On Globalized Traces for the Poisson Sigma Model

A globalized version of a trace formula for the Poisson Sigma Model on the disk is presented by using its formal global picture in the setting of the Batalin–Vilkovisky formalism. This global construction includes the concept of zero modes. Moreover, for the symplectic case of the Poisson Sigma Model with cotangent target, the globalized trace reduces to a symplectic construction which was presented by Grady, Li and Li for 1-dimensional Chern–Simons theory (topological quantum mechanics). In addition, the connection between this formula and the Nest–Tsygan theorem and the Tamarkin–Tsygan theorem is explained.


Introduction
In [36] Kontsevich showed that the differential graded Lie algebra (DGLA) of multidifferential operators on a manifold M is L ∞ -quasi-isomorphic to the DGLA of multivector fields on M. This is known as the formality theorem. The construction of Kontsevich's star product in deformation quantization is given by the special case of the formality theorem for bivector fields and bidifferential operators. In [11] it was shown that this star product can be written as a perturbative expansion of a path integral given by the Poisson Sigma Model [34,38]. In [42] Tsygan formulated a formality conjecture for cyclic chains (which was motivated as a chain version of the Connes-Flato-Sternheimer cyclic cohomology construction [23]), which was partially proven by Shoikhet [40], Dolgushev [24] and Willwacher [46]. In [47] Willwacher and Calaque have proven the cyclic formality conjecture of Kontsevich, which was the formulation for cyclic cochains. A global geometrical picture of the star product coming from the Weyl quantization approach for symplectic manifolds, i.e. for a constant Poisson structure, was given by Fedosov in [26]. There one chooses a(n) (always existing) symplectic connection and its corresponding exponential map. This construction can be generalized to the local picture of Kontsevich's star product to produce a global version on any Poisson manifold [18,19], where one uses notions of formal geometry [9,29]. The symplectic connection (lifted to the Weyl bundle) can be replaced by the (deformed) Grothendieck connection which is constructed by using any (formal) exponential map (see also [17]). A globalized picture in the field theoretic approach using the Poisson Sigma Model in the Batalin-Vilkovisky (BV) formalism [4][5][6] was given in [8] for closed worldsheet manifolds and in [21] for manifolds with bound-ary using the BV-BFV formalism [12][13][14][15]. Here BFV stands for Batalin-Fradkin-Vilkovisky, which is the Hamiltonian approach of the BV formalism developed in [2,3].
An important object to study for closed 1 star products [23] are trace maps. In [37] Nest and Tsygan showed an algebraic version of the Atiyah-Singer index theorem, where they made the link to a trace map with respect to the underlying star product and computed the index as the trace of the constant function 1 (see also [27] for Fedosov's construction). This construction is given for symplectic manifolds together with the globalization construction of the Moyal product. For a general Poisson manifold with Kontsevich's star product, Cattaneo and Felder constructed a trace map in terms of local field theoretic constructions using the Poisson Sigma Model on the disk for negative cyclic chains [16] in the presence of residual fields (a.k.a "slow" fields,"low energy" fields).
We will extend this construction to a global one by using a formal global version of the Poisson Sigma Model. This construction in fact combines Fedosov's globalization construction with field theoretic concepts on Poisson manifolds and the BV formulation. We also give the connection of the obtained globalized trace to the Tamarkin-Tsygan theorem, which can be seen as a cyclic equivariant extension of the Nest-Tsygan theorem for Poisson manifolds using formally extended Poisson structures. The connection can be understood by field theoretic concepts by looking at the Feynman graph expansion for the obtained trace formula, which geometrically gives rise to a deformed version of the Grothendieck connection and its curvature. We would also like to refer to the work of Dolgushev and Rubtsov [25], who proposed a version of an algebraic index theorem for Poisson manifolds using a trace density map.
In [33] a global equivariant trace formula for symplectic manifolds was constructed by using a Fedosov connection and solutions to the Fedosov equation. The field theoretic construction was given by the effective theory of topological quantum mechanics on the circle S 1 . We show that our trace formula reduces to this trace formula if we consider the Poisson sigma model with cotangent target. To show this, we use the fact (Proposition 4.2) that the vertices of our graphs in the expansion which arise from the Grothendieck connection are linear in the fiber coordinates if the underlying manifold is a cotangent bundle.
Main results. We prove that the map Tr D , constructed out of a globalization construction for the Poisson Sigma Model on the disk D, coincides with the map Tr V , constructed out of curvature terms of Kontsevich's formality map and negative cyclic chains, which is the statement of Proposition 6.7. Then we prove that the map Tr V is indeed a trace on (C ∞ c (M) [[ ]], ), which is the statement of Theorem 8.1. Moreover, we show how the map Tr V is actually related to the Tamarkin-Tsygan theorem for the Poisson case and to the Nest-Tsygan theorem for the symplectic case, which are the results of Theorem 8.2 and Theorem 8.3, respectively. Finally, we show that the construction of the trace Tr V coincides with a trace map presented by Grady-Li-Li for cotangent targets, which is the statement of Proposition 9.3.

