a-model and generalized chern–simons theory

7
Physics Letters B 620 (2005) 180–186 www.elsevier.com/locate/physletb A-model and generalized Chern–Simons theory A. Schwarz 1 Department of Mathematics, University of California, Davis, CA 95616, USA Received 5 February 2005; accepted 13 June 2005 Available online 22 June 2005 Editor: L. Alvarez-Gaumé Abstract The relation between open topological strings and Chern–Simons theory was discovered by Witten. He proved that A-model on T M where M is a three-dimensional manifold is equivalent to Chern–Simons theory on M and that A-model on arbitrary Calabi–Yau 3-fold is related to Chern–Simons theory with instanton corrections. In present Letter we discuss multidimensional generalization of these results. 2005 Elsevier B.V. All rights reserved. 1. Introduction In present Letter we analyze the relation between multidimensional A-model of open topological strings and generalized Chern–Simons theory. Such a rela- tion was discovered by Witten [1] in three-dimensional case; we generalize his results. Our approach is based on rigorous mathematical results of [2–6]; in three- dimensional case it gives mathematical justification of some of Witten’s statements. In modern language Witten considers A-model in presence of a stack of N coinciding D-branes wrap- ping a Lagrangian submanifold M. In the neighbor- hood of Lagrangian submanifold a symplectic mani- E-mail address: [email protected] (A. Schwarz). 1 Partly supported by NSF grant No. DMS 0204927. fold V looks like T M. In the case V = T M, dim M = 3 Witten shows that A-model is equivalent to Chern–Simons theory on M. He considers also the case when V is a Calabi–Yau 3-fold and shows that in this case Chern–Simons action functional on M ac- quires instanton corrections. We remark that one can analyze instanton correc- tions to Chern–Simons functional combining results by Fukaya [4] and Cattaneo–Froehlich–Pedrini [2] and that this approach works also in multidimensional case. To study the origin of Chern–Simons functional and its generalizations one can replace the stack of N coinciding D-branes by N Lagrangian submani- folds depending on ε and tending to the same limit as ε 0. This situation was studied by Fukaya–Oh [5] and Kontsevich–Soibelman [6]; we will show that the appearance of Chern–Simons functional follows from their results. 0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.06.030

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b

ensional

Physics Letters B 620 (2005) 180–186

www.elsevier.com/locate/physlet

A-model and generalized Chern–Simons theory

A. Schwarz1

Department of Mathematics, University of California, Davis, CA 95616, USA

Received 5 February 2005; accepted 13 June 2005

Available online 22 June 2005

Editor: L. Alvarez-Gaumé

Abstract

The relation between open topological strings and Chern–Simons theory was discovered by Witten. He proved thatA-modelon T ∗M whereM is a three-dimensional manifold is equivalent to Chern–Simons theory onM and thatA-model on arbitraryCalabi–Yau 3-fold is related to Chern–Simons theory with instanton corrections. In present Letter we discuss multidimgeneralization of these results. 2005 Elsevier B.V. All rights reserved.

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1. Introduction

In present Letter we analyze the relation betwemultidimensionalA-model of open topological stringand generalized Chern–Simons theory. Such a rtion was discovered by Witten[1] in three-dimensionacase; we generalize his results. Our approach is bon rigorous mathematical results of[2–6]; in three-dimensional case it gives mathematical justificationsome of Witten’s statements.

In modern language Witten considersA-model inpresence of a stack ofN coinciding D-branes wrapping a Lagrangian submanifoldM . In the neighbor-hood of Lagrangian submanifold a symplectic ma

E-mail address: [email protected](A. Schwarz).1 Partly supported by NSF grant No. DMS 0204927.

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.06.030

fold V looks like T ∗M . In the caseV = T ∗M ,dimM = 3 Witten shows thatA-model is equivalento Chern–Simons theory onM . He considers also thcase whenV is a Calabi–Yau 3-fold and shows thin this case Chern–Simons action functional onM ac-quires instanton corrections.

We remark that one can analyze instanton cortions to Chern–Simons functional combining resuby Fukaya [4] and Cattaneo–Froehlich–Pedrini[2]and that this approach works also in multidimensiocase.

To study the origin of Chern–Simons functionand its generalizations one can replace the stacN coinciding D-branes byN Lagrangian submanifolds depending onε and tending to the same limit aε → 0. This situation was studied by Fukaya–Oh[5]and Kontsevich–Soibelman[6]; we will show that theappearance of Chern–Simons functional follows frtheir results.

