The Poisson sigma model on closed surfaces
aa r X i v : . [ h e p - t h ] J a n THE POISSON SIGMA MODEL ON CLOSEDSURFACES
FRANCESCO BONECHI, ALBERTO S. CATTANEO, AND PAVEL MNEV
Abstract.
Using methods of formal geometry, the Poisson sigmamodel on a closed surface is studied in perturbation theory. The ef-fective action, as a function on vacua, is shown to have no quantumcorrections if the surface is a torus or if the Poisson structure isregular and unimodular (e.g., symplectic). In the case of a K¨ahlerstructure or of a trivial Poisson structure, the partition function onthe torus is shown to be the Euler characteristic of the target; someevidence is given for this to happen more generally. The methodsof formal geometry introduced in this paper might be applicableto other sigma models, at least of the AKSZ type.
Contents
1. Introduction 22. Formal local coordinates 42.1. Gauge transformations 73. PSM in formal coordinates 84. Some computations of the effective action 134.1. Factorization of Feynman graphs 134.2. Regular Poisson structures 154.3. Axial gauge on the torus Σ = T := S × S A. S. C. acknowledges partial support of SNF Grant No. 200020-131813/1. P. M.acknowledges partial support of RFBR Grants Nos. 11-01-00570-a and 11-01-12037-ofi-m-2011 and of SNF Grant No. 200021-137595. Introduction
In this note we study the Poisson sigma model [17, 20] with world-sheet a connected, closed surface Σ. To do so we treat the Poisson struc-ture on the target manifold M as a perturbation and expand aroundthe vacua (a.k.a. zero modes) of the unperturbed action. As a criticalpoint of the latter in particular contains a constant map, we have firstto localize around its image x ∈ M . To glue perturbations arounddifferent points x , we use formal geometry [16]. Our first result is thatthe perturbative effective action (as a function on the moduli space ofvacua for the unperturbed theory) has no quantum corrections if Σ isthe torus or if the Poisson structure is regular and unimodular (e.g.,symplectic). In the former case, under the further assumption that thePoisson structure is K¨ahler, we can also perform the integration overvacua and show that the partition function is the Euler characteristicof M . For a general Poisson structure we can use worldsheet super-symmetry to regularize the effective action and study it like in [22];this argument is however a bit formal unless some extra conditions onthe Poisson structure are assumed.Notice that on the torus we need not assume unimodularity. Forother genera, the requirement of unimodularity was first remarked in[6] where the leading term of the effective action on the sphere was alsocomputed.The techniques presented in this note, in particular the way of usingformal geometry to get a global effective action, should be applicable toother field theories, in particular of the AKSZ type [1]. The techniquesof subsection 4.3 and of subsections 5.2 and 5.3 should also extend tohigher dimensional AKSZ theories in which the source manifold is aCartesian product with a torus.The torus case may also be understood as follows. Recall that theBV (Batalin–Vilkovisky) action for the Poisson sigma model can begiven in terms of the AKSZ construction [10]. It is a function on theinfinite dimensional graded manifold Map( T [1]Σ , T ∗ [1] M ). On a cylin-der Σ = S × I , the partition function should be interpreted as anoperator on the Hilbert space associated to the boundary S . As thetheory is topological, this operator is the identity and the partition What we compute is then h e i ~ S π i where S π is the interaction part dependingon the target Poisson structure π and h i denotes the expectation value for thePoisson sigma model with zero Poisson structure. We prove that the regularized effective action does not depend on the regular-ization as long as one is present. However, in principle this is not the same theoryas the non regularized one.
HE POISSON SIGMA MODEL ON CLOSED SURFACES 3 function on the torus is just its supertrace. Now, in the case of triv-ial Poisson structure, the BFV (Batalin–Fradkin–Vilkovisky) reducedphase space associated to the boundary is the graded symplectic man-ifold T ∗ T ∗ [1] M = T ∗ T [ − M . If we choose the vertical polarizationin the second presentation, the Hilbert space will be C ∞ ( T [ − M ),i.e., the de Rham complex with opposite grading. It is then to be ex-pected that the partition function on the torus should be the Eulercharacteristic of M . In the perturbative computation, however, thefinal result is usually of the form 0 · ∞ , but in the K¨ahler case we getan unambiguous answer. We might then think of a K¨ahler structureon M , if it exists, as a regularization of the Poisson sigma model withtrivial Poisson structure. Notice that, if such structures exist, theyare open dense in the space of all Poisson structures on M . Anotherregularization, which produces the same result, consists in adding theHamiltonian functions of the supersymmetry generators for the effec-tive action. Formally, this can even been done before integrating overfluctuations around vacua.Finally, notice that apart from the cases mentioned above we do ex-pect the effective action to have quantum corrections. Moreover, thenaively computed effective action in formal coordinates might happennot to be global. We show however that it is always possible to find aquantum canonical transformation which makes it into the Taylor ex-pansion of a global effective action. Its class modulo quantum canonicaltransformations is then the well-defined object associated to the theory.Section 2 is a crash course in formal geometry (essentially following[11, § §
6] to definethe effective action in formal coordinates. Next using the results ofSection 4, we show that, in the two special cases mentioned above,the effective action has no quantum correction and is the expressionin formal coordinates of a global effective action. In Section 5, westudy the effective action for the case of the torus and perform thecomputation of the partition function. Finally, in Section 6 we studythe globalization of the effective action in general.
Acknowledgment.
We thank G. Felder, T. Johnson-Freyd, T. Willwacherand M. Zabzine for useful discussions. A.S.C. thanks the University ofFlorence for hospitality. That is, we regularize STr C ∞ ( T [ − M ) id = h i as h i := lim ǫ → h e i ǫ ~ S π i . We then show that, in the K¨ahler case, h e i ǫ ~ S π i is independent of ǫ and equal tothe Euler characteristic of M . F. BONECHI, A. S. CATTANEO, AND P. MNEV Formal local coordinates
We shortly review the notion of formal local coordinates followingthe simple introduction of [11, §
