ω)n−kF k.

Following [3] we introduce the other form of theLagrangian

(3.33)δL2k+1

δC= (k + 1)F k.

Then we can rewrite (3.31) as

δ

δC

{S2n+1 − (2π)n

√detθ

×(

n∑k=0

(n + 1

k + 1

)(−ω)n−kL2k+1

−n−1∑k=0

(−1)n−k−1 1

n

(n + 1

k + 1

)ωn−k ∧ L2k+1

)}= 0

�⇒ S2n+1 = (2π)n√

detθ

(3.34)

× Tr

(n∑

k=0

(−1)n−k

(n + 1

k + 1

)ωn−kL2k+1

+n−1∑k=0

(−1)n−k−1 1

n

(n + 1

k + 1

)

× ωn−k ∧ L2k+1

).

As a check, forn = 1 we obtain from (3.34)

S3 = (2π)√

detθ Tr(−2ω ∧ L1 + L3 + 2ω ∧ L1

)(3.35)= 2π

√detθ Tr L3,

and using

δL3

δA= 2F = −2i

(dA + Ad + A2)

�⇒ L3 = −iA ∧ d ∧ A

(3.36)− id ∧ A ∧ A − i2

3A ∧ A ∧ A,

we obtain

(3.37)S3 = 2π√

detθ Tr(A ∧ d · A − 2i

3 A ∧ A ∧ A),

which, as we have seen above, is a correct form ofthe noncommutative Chern–Simons action in threedimensions.

248 J. Kluson / Physics Letters B 505 (2001) 243–248

4. Conclusion

In this short note we have shown that simplemodification of the topological matrix model [4] couldlead to the emergence of the noncommutative Chern–Simons action [3]. In order to obtain this action we hadto introduce the antisymmetric matrixθ expressing thenoncommutative nature of the spacetime. It is crucialfact that we must introduce this term into the actionexplicitly which differs from the case of the standardmatrix theory [2], where different configurations withany values of the noncommutative parameters arise asparticular solutions of the matrix theory.

It is also clear that we can find much more con-figurations than we have shown above. The form ofthese configurations depend onω. It is possible to findsuch aθ which leads to the emergence of lower dimen-sional Chern–Simons actions and also which leads tothe emergence of point-like degrees of freedom in theChern–Simons theory. For example, we can considerθ in the form

θµν = 1mn ⊗(0 0 0

0 0 A

0 −A 0

),

(4.1)A =(

0k×k 00 θ1N×N

).

This corresponds to the configuration describingChern–Simons action with the presence ofk point-likedegrees of freedom — “partons”. We could analyse theinteraction between these partons and gauge fields inthe same way as in matrix theory (For more details see[13,15] and reference therein.) It is possible that thissimple model could have some relation to the holo-graphic model of M-theory [11]. In particular, we seethat the partons arise naturally in our approach. On theother hand, the similar analysis as in [11] could deter-mineθ , i.e., requirements of the consistency of the the-ory could chooseθ in some particular form. In short,we hope that the approach given in this Letter couldshine some light on the relation between the matrixmodels and Chern–Simons theory.

Acknowledgements

We would like to thank Rikard von Unge for manyhelpful discussions. This work was supported by theCzech Ministry of Education under Contract No.144310006.

References

[1] N. Seiberg, E. Witten, String theory and noncommuta-tive geometry, J. High Energy Phys. 09 (1999) 032, hep-th/9908142.

[2] N. Seiberg, A note on background independence in noncom-mutative gauge theories, matrix models and tachyon conden-sation, hep-th/0008013.

[3] A.P. Polychronakos, Noncommutative Chern–Simons termsand the noncommutative vacuum, hep-th/0010264.

[4] I. Oda, Background independent matrix models, hep-th/9801051.

[5] T. Banks, W. Fischer, S. Shenker, L. Susskind, M-theory asa matrix model: a conjecture, Phys. Rev. D 55 (1997) 5112,hep-th/9610043.

[6] L. Smolin, M theory as a matrix extension of Chern–Simonstheory, hep-th/0002009.

[7] L. Smolin, The cubic matrix model and duality between stringsand loops, hep-th/0006137.

[8] L. Smolin, Covariant quantisation of membrane dynamics,hep-th/9710191.

[9] L. Alvarez-Gaume, S.R. Wadia, Gauge theory on a quantumphase space, hep-th/0006219.

[10] W. Taylor, D-brane field theory on compact spaces, Phys. Lett.B 394 (1997) 283, hep-th/9611042.

[11] P. Horava, M-theory as a holographic theory, Phys. Rev.D 59 (046004) (1999), hep-th/9712130.

[12] T. Banks, Matrix theory, Nucl. Phys. Proc Suppl. 67 (1998)180, hep-th/9710231.

[13] W. Taylor, Lectures on D-branes, gauge theory and M(atrices),hep-th/9801182.

[14] T. Banks, TASI lectures on matrix theory, hep-th/9911068.[15] W. Taylor, The M(atrix) model of M-theory, hep-th/0002016.[16] A. Konechny, A. Schwarz, Introduction to M(atrix) theory and

noncommutative geometry, hep-th/0012145.[17] J. Brodie, L. Susskind, N. Toumbas, How Bob Laughlin tamed

the giant graviton from Taub-NUT space, hep-th/0010105.[18] L. Susskind, The quantum hall fluid and non-commutative

Chern–Simons theory, hep-th/0101029.[19] R.C. Myers, Dielectric D-branes, J. High Energy Phys. 9912

(1999) 022, hep-th/9910053.