.

A. Schwarz / Physics Letters B 620 (2005) 180–186 181

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2. Generalized Chern–Simons theory

Multidimensional generalization of Chern–Simotheory can be constructed in the following way. Wconsider differential forms ond-dimensional compacmanifold M taking values in Lie algebraG. One as-sumes thatG is equipped with invariant inner producWe will restrict ourselves to the only case we neG = gl(N); then invariant inner product can be definas〈a, b〉 = Trab where Tr denotes the trace in vectrepresentation ofG = gl(n). The graded vector spacΩ∗(M) ⊗ G of such forms will be denoted byE . Thebilinear form〈C,C′〉 = ∫

MTrC · C′ specifies an odd

symplectic structure onE if dim M is odd and evensymplectic structure if dimM is even.

The generalized Chern–Simons functionalCS(C)

is defined by the standard formula

(1)

S(C) = CS(C) = 1

2

∫M

TrC dC + 1

3

∫M

TrC[C,C]

whereC ∈ E = Ω∗(M) ⊗ gl(n) andd stands for thede Rham differential. We can replaced in (1) by thedifferential dA corresponding to flat connectionA;corresponding functional will be denoted bySA. No-tice that the functionalSA for arbitrary flat connectionin trivial vector bundle can be obtained from the funtional (1) with the standard de Rham differential bmeans of shift of variables. It is easy to see thatany solutionA of equationdA+ 1

2[A,A] = 0 we have

S(C + A) = S(A) + 1

2

∫M

Tr(C dC + C[A,C])

(2)+ 1

3

∫M

TrC[C,C].

If A is a 1-form such a solution determines a flat cnection and(2) coincides (up to a constant) with thcorresponding action functional. This remark permus to reduce the study of Chern–Simons functiowith flat connection to the study of functional(1).

In the case when dimM is oddE is an odd sym-plectic space hence we can define an odd Poisbracket on the space of functionals onE (on the spaceof preobservables); the functionalSA obeys the BVclassical master equationSA,SA = 0 and thereforecan be considered as an action functional of class

mechanical system in BV-formalism. Correspondequations of motion have the form

dAC + 1

2[C,C] = 0.

The functionalS determines an odd differentiaδ on the algebra of preobservables by the formδ(O) = S,O; homology of δ are identified withclassical observables.

In the case of even-dimensional manifoldM thefunctional (1) has an interpretation in terms of BFVformalism. The Poisson bracket on the space of futionals onE (on the space of preobservables) is evthe operatorδ can be interpreted as BRST operator aits homology as classical observables.

The generalized Chern–Simons action functio(1) was considered in[7,8] in the framework of BVsigma-model. In the definition of BV sigma-modwe consider the spaceE of maps ofΠT M , whereM is a d-dimensional manifold into (odd or evensymplecticQ-manifold X. (One says that a supemanifold equipped with an odd vector field obeyiQ,Q = 0 is a Q-manifold. De Rham differentiaspecifies the structure ofQ-manifold onΠT M .) Thespace of maps ofQ-manifold into aQ-manifold alsocan be regarded as aQ-manifold. From the other sidusing the volume element onΠT M and symplecticstructure onX we can define odd or even symplecstructure onE . These facts permit us to consider Bor BFV theory where fields are identified with funtionals onE .

Numerous topological theories can be obtainedparticular cases of BV sigma-model. It was shown[7] thatA-model andB-model can be constructed thway.

To obtain generalized Chern–Simons theory frBV-sigma model we should takeX = ΠG in this con-struction. (IfG is a Lie algebra we can considerΠG asaQ-manifold whereQ is a vector field1

2fγαβcαcβ ∂

∂cγ .We use the notationcα for coordinates inΠG cor-responding to the basiseα in G; structure constantof G corresponding to this basis are denoted byf α

βγ .An invariant inner product onG specifies a symplectistructure onΠG.)

In [9] Kontsevich constructed a multidimensiongeneralization of perturbation series for standChern–Simons. It was shown in[8] that the pertur-bation theory for generalized Chern–Simons the

182 A. Schwarz / Physics Letters B 620 (2005) 180–186

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coincides with Kontsevich generalization. It is important to emphasize that usual correlation functiof multidimensional Chern–Simons theory are trial, however, one can define non-trivial cohomoloclasses of some space that play the role of generacorrelation functions. (In[9] this space was relatedthe classifying space of diffeomorphism group ofM ,in [8] it was interpreted as moduli space of gauge cditions in the corresponding BV sigma-model.)