2] (for more on formal geometry see[2, 7, 16]).A generalized exponential map for a manifold M is just a smoothmap φ : U → M , where U is some open neighborhood of the zerosection of M in T M , ( x, y ∈ U x ) φ x ( y ), satisfying φ x (0) = x andd y φ x (0) = id ∀ x ∈ M . As an example, one may take the exponentialmap of a connection.If f is a smooth function on M , then the function φ ∗ f ∈ C ∞ ( U )satisfies d( φ ∗ f ) = d f ◦ d φ . Denoting by d x (d y ) the horizontal (ver-tical) part of the differential, we then get d x ( φ ∗ f ) = d f ◦ d x φ andd y ( φ ∗ f ) = d f ◦ d y φ . Because of the assumptions on φ , there is an openneighborhood U ′ ⊂ U of the zero section of M in T M on which d y φ isinvertible. As a consequence, on U ′ we have the formula(2.1) d x ( φ ∗ f ) = d y ( φ ∗ f ) ◦ (d y φ ) − ◦ d x φ. Notice that, for each x , φ ∗ x f is a smooth function on U x . By T φ ∗ x f ∈ b ST ∗ x M we then denote its Taylor expansion in the y ∈ U x -variablesaround y = 0. In doing this, we associate to f ∈ C ∞ ( M ) a section T φ ∗ f of b ST ∗ M over M . We may now reinterpret (2.1) as a conditionon the section T φ ∗ f simply taking Taylor expansions w.r.t. y on bothsides. Notice that in the definition of T φ ∗ f and in the resulting con-dition only the Taylor coefficients of φ appears. We are thus let toconsidering two generalized exponential maps as equivalent if all theirpartial derivatives in the vertical directions, for each point of the base M , coincide at the zero section. We call formal exponential map anequivalence class of generalized exponential maps.If φ is a formal exponential map, then T φ ∗ f ∈ Γ( b ST ∗ M ) is con-structed as above just by picking any generalized exponential map inthe given equivalence class. Choosing local coordinates { x i } on thebase and { y i } on the fiber, we have explicit expressions(2.2) φ ix ( y ) = x i + y i + 12 φ ix,jk y j y k + 13! φ ix,jkl y j y k y l + · · · , and the class of φ is simply given by the collection of coefficients φ x, • .One can easily see that the coefficients φ ix,jk of the quadratic termtransform as the components of a connection. We will refer to this as Here b S denotes the formal completion of the symmetric algebra. HE POISSON SIGMA MODEL ON CLOSED SURFACES 5 the connection in φ . Also explicitly we may compute(2.3) T φ ∗ x f = f ( x ) + y i ∂ i f ( x ) + 12 y j y k ( ∂ j ∂ k f ( x ) + φ ix,jk ∂ i f ( x )) + · · · . Above we have proved that sections of b ST ∗ M of the form T φ ∗ f satisfy (the Taylor expansion of) equation (2.1). One can easily provethat the converse is also true. In fact, one has even more. We maythink of the Taylor expansion of the r.h.s. as an operator acting on thesection T φ ∗ f . Actually, for every section σ of b ST ∗ M one can definea section R ( σ ) of T ∗ M ⊗ b ST ∗ M by taking the Taylor expansion of − d y σ ◦ (d y φ ) − ◦ d x φ . Notice that R is C ∞ ( M )-linear. As a consequencewe have a connection ( X, σ ) ∈ Γ( T M ) ⊗ Γ( b ST ∗ M ) i X R ( σ ) ∈ Γ( b ST ∗ M )on b ST ∗ M . One can check that this connection is flat. We can alsoregard R as a one-form on M taking values in the bundle End( b ST ∗ M ).Also notice that b ST ∗ M is a bundle of algebras and that R acts asa derivation; so we can regard R as a one-form on M taking valuesin the bundle Der( b ST ∗ M ), which is tantamount to saying the bundleof formal vertical vector fields b X ( T M ) :=
T M ⊗ b ST ∗ M . Notice thatthe flatness of the connection may be expressed as the MC (Maurer–Cartan) equation(2.4) d x R + 12 [ R, R ] = 0 , where [ , ] is the Lie bracket of vector fields. Finally, equation (2.1)may now be expressed by saying that d σ + R ( σ ) = 0 if σ is of the form T φ ∗ f for some f . Below we will see that also the converse is true.We first extend this connection to a differential D on the complexof b ST ∗ M -valued differential forms Γ(Λ • T ∗ M ⊗ b ST ∗ M ). The mainresult is that the cohomology of D is concentrated in degree zero and H D = T φ ∗ C ∞ ( M ). This can be easily seen working in local coordinatesagain: R ( σ ) i = ∂σ∂y k (cid:18) ∂φ∂y (cid:19) − ! kj ∂φ j ∂x i . This is the Grothendieck connection in the presentation given by the choice ofthe formal exponential map φ . Since Γ(Λ • T ∗ M ⊗ b ST ∗ M ) is the algebra of functions on the formal gradedmanifold M := T [1] M ⊕ T [0] M , the differential D gives M the structure of adifferential graded manifold. In particular since D vanishes on the body, we maylinearize at each x ∈ M and get an L ∞ -algebra structure on T x M [1] ⊕ T x M ⊕ T x M . F. BONECHI, A. S. CATTANEO, AND P. MNEV
Using (2.2) we get R = δ + R ′ with δ = − d x i ∂∂y i and R ′ a one-form onthe base taking value in the vector fields vanishing at y = 0. Hence wehave D = δ + D ′ with D ′ = d x i ∂∂x i + R ′ . Notice that δ is itself a differential and that it decreases the polynomialdegree in y , whereas the operator D ′ does not decrease this degree. Thefundamental remark is that the cohomology of δ consists of zero formsconstant in y . This is easily shown by introducing δ ∗ := y i ι ∂∂xi andobserving that ( δδ ∗ + δ ∗ δ ) σ = kσ if σ is an r -form of degree s in y and r + s = k . By cohomological perturbation theory the cohomology of D is isomorphic to the cohomology of δ , which is what we wanted toprove.Finally, observe that, if σ is a D -closed section, we can immediatelyrecover the function f for which σ = T φ ∗ f simply by setting y = 0, f ( x ) = σ x (0), as follows from (2.3).We can now extend the whole story to other natural objects. Let V ( M ) denote the multivector fields on M (i.e., sections of Λ T M ), Ω( M )the differential forms, W j ( M ) := Γ( S j T M ) and O j ( M ) := Γ( S j T ∗ M ).We use similar symbols for formal vertical vector fields b V ( T M ) :=Γ(Λ
T M ⊗ b ST ∗ M ) and formal vertical differential forms b Ω( T M ) :=Γ(Λ T ∗ M ⊗ b ST ∗ M ). We have injective maps T φ ∗ := T ( φ ∗ ) − : V ( M ) → b V ( T M ) , T φ ∗ : Ω( M ) → b Ω( T M ) . Similarly, we set c W j ( T M ) := Γ( S j T M ⊗ b ST ∗ M ) and b O j ( T M ) :=Γ( S j T ∗ M ⊗ b ST ∗ M ) and get T φ ∗ := T ( φ ∗ ) − : W j ( M ) → c W ( T M ) , T φ ∗ : O j ( M ) → b O j ( T M ) . We can now let R naturally act on b V ( T M ), b Ω( T M ), c W j ( T M ) and b O j ( T M ) by Lie derivative and hence get a differential D on the cor-responding complexes of differential forms. Notice that D respectsthe Gerstenhaber algebra structure (by the vertical Schouten–Nijenhuisbracket) of b V ( T M ) and the differential complex structure (by the ver-tical differential) of b Ω( T M ), so that these structures are induced incohomology. By the same argument as above, we get that all thesecohomologies are concentrated in degree zero with H D ( b V ( T M )) =
T φ ∗ V ( M ), H D ( b Ω( T M )) =
T φ ∗ Ω( M ), H D ( c W j ( T M )) =
T φ ∗ W j ( M ),and H D ( b O j ( T M ) =
T φ ∗ O j ( M ). Notice in particular that a section is HE POISSON SIGMA MODEL ON CLOSED SURFACES 7 in the image of
T φ ∗ if and only ifd x σ + L R σ = 0 . In order to recover, in local coordinates, the global object correspond-ing to a solution to the above equation, we should only observe thatby assumption d y φ x (0) = id, so that it is enough to evaluate the com-ponents of σ at y = 0 and to replace formally each d y i by d x i and each ∂∂y i by ∂∂x i . More explicitly, if σ x ( y ) = σ x ; i ,...,i n ( y ) d y i · · · d y i n is equalto T φ ∗ ω , then ω ( x ) = σ x ; i ,...,i n (0) d x i · · · d x i n . If on the other hand, σ x ( y ) = σ i ,...,i n x ( y ) ∂∂y i · · · ∂∂y in is equal to T φ ∗ ( Y ),then Y ( x ) = σ i ,...,i n x (0) ∂∂x i · · · ∂∂x i n . One can immediately extend these results to direct sums of thevector bundles above. Notice that cohomology also commutes withdirect limits. This implies that the cohomology of Q j c W j ( T M ) isalso concentrated in degree zero and coincides with
T φ ∗ Q j W j ( M ).Now we have that Q j W j ( M ) = Γ( b ST M ) whereas Q j c W j ( T M ) =Γ( b S ( T M ⊗ T ∗ M )). Similarly, we see that the cohomology with valuesin b S ( T ∗ M ⊗ T ∗ M ) is concentrated in degree zero and coincides with T φ ∗ Γ( b ST ∗ M ). To summarize: H • D (Γ( b S ( T M ⊗ T ∗ M ))) = H D (Γ( b S ( T M ⊗ T ∗ M ))) = T φ ∗ Γ( b ST M )and H • D (Γ( b S ( T ∗ M ⊗ T ∗ M ))) = H D (Γ( b S ( T ∗ M ⊗ T ∗ M ))) = T φ ∗ Γ( b ST ∗ M ) . Gauge transformations.
We now wish to consider the effects ofchanging the choice of formal exponential map. Namely, let φ be a fam-ily of formal exponential maps depending on a parameter t belonging toan open interval I . We may associate to this family a formal exponen-tial map ψ for the manifold M × I by ψ ( x, t, y, τ ) := (( φ ) x,t ( y ) , t + τ ),where τ denotes the tangent variable to t . We want to define the as-sociated connection e R : on a section e σ of b ST ∗ ( M × I ) we have, bydefinition, e R ( e σ ) = − (d y e σ, d τ e σ ) ◦ (cid:18) (d y φ ) −
00 1 (cid:19) ◦ (cid:18) d x φ ˙ φ (cid:19) . So we can write e R = R + C d t + T with R defined as before (but now t -dependent), C ( e σ ) = − d y e σ ◦ (d y φ ) − ◦ ˙ φ F. BONECHI, A. S. CATTANEO, AND P. MNEV and T = − d t ∂∂τ . We now spell out the MC equation for e R observingthat d x T = d t T = 0 and that T commutes with both R and C . The(2 , M × I yields again the MC equation for R ,whereas the the (1 , R = d x C + [ R, C ] . Hence, under a change of formal exponential map, R changes by agauge transformation with generator the section C of b X ( T M ).Finally, if σ is a section in the image of T φ ∗ , then by a simple com-putation one gets ˙ σ = − L C σ, which can be interpreted as the associated gauge transformation forsections. 3. PSM in formal coordinates
The Poisson sigma model (PSM) [17, 20] is a topological field theorywith source a two-manifold Σ and target a Poisson manifold M . Beforegetting to the BV action for the PSM, we fix the notations and intro-duce the AKSZ formalism [1] (for a gentle introduction to it, especiallysuited to the PSM, see [10]). Let Map( T [1]Σ , T ∗ [1] M ) be the infinitedimensional graded manifold of maps from T [1]Σ to T ∗ [1] M . It fibersover Map( T [1]Σ , M ). We denote by X a “point” of Map( T [1]Σ , M ) andby η a “point” of the fiber. In local target coordinates, the super fields X and η have simple expressions: X i = X i + η i + + β i + , η i = β i + η i + X + i , where we have ordered the terms in increasing order of form degree onΣ. The ghost number is 0 for X and η , 1 for β , − η + and X + ,and − β + . As unperturbed BV action one considers S := Z Σ η i d X i . Notice that it satisfies the classical master equation (CME) ( S , S ) = 0if Σ has no boundary or if appropriate boundary conditions are taken(which we assume throughout). Here ( , ) is the BV bracket corre-sponding to the odd symplectic structure on the space of fields forwhich the superfield η is the momentum conjugate to the superfield X .Formally one may also assume ∆ S = 0 where ∆ is the BV operator,so S satisfies also the quantum master equation (QME). HE POISSON SIGMA MODEL ON CLOSED SURFACES 9
To perturb this action, we pick a multivector field Y on M . We mayregard it as a function on T ∗ [1] M . We then define S Y as the integralover T [1]Σ of the pullback of Y by the evaluation mapev : T [1]Σ × Map( T [1]Σ , T ∗ [1] M ) → T ∗ [1] M. Explicitly, for a k -vector field Y , we have S Y = 1 k ! Z Σ Y i ,...,i k ( X ) η i . . . η i k . This construction has several interesting properties. First, ( S , S Y ) = 0(for ∂ Σ = ∅ or with appropriate boundary conditions). Second, for anytwo multivector fields Y and Y ′ , we have ( S Y , S Y ′ ) = S [ Y,Y ′ ] . The BVaction for the PSM with target Poisson structure π is recovered as S = S + S π . Notice that by the above mentioned properties it satisfiesthe CME. As for the quantum master equation, we refer to [12], whereit is shown that one can assume ∆ S π = 0 if the Euler characteristic ofΣ is zero or if π is unimodular. In the latter case, one picks a volumeform v on M such that div v π = 0 and defines ∆ according to it.We consider S π as a perturbation, so we expand the functional inte-gral around the critical points of S . They consist of closed superfields.In particular, the component X of X will be a constant map, say withimage x ∈ M . Fluctuations will explore only a formal neighborhood of x in M , so as in [11, § X = φ x ( A ) , η = d φ x ( A ) ∗ , − B , where φ is a formal exponential map and the new superfields ( A , B ) arein Map( T [1]Σ , T ∗ [1] T x M ). Notice that this change of variables φ x : Map( T [1]Σ , T ∗ [1] T x M ) → Map( T [1]Σ , T ∗ [1] M )is a local symplectomorphism and that T φ ∗ x S = Z Σ B i d A i . The moduli space of vacua (i.e., the space of critical points modulogauge transformations) is now H x := H • (Σ) ⊗ T x M ⊕ H • (Σ) ⊗ T ∗ x M [1].(Here H • (Σ) is regarded as a graded vector space with its naturalgrading.) We should regard H = S x H x as a vector bundle over M ,but for the moment we concentrate on a single x . Later on we will alsoconsider the remaining integration over vacua and, in particular, over M (which actually shows up as the space of constant maps Σ → M );see Section 5. We may repeat the AKSZ construction on Map( T [1]Σ , T ∗ [1] T x M ).In particular, if Y is a function of degree k on T ∗ [1] T x M (i.e., a formalvertical k -vector field), we may construct a functional S Y = 1 k ! Z Σ Y i ,...,i k ( A ) B i . . . B i k . In particular, we have
T φ ∗ x S π = S T φ ∗ x π . As a result, we have a solution S x := T φ ∗ x S of the QME and maycompute its partition function Z x (as a function on H x ) upon inte-grating over a Lagrangian submanifold L of a complement of H x inMap( T [1]Σ , T ∗ [1] T x M ): Z x := Z L e i ~ S x . Notice [19, 12] that there is an induced BV operator ∆ on H x and that Z x satisfies ∆ Z x = 0. Moreover, upon changing the gauge fixing L , Z x changes by a ∆-exact term. We wish to compare the class of Z x withthe globally defined partition function morally obtained by integratingin Map( T [1]Σ , T ∗ [1] M ). For this we have to understand the collection { Z x } x ∈ M as a section b Z : x Z x of b S H ∗ (we hide the dependency on ~ here) and compute how it changes over M . Using all properties aboveand setting b S : x S x , we get d x b Z = i ~ Z L e i ~ b S S d x T φ ∗ π = − i ~ Z L e i ~ b S ( S R , b S ) . Notice that this may also be rewritten asd x b Z = − ∆ Z L e i ~ b S S R if we assume ∆ S R = 0. This is correct if Σ has zero Euler characteristicor if div T φ ∗ v R = 0. From the equation d x T φ ∗ v + L R T φ ∗ v = 0, we seethat the latter condition is satisfied if and only if d x T φ ∗ v = 0. Given avolume form v , it is always possible to find a formal exponential map φ satisfying this condition; actually, one can even get T φ ∗ x v = d y . . . d y d ∀ x .We can collect the above identities nicely if we define e S := b S + S R . The choice of L might be different for different x s, but for simplicity we assumeit not to be the case. For π = 0, e S is also the BV action for the BF ∞ -theory [19] with target the L ∞ algebra of footnote 6. See also [14]. HE POISSON SIGMA MODEL ON CLOSED SURFACES 11