Notice that one can construct Chern–Simons futional for every differential associativeZ2-graded al-gebraA equipped with invariant inner product〈 , 〉.(We assume that the algebra is unital; then the invant inner product can be written in terms of tra〈a, b〉 = trab.) For everyN we define the associative algebraAN as tensor productA ⊗ MatN whereMatN stands for the matrix algebra. We define CheSimons functional forA ∈AN by the formula

CS(A) = 1

2trAdA + 2

3trA3

= 1

2trAdA + 1

3trA[A,A].

(Notice that we need really only the super Lie algbra structure defined by the super commutator inassociative algebraAN .)

The functionalCS coincides with(1) in the casewhenA is the algebraΩ(M) of differential forms onmanifoldM equipped with a trace trC = ∫

MC.

The construction ofCS functional can be generaized to the case whenA is anA∞-algebra equippedwith invariant inner product. Recall that the structuof A∞-algebraA on a Z2-graded space is specifieby means of a sequence(k)m of operations; in a coordinate system the operation(k)m is specified by atensor(k)ma

a1,...,akhaving one upper index andk lower

indices. Having an inner product we can lower tupper index; invariance of inner product means tthe tensor(k)µa0,a1,...,ak

= ga0amaa1,...,ak

is cyclicallysymmetric (in graded sense). The Chern–Simons futional can be defined onA⊗MatN by means of tensor(k)µ; see[10] for details. Notice that two quasiisomorphic A∞-algebras are physically equivalent, i.corresponding Chern–Simons functionals lead tosame physical results. (In this statement we consaction functionals at the level of classical theory.)

A differential associative algebra can be consideas anA -algebra where only operations(1)m and(2)m

do not vanish; in this case both definitions of CheSimons functional coincide.

3. Observables of Chern–Simons theory

If Chern–Simons theory is constructed by meaof associative graded differential algebraA with innerproduct it is easy to check that classical observaof this theory correspond to cyclic cohomology ofA.This fact is equivalent to the statement that infinitimal deformations ofA into A∞-algebra with innerproduct are labeled by cyclic cohomologyHC(A) ofA [11]. (Recall, that classical observables are relato infinitesimal deformations of the theory.) AlgebraAdetermines Chern–Simons theory for allN the observ-ables we were talking about were defined for everyN .

As we mentioned the generalized Chern–Simtheory corresponds to the algebra of differential forΩ∗(M) with de Rham differential. It is well known[12,13] that cyclic cohomology of this algebra are rlated to equivariant homology of loop spaceL(M).More precisely, there exists a map of equivarihomology HS1(L(M)) into cyclic cohomologyHC(Ω∗(M), d); if M is simply connected this mais an isomorphism.

Recall that the loop spaceLM is defined as a spacof all continuous maps of the circleS1 = R/Z into M ;the groupS1 acts onLM in obvious way:γ (t) →γ (t + s). It will be convenient to modify the definitionof LM considering only piecewise differential mapthis modification does not change the homology.

Instead of equivariant homology ofLM one canconsider homology of the space of closed cur(string space)SM obtained fromLM by means offactorization with respect toS1. The manifoldM isembedded inLM and in SM = LM/S1 as the spaceof constant loops; excluding constant loops from csideration we can identifyS1-equivariant homologyof LM \ M with homology of SM \ M . (In generalS1-equivariant homology of over real numbers canidentified with homology of quotient space if all stablizers are finite.)2

2 The spaceSM \M is an infinite-dimensional orbifold with orbifold points corresponding ton-fold curves. See[14] for the firstanalysis of orbifold structure ofSM and calculation of homology oSM in simple cases.

A. Schwarz / Physics Letters B 620 (2005) 180–186 183

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Following [3] we will use the term “string homology” and the notationH∗M for the homology of stringspaceSM.

The homomorphism ofH∗M into the space of observables of Chern–Simons theory can be describethe following way[2].

Let us consider the standard simplex∆n = (t1,. . . , tn) ∈ Rn | 0 t1 · · · tn 1 and evaluationmapsevn,k :∆n × LM → M that transform a poin(t1, . . . , tn, γ ) ∈ ∆n × LM in γ (tk) (here 1 k n).Using these maps we can construct a differential foon LM by the formula

h(C) = Tr∫∆n

ev∗n,1C · · · ev∗

n,nC,

whereC ∈ E = Ω∗(M) ⊗ MatN is a differential formon M taking values inN × N matrices. We obtain amap of the space of fields of Chern–Simons theoryΩ∗(LM).