Notice that e S is of total degree zero (the term S R has ghost numberminus one but is a one-form on M ) and satisfies the modified CMEd x e S + 12 ( e S, e S ) = 0and by assumption also ∆ e S = 0 (so it satisfies a modified QME aswell). We then define(3.1) e Z := Z L e i ~ e S as a nonhomogeneous differential form on M taking values in H . Itsatisfies d x e Z − i ~ ∆ e Z = 0 . Remark . We are now also in a position to understand the changeof e Z under a change of the formal exponential map. Using the resultsof subsection 2.1, we immediately see that˙ e Z = (d x − i ~ ∆) Z L e i ~ e S i ~ S C , assuming ∆ S C = 0. The assumption is verified if Σ has zero Eulercharacteristic or if we let φ vary only in the class of formal exponentialmaps that make T φ ∗ v constant. Notice that the space of such formalexponential maps is connected. Therefore, the class of e Z under thesetransformations is independent of all choices needed to compute it.Finally, we consider the effective action e S eff defined by the identity e Z = e i ~ e S eff . It is a differential form taking values in b S H ∗ [[ ~ ]] and satisfiesthe modified QME(3.2) d x e S eff + 12 ( e S eff , e S eff ) − i ~ ∆ e S eff = 0 . This equation formally follows from the properties of BV integrals.In the case when Σ has no boundary, it may be proved directly byconsidering the expansion of e S eff in Feynman diagrams and applying theusual Stokes theorem techniques on integrals over configuration spaces,see [18, 13]. If Σ has a boundary, additional terms corresponding toseveral points collapsing to the boundary together may appear andspoil (3.2). From now on we therefore assume that Σ is closed.Equation (3.2) contains both information on the QME satisfied bythe zero-form component e S (0)eff and on its global properties. The equa-tions are in general mixed: we do not simply get a flat connection withrespect to which e S (0)eff is covariantly constant. However, it is possible tofind a modified quantum BV canonical transformation that produces a flat connection with respect to which the zero form part of the effec-tive action is horizontal and hence global; we postpone this discussionto Section 6. In the remaining of this Section, we concentrate on twospecial cases where the general theory is not needed.The first special case is when Σ is a torus. In Section 4, see Lemma 4.4,we will show that, in an appropriate gauge, there are no quantum cor-rections, so e S eff = e S (0)eff + e S (1)eff , where e S (0)eff is the zero-form obtained by restricting S T φ ∗ π to vacua and e S (1)eff is the one-form obtained by restricting S R to vacua. One canexplicitly check, see Section 5, that ∆ e S (0)eff = ∆ e S (1)eff = 0. Hence themodified QME now simply yields the CME for e S (0)eff ,( e S (0)eff , e S (0)eff ) = 0 , the flatness condition for e S (1)eff ,d x e S (1)eff + 12 ( e S (1)eff , e S (1)eff ) = 0 , and the fact that e S (0)eff is covariantly constant,d x e S (0)eff + ( e S (1)eff , e S (0)eff ) = 0 . Now notice that ( e S (1)eff , ) is just the natural action of R on the sectionsof H . Hence we can conclude that e S (0)eff is just T φ ∗ ( S | vacua ).The second special case is when π is regular and unimodular (andΣ is any two-manifold). Also in Section 4 we will show that, uponchoosing an appropriate formal exponential map, S (0)eff and S (1)eff haveno quantum corrections. Therefore, we may use the same reasoning asabove and conclude that e S (0)eff is just T φ ∗ ( S | vacua ).A final important remark is that in the two cases above the effectiveaction depends polynomially on all vacua but, possibly, for those re-lated to the X field; therefore, S (0)eff is a section of S e H ∗ ⊗ b ST ∗ M , where e H x = H > (Σ) ⊗ T x M ⊕ H • (Σ) ⊗ T ∗ x M [1] is the moduli space of vacuaexcluding those for X . The corresponding global effective action S | vacua will then be a section of S e H ∗ , i.e., a function on the vector bundle e H (polynomial in the fibers). Notice that this vector bundle is diffeomor-phic, by choosing a connection (e.g., the one contained in the choiceof φ ), to the natural global definition of the moduli space of vacua aspresented, e.g., in [5]. HE POISSON SIGMA MODEL ON CLOSED SURFACES 13 Some computations of the effective action
In this Section we discuss the perturbative computation for the ef-fective action and show that it has no quantum corrections in twoimportant cases.4.1.
Factorization of Feynman graphs.
Consider the effective ac-tion e S eff defined in (3.1) by the identity e Z = e i ~ e S eff . Here the Lagrangiansubspace L in the complement of H x inside the space of fields(4.1) F x = Map( T [1]Σ , T ∗ [1] T x M ) ∼ = Ω • (Σ) ⊗ ( T x M ⊕ T ∗ x [1] M )accounts for the gauge fixing. Let L have the factorized form(4.2) L = L K ⊗ ( T x M ⊕ T ∗ x [1] M )where L K ⊂ Ω • (Σ) is defined as L K = ker P ∩ ker K with P the projector from differential forms on Σ to (the chosen rep-resentatives of) de Rham cohomology of Σ and K : Ω • (Σ) → Ω •− (Σ)a linear operator satisfying(4.3)d K + K d = id − P , P K = K P = 0 , K T = K, K = 0(i.e., K is the chain homotopy between identity and the projection tocohomology, also known as a parametrix). The transpose is w.r.t. thePoincar´e pairing on forms R Σ • ∧ • . We assume the operator K (whichnow determines the gauge fixing) to be an integral operator with adistributional integral kernel ω ∈ Ω (Σ × Σ) – the propagator. Anexplicit construction may be done along the same lines as in [8, 9]. Letus introduce a basis { χ α } in the cohomology space H • (Σ); denote thematrix of the Poincar´e pairing by Π αβ = R Σ χ α ∧ χ β . In terms of ω ,properties (4.3) read:(1) d ω = δ diag − P α,β (Π − ) αβ χ α ⊗ χ β , where δ diag is the delta-formsupported on the diagonal of Σ × Σ;(2) R Σ (1) ω π ∗ χ α = R Σ (2) ω π ∗ χ α = 0 , ∀ α , where Σ (1) and Σ (2) denotethe two factors of Σ × Σ, and π and π are the two projectionsfrom Σ × Σ to its factors;(3) t ∗ ω = ω , where t : Σ × Σ → Σ × Σ is the map swapping the twocopies of Σ;(4) R Σ (2) π ∗ ω π ∗ ω = 0, where Σ (2) denotes the middle factor inΣ × Σ × Σ, whereas π and π are the projections from Σ × Σ × Σto the first two and last two factors, respectively.
Notice that the restriction of ω to the configuration space C (Σ) := { ( u, v ) ∈ Σ : u = v } is smooth and it extends to the Fulton–MacPherson–Axelrod–Singer compactification as a smooth form. In [13] it is shownhow to implement the property P K = K P = 0 on the propagator.Once this is done, the propagator will also satisfy the property K = 0.This is proved exactly as in [12, Lemma 10].The perturbation expansion for the effective action (3.1) has the form(4.4) S eff ( A z . m . , B z . m . ; ~ ) = X Γ ( i ~ ) l (Γ) | Aut(Γ) | W targetΓ ( A z . m . , B z . m . ) · W sourceΓ where the sum is over connected oriented graphs Γ with leaves deco-rated by basis cohomology classes { χ α } ; l (Γ) stands for the number ofloops, | Aut(Γ) | is the number of graph automorphisms; { A z . m . , B z . m . } = { A αi , B αi } are the coordinates on the moduli space of vacua H x .The “target part” W targetΓ of the contribution of a graph to e S eff is ahomogeneous polynomial function on H x of degree equal to the numberof leaves, computed using the following set of rules:(1) to an incoming leaf of Γ decorated by χ α one associates A αi (2) to an outgoing leaf decorated by χ α one associates B αi (3) to a vertex with m inputs and n outputs one associates theexpression ∂ i · · · ∂ i m Y j ··· j n – the m -th derivative of n -vector contribution to the action S .(4) for every edge contract the dummy Latin indices for the twoconstituent half-edges.The result of contraction is a polynomial function on H x .The “source” (or “de Rham”) part W sourceΓ is a number defined as(4.5) W sourceΓ = Z Σ × V (Γ) Y edges ( h in ,h out ) π ∗ v ( h in ) ,v ( h out ) ω · Y leaves l π ∗ v ( l ) χ α l ! A leaf for us is a loose half-edge, i.e., one not connected to another half-edge toform an edge. In the standard setup for the Poisson sigma model, only the perturbation byPoisson bivector field Y = π ij ∂ i ∧ ∂ j is present in the action, hence all verticeshave to have exactly two outputs, otherwise the graph does not contribute. In thepresent case, we also have the vector field R . HE POISSON SIGMA MODEL ON CLOSED SURFACES 15 where V (Γ) is the number of vertices, π v : Σ × V (Γ) → Σ is the projectionto v -th copy of Σ, π u,v : Σ × V (Γ) → Σ × Σ is the projection to u -th and v -th copies of Σ; v ( h ) is the vertex incident to the half-edge h . Notice thatall these integrals converge. The usual way to show this is to observethat the integrals are actually defined on configuration spaces (i.e., thecomplements of all diagonals in the Cartesian products of copies of Σ)and that the propagators ω extend to their compactifications. Remark . The factorization into source- and target contributions forFeynman diagrams in the expansion (4.4) is due to the factorization ofthe space of fields (4.1) and to the fact that our ansatz for the gaugefixing (4.2) is compatible with this factorization.