The formh(C) descends to the string spaceSM. Ifa is a singular chain inSM, then

(3)ρa(C) =∫a

h(C)

specifies a functional on the spaceE of fields (a pre-observable of Chern–Simons theory). It follows froresults of[2] that

(4)δρa = (−1)nρ∂a,

where∂a stands for the boundary of the chaina. Thismeans, in particular, that in the case whena is a cy-cle in the homology ofSM (in the string homology)ρa is an observable and that two homologous cycspecify equivalent observables. We obtain a mapstring homologyH∗M into the space of observableof Chern–Simons theory onM .

4. String bracket

Let us describe some operations in homologyloop spaceLM and string spaceSM that were intro-duced in[3].

The most fundamental of these operation isloop product on the loop space. It assigns (under stransversality assumptions) an(i+j −d)-dimensional

chain a • b in LM to i-dimensional chaina and j -dimensional chainb. To constructa • b one first in-tersects inM the chain of marked points ofa withthe chain of marked points ofb to obtain an(i + j −d)-dimensional chain inM along which the markedpoints ofa coincides with the marked points ofb. Nowone defines the chaina • b by means of concatenatioof the loops ofa and the loops ofb having commonmarked points.

The operator∆ on the chains of the loop spacLM transforms ani-dimensional chaina into (i + 1)-dimensional chain∆a obtained by means of circlaction onLM.

The bracketa, b of i-dimensional chaina in LMand j -dimensional chainb in LM is an (i + j + 1)-dimensional chain that can be defined by the formu

(5)a, b = (−1)i∆(a • b) − (−1)i∆a • b − a • ∆b.

All these operations descend to homology ofLM; thehomology becomes a Batalin–Vilkovisky algebra[15,16] with respect to them.

A natural map ofLM ontoSM (erasing the markepoint) determines a homomorphism proj of chain coplexes. Ani-dimensional chain inSM can be lifted to(i + 1)-dimensional chain inLM (we insert markedpoints in all possible ways); corresponding homomphism of chain complexes will be denoted by lift.

The string bracket of two chains inSM can be de-fined by the formula

(6)[a, b] = proj(lift b • lift a).

If dim a = i, dimb = j , then dim[a, b] = i+j −d +2.This bracket descends to homology ofSM (to stringhomology), defining a graded Lie algebra. The abdefinition of bracket agrees with[2]; in the definitionof [3] a andb are interchanged.

As we know, there exists a map of string homolointo the space of observables. The main result of[2] isa theorem that this map is compatible with Lie algbra structures on string homology and on the spacobservables:

(7)ρa,ρb = ρ[a,b],

where , stands for the Poisson bracket.It is important to notice that(7) remains correct ifa

andb are arbitrary chains not necessary cycles obeysome transversality conditions. Thenρ andρ are in

a b

184 A. Schwarz / Physics Letters B 620 (2005) 180–186

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general preobservables. This fact follows immediatfrom the considerations of[2].

Notice that the action of the group Diff(S1) of ori-entation preserving diffeomorphims of circleS1 deter-mines an action of this group onLM. FactorizingLMwith respect to this action we obtain a spaceSMnewthat is homotopically equivalent toSM. (This followsfrom the fact that Diff(S1) is homotopically equivalent toS1.) Similarly, instead ofLM we can considea spaceLMnew obtained fromLM by means of factorization with respect to the contractible group Diff0 S1

defined as a subgroup of Diff(S1) consisting of mapsleaving intact the point 1∈ ∂D.

5. A-model and string bracket

In this section we review some results of Fuka[4]. We will give also modification of these resultsthe form that allows us to relate them with the costructions of[2].

Let us consider a symplectic manifoldV and aLagrangian submanifoldM ⊂ V . Correlation func-tions ofA-model onV can be calculated by meanslocalization to moduli spaces of (psedo)holomorpmaps of Riemann surfaces; in the case of open strone should consider maps of bordered surfaces trforming the boundary intoM [1]. We restrict our-selves to the genus zero case; then one should conholomorphic mapsϕ of the diskD into V obeyingϕ(∂D) ⊂ M . Every such map specifies an elemeof π2(V ,M). One denotes byM(M,β) the mod-uli space of holomorphic mapsϕ : (D, ∂D) → (V ,M)

that have a homotopy typeβ ∈ π2(V ,M). We usethe notationsM(M,β) = M(M,β)/Aut(D2,1) andM(M,β)/PSL(2,R) where PSL(2,R) is the groupof fractional linear transformations identified with bholomorphic mapsD → D and Aut(D,1) denotes itssubgroup consisting of maps leaving intact the po1 ∈ ∂D. The spacesM(M,β) andM(M,β) shouldbe compactified by including stable maps from opRiemann surfaces of genus 0; we will use the sanotation for compactified spaces.