Remark . The orientation of Γ is irrelevant for the source parts W sourceΓ .4.2. Regular Poisson structures. If π is nondegenerate, it is alwayspossible to find a formal exponential map φ such that T φ ∗ π is constant(in the y variables). One simply has to go to formal Darboux coordi-nates. Notice, moreover, that div v π = 0 if for v one chooses v to bethe Liouville volume form ω k /k !, k = dim M/
2. It then follows that
T φ ∗ v is also constant and that div T φ ∗ v R = 0. A slight generalizationoccurs when π is regular (i.e., its kernel has constant rank) and uni-modular (notice that this is not guaranteed if π is degenerate [21]).After choosing v such that div v π = 0, it is again possible to find aformal exponential map φ such that T φ ∗ π and T φ ∗ v are both constantand hence div T φ ∗ v R = 0.In the perturbative expansion, we may thus assume that we havea bivalent vertex, corresponding to T φ ∗ π , with no incoming arrows.If one of the outgoing arrows is replaced by a vacuum mode (i.e., acohomology class), the result is zero by the property P K = K P = 0,otherwise it is zero by the property K = 0. As a result, every graphcontaining a T φ ∗ π -vertex will vanish, apart from the one with bothoutgoing arrows evaluated on vacua. As a consequence S (0)eff and S (1)eff have no quantum corrections.4.3. Axial gauge on the torus
Σ = T := S × S . In the case of atorus, differential forms have a bigrading with respect to the two circles.One may choose the axial gauge by setting the superfields to vanishif they have nonzero degree with respect to the first circle. Na¨ıvely thisimplies the propagator to be the product of a propagator for the de The axial gauge for topological field theories was originally proposed in thecontext of Chern–Simons theory in [15].
Rham differential on the first circle and the identity operator on thesecond circle (just plug in the gauge fixed fields into in the unperturbedaction to realize this). This argument however does not take vacua intoaccount nor the fact that the axial gauge fixing does not fix all the gaugefreedom. In fact, one can prove that the propagator in the axial gaugehas one additional term, see (4.7) below.To start with a rigorous construction of the propagator, observe thatdifferential forms on a circle admit the Hodge decompositionΩ • ( S ) = Ω • Harm ( S ) | {z } Span(1 ,dτ ) ⊕ ˜Ω ( S ) | {z } { f ( τ ) | R S f ( τ ) dτ =0 } ⊕ ˜Ω ( S ) | {z } { g ( τ ) dτ | R S g ( τ ) dτ =0 } (In our convention the coordinate τ on the circle runs from 0 to 1).The associated chain homotopy operator is K S : g ( τ ) dτ Z S ω S ( τ, τ ′ ) g ( τ ′ ) dτ ′ with the integral kernel ω S ( τ, τ ′ ) = θ ( τ − τ ′ ) − τ + τ ′ − P S : f ( τ ) + g ( τ ) dτ Z S ( dτ ′ − dτ ) ∧ ( f ( τ ′ ) + g ( τ ′ ) dτ ′ )For the torus we may decompose the de Rham complex in the fol-lowing way:(4.6) Ω • ( S × S ) = Ω • ( S ) ˆ ⊗ Ω • ( S ) == Ω • Harm ( S ) ⊗ Ω • Harm ( S ) | {z } ∼ = H • ( S × S ) ⊕ ˜Ω ( S ) ˆ ⊗ Ω • ⊕ Ω • Harm ( S ) ⊗ ˜Ω ( S ) | {z } L K ⊕⊕ ˜Ω ( S ) ˆ ⊗ Ω • ⊕ Ω • Harm ( S ) ⊗ ˜Ω ( S )The associated chain homotopy operator is(4.7) K = K S ⊗ id S | {z } K I + P S ⊗ K S | {z } K II : Ω • ( S × S ) → Ω •− ( S × S ) HE POISSON SIGMA MODEL ON CLOSED SURFACES 17
Its integral kernel (the propagator) is(4.8) ω = (cid:18) θ ( σ − σ ′ ) − σ + σ ′ − (cid:19) · δ ( τ − τ ′ ) · ( dτ ′ − dτ ) | {z } ω I ++ ( dσ ′ − dσ ) · (cid:18) θ ( τ − τ ′ ) − τ + τ ′ − (cid:19)| {z } ω II where we denote by σ, τ ∈ R / Z the coordinates on the first and thesecond circles, respectively. Remark . The chain homotopy (4.7) arises from the composition oftwo quasi-isomorphisms: Ω • ( S ) ˆ ⊗ Ω • ( S ) Ω • Harm ( S ) ⊗ Ω • ( S ) ⊕ ˜Ω ( S ) ˆ ⊗ Ω • ( S ) ⊕ ˜Ω ( S ) ˆ ⊗ Ω • ( S ) y Ω • Harm ( S ) ⊗ Ω • ( S ) Ω • Harm ⊗ Ω • Harm ⊕ Ω • Harm ( S ) ⊗ ˜Ω ( S ) ⊕ Ω • Harm ( S ) ⊗ ˜Ω ( S ) y Ω • Harm ( S ) ⊗ Ω • Harm ( S ) i.e., we first contract the first circle to cohomology, then the secondone.4.4. Vanishing of quantum corrections.Lemma 4.4.
For the Poisson sigma model in the axial gauge on torus,the source parts W sourceΓ vanish for all connected graphs Γ except fortrees with one vertex (“corollas”). Proof.
Let us introduce the basis in cohomology of the torus: χ (0 , = 1 , χ , = dσ, χ (0 , = dτ, χ (1 , = dσ ∧ dτ Define a decoration c of Γ as an assignment of bidegree c ( h ) ∈ { (0 , , (1 , , (0 , , (1 , } to each half-edge h of Γ (so that on leaves the bidegree coincides withthe prescribed leaf decoration α ) together with an assignment of anindex c ( e ) ∈ { I, II } to each edge e . Define the source part for adecorated graph Γ as(4.9) W sourceΓ ,c = Z Σ × V (Γ) Y edges e =( h in ,h out ) π ∗ v ( h in ) ,v ( h out ) ω c ( e ) | c · Y leaves l π ∗ v ( l ) χ α l ! where the ω | c symbol means the component of the propagator (as an el-ement of Ω • ( S × S ) ˆ ⊗ Ω • ( S × S )) of de Rham bidegrees c ( h in ) , c ( h out )where h in , h out are the constituent half-edges of the edge; ω c ( e ) is one ofthe two pieces of propagator, ω I or ω II , as defined in (4.8). Then wehave W sourceΓ = X decorations c W sourceΓ ,c The source part W sourceΓ ,c vanishes automatically unless the followingconditions are satisfied simultaneously:(i) At every vertex there is exactly one incident half-edge decoratedby (1 , • ), all others are (0 , • ).(ii) At every vertex there is exactly one incident half-edge decoratedby ( • , • , e = ( h , h ) we have( c ( e ) = I ) = ⇒ (cid:20) c ( h ) = (0 , , c ( h ) = (0 ,
1) or c ( h ) = (0 , , c ( h ) = (0 , c ( e ) = II ) = ⇒ (cid:20) c ( h ) = (0 , , c ( h ) = (1 ,
0) or c ( h ) = (1 , , c ( h ) = (0 , I adjacent to any given vertexshould be different from one.(v) If a vertex has no adjacent I -edges, then the number of adjacent II -edges should be different from one.Requirements (i,ii) follow directly from degree counting in (4.9); (iii)follows from the formula for propagator (4.8); (iv,v) follow from theproperty K S P S = 0 and from the fact that harmonic forms on acircle are closed under wedge multiplication.Fix some decoration c of Γ satisfying (i–v). Consider the subgraphΓ I of Γ obtained by deleting all II -edges in Γ; Γ I may be disconnected.Let Γ I = ⊔ a Γ aI where Γ aI are the connected components of Γ I . Due to(ii), the number of vertices V aI of Γ aI is equal to the number of (0 , aI which is in turn greater or equal to the number of edges E aI due to (iii). Hence the Euler characteristic of Γ aI non-negative: V a Γ − E a Γ ≥
0. Therefore Γ aI is either a tree or a 1-loop graph. Next,property (iv) shows that Γ aI has to be a wheel graph, with arbitrarynumber of leaves attached at vertices, or a corolla. On the other hand, HE POISSON SIGMA MODEL ON CLOSED SURFACES 19 if Γ I contains a wheel then the corresponding source part vanishes:(4.10) W sourceΓ ,c = Z ( S × S ) × V ( dτ − dτ ) δ ( τ − τ ) ∧· · ·∧ ( dτ n − dτ ) δ ( τ − τ n ) ∧ F = Z ( S × S ) × V ( dτ ∧ dτ ∧ · · · ∧ dτ n + ( − n dτ ∧ · · · dτ n ∧ dτ ) | {z } =0 ∧∧ δ ( τ − τ ) · · · δ ( τ − τ n ) ∧ F = 0where n is the length of the wheel and F ∈ Ω • (( S × S ) × V ) is somedifferential form. Remark . Argument (4.10) has the fault that the integrand is singu-lar and the result is 0 · δ (0). This can be remedied by regularizing thepropagator ω , e.g., by changing δ ( τ − τ ′ ) in (4.8) to a smeared delta-function. Notice that the source parts of all diagrams except corollasstill vanish exactly: in this vanishing argument the chain homotopyequation is never used; we only use the de Rham bigrading properties, P K = 0 and the fact that harmonic forms on a circle are closed undermultiplication.Thus we have shown that W sourceΓ ,c vanishes unless Γ I is a collectionof corollas (i.e. there are no I -edges).Now fix a decoration c satisfying (i–v) with c ( e ) = II for all edges.Repeating the Euler characteristic argument as above (using properties(i,iii)), we show that Γ has to be either a tree or a 1-loop graph andusing property (v) we show that it has to be either a wheel or a corolla.If it is a wheel then(4.11) W sourceΓ ,c = Z ( S × S ) × V ( dσ − dσ ) ∧ · · · ∧ ( dσ V − dσ ) ∧ F == Z ( S × S ) × V ( dσ ∧ · · · ∧ dσ V + ( − V dσ ∧ · · · ∧ dσ V ∧ dσ ) | {z } =0 ∧ F == 0Therefore W sourceΓ ,c vanishes for any decoration c unless Γ is a corolla.This concludes the proof of the Lemma. (cid:3) An immediate consequence of the Lemma is that the effective ac-tion S eff x is just the restriction of the action S x to vacua: there are noquantum corrections. The partition function on the torus
Let S eff be the global effective action on the moduli space of vacua forthe torus T . In Lemma 4.4, we have shown that it has no quantum cor-rections. The moduli space of vacua can be viewed as Map( R [1] , T ∗ [1] M )and in [5] it has been remarked that the action restricted to vacua isthe AKSZ action for this mapping space. In local coordinates the su-perfields are(5.1) x µ = x µ + e η + µ + e η + µ − sb + µ , e ν = b ν + e η ν + e η ν + sx + ν , where s = e e is the generator of H dR ( T ) normalized to R R [1] ds s = 1.If π is the Poisson bivector field on M , then(5.2) S eff = 12 Z R [1] ds π µν ( x ) e µ e ν . There exists a canonical Berezinian given by the coordinate volumeform(5.3) ν = dx · · · dx + · · · db · · · db + · · · dη i · · · dη + i · · · . If we denote with ∆ the corresponding Laplacian, the AKSZ actionsatisfies ∆e i ~ S eff = 0and defines a class in ∆-cohomology.5.1. K¨ahler gauge fixing and Euler class.
Now let π be symplecticsuch that π − is the K¨ahler form of the hermitian structure ( J, g ). Inthe complex coordinates { x i } of M we have π i ¯ = ig i ¯ . Let us fix acomplex structure on T defined by z = θ + τ θ , for τ = τ + iτ and τ >
0. Let η + µz = ( η + µ − ¯ τ η + µ ) / iτ , η zµ = ( η µ − ¯ τ η µ ) / iτ . Let L ε,τ be the following Lagrangian submanifold of Map( R [1] , T ∗ [1] M ):(5.4) η + iz = η +¯ ı ¯ z = η zi = η ¯ z ¯ ı = x + = b + = 0 . Let us define p k = η ¯ zk + Γ jki η + i ¯ z b j , where Γ are the Christoffel symbols of the Levi-Civita connection. Allfiber coordinates b, p, η + transform tensorially with respect to a trans-formation of coordinates on M so that L ε,τ = ( T ∗ [1]+ T ∗ M + T [ − M .After a straightforward computation we get S eff = τ (cid:16) R js ¯ lk g s ¯ r η + k ¯ z η +¯ lz b j b ¯ r + g i ¯ j p i p ¯ j (cid:17) HE POISSON SIGMA MODEL ON CLOSED SURFACES 21 = 12 τ (cid:0) R µλ b µ b λ + g µν p µ p ν (cid:1) . The induced Berezinian on L ε,τ reads √ ν = dx µ · · · db µ · · · dp µ · · · dη + µ · · · (2 π ) m , with m = dim M . If we perform the fiberwise integration with respectto the fibration L ε,τ → T [ − M , we get1(2 π ) m Z dp µ · · · db µ · · · e i ~ S eff == 1(2 π ) m ~ ) m Z db µ · · · τ g µν ) / e i2 ~ τ R µν b µ b ν == 1(2 π ) m Z db ′ µ · · · g µν ) / e − R µν b ′ µ b ′ ν == 1(2 π ) m √ g Pf( R ) ∈ C ∞ ( T [ − M ) = Ω M, which, by the Chern–Gauss-Bonnet theorem, is a representative of theEuler class. Notice that ~ and τ disappear in the final formula. (Themain reason for this is that scaling the b and p variables by the samefactor preserves the Berezinian since the former are odd and the latterare even variables.) Finally, we can integrate over M getting Z = χ ( M ) , the Euler characteristic of M . (Actually, by the argument in the Intro-duction that the partition function should be the Euler characteristic,we might in reverse think of this result as one more physical proof ofthe Chern–Gauss–Bonnet theorem, in the case of K¨ahler manifolds.) Remark . In this Section we have assumed the existence of a K¨ahlerstructure on M . We expect the above results to hold if we just usean almost K¨ahler structure, but computations become much more in-volved. We choose here the standard convention that the measure for a pair of evenconjugate coordinates p, q is dp dq/ (2 π i ~ ), whereas the measure for a pair of oddconjugate coordinates b, η + is i ~ db dη + . This is consistent with the standard nor-malization Z e i ~ pq dp dq π i ~ = Z e i ~ bη + i ~ db dη + = 1 . Remark . Another possible gauge fixing consists in setting all +variables to zero. The effective action then reduces to π µν ( x ) η µ η ν andis independent of the b variables. If M is compact, the integrals over the η and x variables is finite (and proportional to the symplectic volume of M ); because of the b -integration, the partition function then vanishes.If M is not compact, the partition function is ambiguous and of theform 0 · ∞ . This gauge fixing is then in general not equivalent to theK¨ahler one used above. From the considerations in the Introduction,the K¨ahler gauge fixing is the one compatible with the Hamiltonianinterpretation of the theory.5.2. Regularized effective action.
We now show that the symme-tries of S eff induce a regularization which allows one to compute thepartition function for every Poisson structure and to show that, inde-pendently of the Poisson structure, one gets the Euler characteristic ofthe target. The main remark is that the effective action and the symplecticform are invariant under the action of the Lie algebra of divergencelessvector fields of R [1] on the moduli space of vacua. This Lie algebrais spanned by the vector fields ∂∂e , ∂∂e , e ∂∂e , e ∂∂e and e ∂∂e − e ∂∂e .The fifth vector field is generated by the previous ones and we are notgoing to need it in the following. We lift the first four vector fieldsfirst to Map( R [1] , M ) and next to its cotangent bundle shifted byone. We will denote the resulting vector fields by δ , δ , K and K ,respectively. Since they have degree − −
1, they are also automatically Hamiltonian with uniquelydefined Hamiltonian functions τ and τ (of degree − ρ and ρ (of degree − τ , ρ ) = τ , ( τ , ρ ) = τ , ( τ , ρ ) = 0 , ( τ , ρ ) = 0 . Also notice that we have(5.6) K ◦ δ = K ◦ δ = 0 , which implies that ρ i Poisson commutes with every δ i -exact function.Finally, since we started with divergenceless vector fields, we get(5.7) ∆ τ = ∆ τ = ∆ ρ = ∆ ρ = 0 . Remark . Even though we do not need the explicit form of thesevector fields and their Hamiltonian functions, we give them for com-pleteness of our presentation. From the defining formulae δ i x = ∂ x ∂e i , We thank T. Johnson-Freyd for pointing out this approach.