Notice thatM(M,β) specifies a chainMβ inthe string spaceSM. (We define a mapM(M,β) →SMnew restricting every mapϕ :D → V belonging toM(M,β) to the boundary of the diskD. We use in thisconstruction the modified definition ofSM discussed

r

at the end of Section4. To obtain a chain inSM weuse a map ofSMnew ontoSM that specifies homotopequivalence of these two spaces.) Similarly,M(M,β)

specifies a chainMβ in the loop spaceLM; the chainMβ can be considered as a lift ofMβ . (Again weare using modified definition ofLM at the intermedi-ate step.)

Fukaya[4,17] proved the following relation

(8)∂Mβ + 1

2

∑β=β1+β2

Mβ1, Mβ2 = 0,

where, stands for the loop bracket inLM. We willderive from(8) the relation

(9)∂Mβ + 1

2

∑β=β1+β2

[Mβ1,Mβ2] = 0,

where[ , ] denotes the string bracket inSM.The derivation is based on relationMβ = lift Mβ .

We notice that

(10)∂Mβ = ∂(lift Mβ) = lift (∂Mβ).

From the other side

∂Mβ = −1

2

∑β1+β2=β

Mβ1,Mβ2

= −1

2

∑β1+β2=β

lift Mβ1, lift Mβ2

= −1

2

∑β1+β2=β

((−1)(dimMβ1+1)

× ∆(lift Mβ1 • lift Mβ2)

− (−1)(dimMβ1+1)∆(lift Mβ1) • lift Mβ2

− (lift Mβ1) • ∆(lift Mβ2))

(11)= lift

(−1

2

∑β1+β2=β

[Mβ1,Mβ2])

.

In the derivation of this formula we used(5)–(7)and relations∆ • lift = 0, ∆ = lift •proj.

We obtain(9) comparing(8) and(11).Let us fix a ringΛ and a mapα :H2(V ,M) → Λ

obeyingα(β1 + β2) = α(β1) · α(β2).We can construct aΛ-valued chainM on SM tak-

ing

(12)M =∑

αβMβ.

β

A. Schwarz / Physics Letters B 620 (2005) 180–186 185

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It follows immediately from(11) that

(13)∂M+ 1

2[M,M] = 0.

Usually one takes asΛ the Novikov ring (a ring offormal expressions of the form

∑aiT

λi whereai ∈ R,λi ∈ R, λi → +∞). The mapα should be fixed in away that guarantees finiteness of all relevant expsions. Our considerations will be completely formwe refer to[18] for an appropriate choice ofα.

6. A-model and Chern–Simons theory

Let us start with the chainM on SM constructed athe end of Section5.

We can construct the corresponding preobservof generalized Chern–Simons theory using(13). It fol-lows immediately from(7) and (13) that the preob-servableρ = ρM obeys

δρ + 1

2ρ,ρ = 0.

We can modify the Chern–Simons functional ading ρ. The new functionalS + ρ verifies

S + ρ,S + ρ = 0.

This means thatS + ρ can be considered as a sotion of classical master equation (an action functioin BV formalism) if dimM is odd and as a BRSgenerator if dimM is even. In the case dimM = 3the functionalρ represents instanton corrections toChern–Simons action; one can argue that this is truany dimension.

The above consideration is not completely rigous. We used the results of[2,3,12] about the stringbracket on the space of chains inSM. These paperuse different definitions of string bracket; all of theagree on homology, however, it is essential for usconsider the bracket of chains that are not necesscycles. To give a rigorous proof one has to check tall results we are using can be verified with the sadefinition of string bracket; this should not be a prolem.

We have seen thatA-model instanton correctionto Chern–Simons functional can be generalized v

naturally to any dimension. This is a strong indiction that Chern–Simons functional by itself also apears in multidimensionalA-model. Indeed, analyzing Witten’s arguments[1] based on the applicatioof string field theory one can reach a conclusion tA-model on T ∗M is equivalent to the generalizeChern–Simons theory onM . (One can understanfrom Witten’s paper, that he was aware of possibity of multidimensional generalization of his construtions.)