HE POISSON SIGMA MODEL ON CLOSED SURFACES 23 δ i e = ∂ e ∂e i K x = e ∂ x ∂e , K e = e ∂ e ∂e , K x = e ∂ x ∂e and K e = e ∂ e ∂e , weget δ = η + µ ∂∂x µ + b + µ ∂∂η + µ + η µ ∂∂b µ − x + µ ∂∂η µ ,δ = η + µ ∂∂x µ − b + µ ∂∂η + µ + η µ ∂∂b µ + x + µ ∂∂η µ ,K = η µ ∂∂η µ + η + µ ∂∂η + µ ,K = η µ ∂∂η µ + η + µ ∂∂η + µ . The corresponding Hamiltonian functions, with respect to the symplec-tic structure ω = Z d e d e δ x µ δ e µ = δx µ δx + µ + δη + µ δη µ − δη + µ δη µ − δb + µ δb µ , are given by τ = x + µ η + µ − η µ b + µ ,τ = x + µ η + µ − η µ b + µ ,ρ = − η µ η + µ ,ρ = η µ η + µ . We now turn back to the effective action. It turns out that it is notonly δ - and δ -closed, but actually exact: S eff = δ δ σ, σ := 12 π µν ( x ) b µ b ν . From all the above it follows that S eff Poisson commutes not only with τ and τ , but also with ρ and ρ . Notice that the Jacobi identity for π implies ( S eff , σ ) = 0.Now consider the regularized effective action S ǫ,t ,t eff := ǫS eff − i ~ ( t τ + t τ ) , which satisfies the QME for all ǫ, t , t . By all the above it follows that ∂∂t e i ~ S ǫ,t ,t = ∆ (cid:18) t e i ~ S ǫ,t ,t ρ (cid:19) ,∂∂t e i ~ S ǫ,t ,t = ∆ (cid:18) t e i ~ S ǫ,t ,t ρ (cid:19) , which means that, as long as the parameters t and t are differentfrom zero, the regularized effective action is independent of them up toquantum canonical transformations. We also have ∂∂ǫ e i ~ S ǫ,t ,t = ∆ (cid:18) i ~ t e i ~ S ǫ,t ,t δ σ (cid:19) = − ∆ (cid:18) i ~ t e i ~ S ǫ,t ,t δ σ (cid:19) , which implies that, as long as one of the parameters t and t is differentfrom zero, the regularized effective action is independent of ǫ up toquantum canonical transformations. This in particular means that thepartition function is independent of ǫ and that, in order to compute it,we may simply set ǫ to zero.To perform the final computation we further deform the regularizedeffective action by adding one more irrelevant term. Namely, let G bea function on Map( R [1] , M ). Then S ,t ,t ,G eff := − i ~ ( t τ + t τ + δ δ G )satisfies the QME for all t , t , G . Moreover, if we take a path G ( t ) ofsuch functions, we get ∂∂t e i ~ S ,t ,t ,G ( t )eff = ∆ e i ~ S ,t ,t ,G ( t )eff δ ˙ G ( t ) t ! = − ∆ e i ~ S ,t ,t ,G ( t )eff δ ˙ G ( t ) t ! , which means that, as long as one of the two parameters t and t is different from zero, adding the new term is irrelevant up to quan-tum canonical transformations. We are now ready to compute thepartition function. Namely, we choose L := Map( R [1] , M ) as the La-grangian submanifold of Map( R [1] , T ∗ [1] M ) over which we integrate.Since τ | L = τ | L = 0, we get Z = Z L e i ~ S ,t ,t ,G eff = Z Map( R [1] ,M ) e δ δ G and we already know that the last integral is independent of G . Weonly have to make sure that G is chosen is such a way that the integralis well defined (choosing G = 0, e.g., would lead to ∞· G := g µν ( x ) η + µ η + ν where g µν is a Riemannian metric on target. Anexplicit computation [3, 4] then shows that Z = χ ( M ). Remark . Switching ǫ to zero first and then turning on the regular-izing term in G is a bit formal since we pass through the solution tothe QME where both terms are absent. This solution has a singularintegral (of the type 0 · ∞ ) on L . In order to find a non formal reg-ularization it is necessary to have additional structure on the Poissonmanifold. For instance let us look for G such that S ǫ,t ,t ,G eff satisfies the HE POISSON SIGMA MODEL ON CLOSED SURFACES 25
QME for any ǫ , preserving the property that the variation of G pro-duces a quantum canonical transformation. Indeed, let us assume G asabove but let g be possibly degenerate. If ( S eff , G ) = 0 then S ǫ,t ,t ,G eff satisfies the QME and the change of G is a quantum canonical trans-formation. This property is equivalent to require that π ◦ g = 0 and L V g = 0 for every vector field V tangent to the symplectic leaves. Inspecial cases we may find such a g and in addition a K¨ahler structureon the leaves, compatible with the symplectic structure, such that agauge fixing given by a mixture of what we discussed in this subsectionand the K¨ahler one is available. We plan to investigate the geometricalconditions needed for this gauge fixing in the future.5.3. Regularization on the space of fields.
The argument of theprevious subsection may formally be lifted to the space of fields to showthat the regularized action is actually independent, up to quantumcanonical transformations, of the Poisson structure. Let s and s denote the coordinates on the two S factors of the torus T , and let e and e denote the corresponding fiber coordinates on T [1] T . Wenow denote by δ , δ , K and K the lifts of the vector fields ∂∂e , ∂∂e , e ∂∂e and e ∂∂e to the space of fields F = Map( T [1] T , T ∗ [1] M ). Wedenote by τ , τ , ρ and ρ their Hamiltonian functions. They satisfy(5.5) and (5.6), and formally also (5.7). Remark . For completeness, we give explicit expressions also in thiscase, even if we are not going to need them. If we write X = X + η +1 e + η +2 e + β + e e , η = β + η e + η e + X + e e , we then have δ X = − η +1 , δ η +2 = β + , δ β = η , δ η = − X + ,δ X = − η +2 , δ η +1 = − β + , δ β = η , δ η = X + , and K η +2 = η +1 , K η +1 = η +2 ,K η = η , K η = η . With respect to the symplectic formΩ = Z F δ X δ η = Z T ( δX µ δX + µ − δη + µ δη µ + δη + µ δη µ + δβ + µ δβ µ ) d s d s , the corresponding Hamiltonian functions are τ = Z T ( − η + µ X + µ + β + µ η µ ) d s d s ,τ = Z T ( − η + µ X + µ + β + µ η µ ) d s d s ,ρ = Z T η + µ η µ d s d s ,ρ = − Z T η + µ η µ d s d s . Notice that despite their non covariant look the above formulae areactually globally well defined.The action S = S + S π is δ - and δ -closed; it turns out that theinteraction part S π is actually exact: S π = δ δ σ π , σ π := Z T π µν ( X ) β µ β ν d s d s . From all the above it follows that S Poisson commutes not only with τ and τ , but also with ρ and ρ . Notice that the Jacobi identity for π implies ( S, σ π ) = 0.Now consider the regularized action S ǫ,t ,t := S + ǫS π − i ~ ( t τ + t τ ) , which satisfies the CME and formally also the QME for all ǫ, t , t . Byall the above it follows that, formally, ∂∂t e i ~ S ǫ,t ,t = ∆ (cid:18) t e i ~ S ǫ,t ,t ρ (cid:19) ,∂∂t e i ~ S ǫ,t ,t = ∆ (cid:18) t e i ~ S ǫ,t ,t ρ (cid:19) , which means that, as long as the parameters t and t are differentfrom zero, the regularized action is independent of them up to quantumcanonical transformations. We also have, again formally, ∂∂ǫ e i ~ S ǫ,t ,t = ∆ (cid:18) i ~ t e i ~ S ǫ,t ,t δ σ (cid:19) = − ∆ (cid:18) i ~ t e i ~ S ǫ,t ,t δ σ (cid:19) , which implies that, as long as one of the parameters t and t is differentfrom zero, the regularized action is independent of ǫ up to quantumcanonical transformations. This in particular means that the partitionfunction is independent of ǫ and that, in order to compute it, we maysimply set ǫ to zero. It is now easy to see that, for a reasonable choiceof propagators, the effective action for S ,t ,t is simply the restriction HE POISSON SIGMA MODEL ON CLOSED SURFACES 27 to vacua, that is the the regularized effective action S ,t ,t eff consideredin the previous subsection.6. Globalization of the effective action
We now go back to the problem of globalizing e S (0)eff in the general case.Recall that e S eff ∈ Γ(Λ T ∗ M ⊗ b S H ∗ [[ ~ ]]) satisfies the modified QME(3.2). We write e S eff = P mi =0 e S ( i )eff , where e S ( i )eff is the i -form componentand m = dim M . In form degree zero, we have12 ( e S (0)eff , e S (0)eff ) − i ~ ∆ e S (0)eff = 0 , which is the usual QME.The modified QME is preserved under modified quantum canonicaltransformations. Namely, T ∈ Γ(Λ T ∗ M ⊗ b S H ∗ [[ ~ ]]) of total degree − δ e S eff = d x T + ( e S eff , T ) − i ~ ∆ T which preserves the modified QME at the infinitesimal level. Noticethat, setting T = P mi =0 T ( i ) , we get in form degree zero δ e S (0)eff = ( e S (0)eff , T (0) ) − i ~ ∆ T (0) , which is a usual infinitesimal quantum canonical transformation. Thegoal of this Section is to prove the following Theorem 6.1.