It seems that the mathematical justification of tstatement can be based on the idea that a stacN coinciding D-branes can be replaced byN La-grangian submanifolds that depend on some paramand coincide when the parameter tends to 0. This sation was studied by Fukaya–Oh[5] and Kontsevich–Soibelman[6].

Let us considerN transversal Lagrangian submanfoldsM1, . . . ,MN in symplectic manifoldV . One canconstruct correspondingA∞-category (Fukaya category) [18]. The construction of operations in this cagory is based on the consideration of moduli spacepseudoholomorphic maps of a diskD into V . (One as-sumes thatV is equipped with almost complex struture J ; in the case whenV = T ∗M one assumethat almost complex structure is induced by a meon M .) One fixes the intersection pointsxi ∈ Mi ∩Mi+1 for 1 i N − 1 andxN ∈ MN ∩ M1. TheFukaya category is defined in terms of moduli spaMz

J (V,Mi, xi) of J -holomorphic mapsv :D → V

transforming given pointszi ∈ ∂D into pointsxi .One should consider also the union of all spa

MzJ where zi run over all cyclically ordered sub

sets of∂D and factorize this union with respectthe group PSL(2,R) acting as a group of biholomorphic automorphisms of the disk; one obtathe moduli spacesMJ (V ,Mi, xi). The definition ofoperations in Fukaya category involves summatoverMJ .

Following [5] we can consider the case whV = T ∗M and the Lagrangian submanifoldsMi aredefined as graphsMi = (x, ξ) ∈ T ∗M|ξ = εdfi(x)

wheref1, . . . , fN are such functions onM that dif-ference between any two of them is a Morse functithen the corresponding Lagrangian submanifoldstransversal and intersection pointsxi ∈ Mi ∩Mi+1 arecritical points of functionsfi − fi+1. Fukaya and Oh[5] have studied the moduli spacesM (V ,M ,x ) for

J i i

186 A. Schwarz / Physics Letters B 620 (2005) 180–186

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this choice of Lagrangian submanifolds. They haproved that for smallε these moduli spaces are dfeomorphic to moduli spacesMg(M,fi,pi) of graphflows. (An element of moduli spacesMg(M,fi,pi)

wherepi are critical points offi − fi+1 is a map of ametric graphγ intoM transforming edges of the grapγ into trajectories of negative gradient flow of the dference of two of the functions. It is assumed thatgraphγ is a rooted tree embedded into the diskD andthe exterior vertices are mapped into∂D.)

This picture is very close to the Witten’s pictu[1] where graphs appear as degenerate instantonis clear from it thatA-model onT ∗M can be reducedto quantum field theory—summation over embeddholomorphic disks can be replaced by the summaover graphs. However, it is not clear yet that this qutum field theory coincides with Chern–Simons theoTo establish this one can apply the results of[6].

The papers[5,6] use the language ofA∞-categoriesIn this language the results of[5] can be formulated inthe following way: FukayaA∞-category constructeby means of Lagrangian submanifolds ofT ∗M isequivalent to MorseA∞-category of smooth functions on M . It is proved in [6] under certain conditions that the MorseA∞-category is equivalent tde Rham category. AllA∞-categories (or, more precisely, A∞-precategories) in question are equippwith inner product; the equivalence is compatible winner product.

The minimal model of FukayaA∞-category isrelated to tree level string amplitudes; the relatof these amplitudes to Chern–Simons theory canderived from the remark that quasiisomorphicA∞-algebras with inner product specify equivalent CheSimons theories.

It is important to emphasize thatA-model for anygenus is related to Chern–Simons theory. It was mtioned in [5] that not only moduli spaces of pseudholomorphic disks onT ∗M but also moduli spaceof higher genus pseudoholomorphic curves can bescribed in terms of graphs. Again, this is consistwith equivalence ofA-model to quantum field theoryIn simplest case the relation to Chern–Simons thewas studied in[19].

t

Acknowledgements

I am indebted to K. Fukaya, M. KontsevicM. Movshev, Y.G. Oh and Y. Soibelman for interesing discussions. I appreciate the hospitality of KIAwhere I learned from Y.G. Oh about unpublishedsults by K. Fukaya used in this Letter.

References

[1] E. Witten, Chern–Simons gauge theory as a string theoryThe Floer Memorial Volume, in: Progress in Mathemativol. 133, Birkhauser, Boston, MA, 1995, pp. 637–678.

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