There is a quantum canonical transformation startingat order in ~ that makes the form degree zero part e S (0)eff of the effectiveaction closed with respect to the induced Grothendieck differential D =d x + ( S R | vacua , ) on Γ(Λ T ∗ M ⊗ b S H ∗ [[ ~ ]]) , where S R | vacua denotes theevaluation of S R on vacua. This will ensure that the so obtained effective action, call it ˇ S (0)eff ,is the image under T φ ∗ of a global effective action S eff . Since ˇ S (0)eff ∈ Γ( b S H ∗ [[ ~ ]]), it follows from the discussion just before subsection 2.1that S eff is a section of b S e H ∗ [[ ~ ]], i.e., a formal power series in ~ of functions on e H (formal in the fiber coordinates). Again, we mayidentify e H with the canonical global moduli space of vacua by using aconnection (e.g., the one in φ ). By Remark 3.1, we conclude that theclass of S eff under quantum canonical transformations is a well-definedobject independent of all choices. Recall that e H x = H > (Σ) ⊗ T x M ⊕ H • (Σ) ⊗ T ∗ x M [1]. Proof of Theorem 6.1.
We start with a simple observation:
Lemma 6.2.
Write e S ( i )eff = P ∞ k =0 ~ k S ( i ) k . If the propagator satisfies theproperties in (4.3) , then S ( i )0 = 0 ∀ i > , whereas S (0)0 and S (1)0 areobtained by the evaluation on vacua of b S and S R , respectively.Proof. The terms for k = 0 correspond to trees in the expansion inFeynman diagrams; so, using the notations of Section 4, what we haveto prove is that the source part W sourceΓ vanishes for any tree Γ contain-ing more than one vertex.This is checked by the following degree counting argument. Considera tree Γ containing more than one vertex. Let V k be the number ofvertices in Γ of internal valence (i.e., not counting the leaves) equal to k ≥
1. Then the total number of vertices is V = X k ≥ V k , the number of internal edges is(6.1) E = 12 X k ≥ kV k , and the Euler characteristic of Γ is(6.2) 1 = V − E = X k ≥ − k V k . Next, the source part W sourceΓ vanishes automatically due to K P = P K = 0 and K = 0, unless the following two properties hold for thedecoration of leaves by cohomology classes χ α ∈ H • (Σ):(i) At every vertex of internal valence 1 there are at least two inci-dent leaves decorated by cohomology classes of non-zero degree.(Otherwise W sourceΓ vanishes due to K P = 0.)(ii) At every vertex of internal valence 2 there is at least one incidentleaf decorated by a cohomology class of non-zero degree. (Other-wise W sourceΓ vanishes due to K = 0.)This gives a lower bound E + 2 V + V for the form degree of theintegrand in (4.5); since it should coincide with the dimension of thespace Σ × V it is integrated against, we have the inequality(6.3) E + 2 V + V ≤ V. By (6.1) this is equivalent to12 V + X k ≥ k − V k ≤ . HE POISSON SIGMA MODEL ON CLOSED SURFACES 29
Subtracting (6.2) from this inequality, we get X k ≥ ( k − V k ≤ − W sourceΓ vanishes for any decoration. (cid:3) We now set S (1) ′ = e S (1)eff − S (1)0 and S ( i ) ′ = e S ( i )eff for i > Lemma 6.3.
There is a modified quantum canonical transformationstarting at order in ~ after which all S ( i ) ′ for i ≥ vanish.Proof. We work by induction on the order of ~ . At order zero thestatement holds by Lemma 6.2. Assume that S ( i ) ′ r = 0 ∀ i ≥ ∀ r 1. Integrating this transfor-mation up to time 1, we make S ( m ) ′ k vanish; as a result the new S ( m − ′ k will be D -closed. We may then proceed like this until we make all the S ( i ) ′ vanish. This proves our claim.Notice that these transformations may change the S ( i ) ′ r for r > k .Moreover, the generator used to kill S (1) ′ k will act on S (0) r for r ≥ k bya quantum canonical transformation. (cid:3) This completes the proof of Theorem 6.1.As a final remark, observe that in the case when π is regular and uni-modular we start with S (1) ′ = 0, so we have two different but equivalentways of getting the global action. One consists in taking the original e S (0)eff , the other in applying the method described in this Section sincenothing guarantees that the remaining S ( i ) ′ vanish at the start. Afterapplying the method we get another effective action ˇ S (0)eff that simply differs from e S (0)eff by a quantum canonical transformation and is alsothe image of T φ ∗ of a global action which we denote by ˇ S . Eventually,the two global effective actions S and ˇ S simply differ by a quantumcanonical transformation starting at order ~ .7. Conclusions and perspectives In this paper we have studied the effective action of the Poissonsigma model on a closed surface Σ, where the Poisson structure π onthe target M is treated perturbatively and, for consistency, has to beassumed to be unimodular unless Σ is a torus. We have shown howto obtain a global effective action S eff as an ~ -dependent function onthe moduli space of vacua of the theory with zero Poisson structure,around which we are perturbing. Because of the freedom in the choiceof gauge fixing and the details of globalization, S eff is, as usual, onlywell-defined up to quantum canonical transformations. By a reasonablechoice of the class of allowed gauge fixings—namely, those for whichthe propagator enjoys properties (4.3)—we make sure that the orderzero S eff,0 of the effective action is fixed and equal to the evaluation onvacua of the Poisson-dependent part S π of the action; moreover, theremaining quantum canonical transformations will start at order 1.In the cases when Σ is a torus or π is regular and unimodular, wehave shown that S eff has (a representative with) no quantum correc-tions. In the particular case when Σ is a torus, π is nondegenerateand there is a compatible complex structure, we can use the latter togauge-fix the remaining integration over vacua: the final result is that,as expected from the Hamiltonian formulation and from comparisonwith with the A-model, the partition function is the Euler characteris-tic of the target. An alternative approach that produces the same resultconsists in regularizing the effective action by adding the Hamiltonianfunctions of supersymmetry generators. In general, the effective ac-tion modulo quantum canonical transformations is an invariant of thePoisson structure.Recall that each order in ~ of S eff is actually a section of a vectorbundle Z g := b S e H ∗ over the target M whose structure is fixed by thegenus g of the source Σ. These sections are just tensors of a particularsort. 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Sansone 1, I-50019 SestoFiorentino - Firenze, Italy E-mail address : [email protected] Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthur-erstrasse 190, CH-8057 Z¨urich, Switzerland E-mail address : [email protected] Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthur-erstrasse 190, CH-8057 Z¨urich, Switzerland E-mail address ::