The Kontsevich formality. Let (T •
poly (R d ), [ , ] SN , d = 0) be the DGLA of multivector fields on R d endowed with the Schouten-Nijenhuis bracket and the zero differential and let (D • poly (R d ), [ , ] G , b) be the DGLA of multidifferential operators on R d endowed with the Gerstenhaber bracket and the Hochschild differential. In [36] Kontsevich proved the celebrated formality theorem, which states that these two complexes are quasi-isomorphic as L ∞ -algebras. Theorem 2.1 (Kontsevich [36]). There exists an L ∞ -quasi-isomorphism For the case of degree two, Theorem 2.1 implies a star product on R d endowed with any Poisson structure. Moreover, Theorem 2.1 can be extended to a global version, where R d can be replaced by any finite-dimensional manifold M as we will also describe in Sect. 4.6. Let us briefly recall the main objects to understand the formality theorem.

The Hochschild complex and the Gerstenhaber bracket Let
A be a unital algebra with unit 1. One can consider the graded algebra One can check that b : C • (A) −→ C •−1 (A) is a differential, called the Hochschild differential and the tuple (C • (A), b) is called the Hochschild chain complex of A. Here we denote by [a 0 ⊗ · · · ⊗ a m ] the class of a 0 ⊗ · · · ⊗ a m in C • (A). Moreover, we define C m (A) = 0 for all m < 0. The DGLA of multidifferential operators D • poly (M), for a manifold M, can thus be seen as the subcomplex of the shifted complex C • (A) := Hom(A ⊗•+1 , A), where A = C ∞ (M), consisting of multilinear maps which are differential operators in each argument. The Gerstenhaber bracket of two multidifferential operators D, D is given by where |D| denotes the degree of the multidifferential operator D and the Gerstenhaber product • G is given by The differential on C • (A) is given in terms of the Gerstenhaber bracket by [μ, ] G for μ ∈ Hom(A ⊗ A, A) being the multiplication map of A. In fact, in [30] it was shown that the Hochschild cohomology H H • (A) together with • G and [ , ] G is a Gerstenhaber algebra.

Multivector fields and the Schouten-Nijenhuis bracket
The space of multivector fields on a manifold M is given by The Schouten-Nijenhuis bracket [ , ] SN is given by the usual Lie bracket extended to multivector fields by the Leibniz rule, i.e. for multivector fields α, β, γ we have

The Hochschild-Kostant-Rosenberg map
Consider vector fields ξ 1 , . . . , ξ n ∈ T 0 poly (M) and f 1 , . . . , f n ∈ A. One can construct a map, which for n ≥ 1 is given by and for n = 0 it is given by the identity on C ∞ (M). Here S n denotes the symmetric group of order n. This map is called Hochschild-Kostant-Rosenberg (HKR) map. One can check that it is indeed a chain map and a quasi-isomorphism of complexes, but does not respect the Lie bracket on the level of complexes. In fact Kontsevich's L ∞ -quasiisomorphism U gives a solution to this problem as a certain extension of the HKR map.
In particular, the first Taylor component U 1 of U is precisely the HKR map.

The Kontsevich-Tsygan formality.
One can generalize the formality construction to a cyclic version by considering cyclic chains. There is another differential, called the Connes differential [22,23], of degree +1 on the Hochschild complex given by Note that there is an HKR chain map This map is also called the Connes map [23], which identifies cyclic and de Rham cohomology. Following Getzler [32], the negative cyclic chain complex is then given by Consider a module W over the graded algebra R[u] of finite projective dimension and The formality for cyclic chains is given by the following theorem. ]. There exists an L ∞ -quasi-isomorphism

Theorem 2.2
This was proven by Shoikhet, Willwacher and globally extended by Dolgushev using Fedosov resolution. Using Shoikhet's L ∞ -quasi-isomorphism U Sh , one can obtain Theorem 2.2 as a corollary by obtaining [46].

Remark 2.3.
This construction leads to a field theoretic construction using the Poisson Sigma Model on the disk as we will see in Sect. 6. One can construct a trace map which uses an R[u]-linear morphism of L ∞ -modules over some suitable algebra.

Fedosov's Approach to Deformation Quantization
In this section we want to recall the most important notions and constructions of [26]. associated to M, where Sym denotes the completed symmetric algebra. The Weyl bundle can be regarded as a deformation of the bundle of formal functions on T * M. We will write W instead of W(M) whenever it is clear. A section a ∈ (W) is locally given by 2 where a k,i 1 ,...,i ∈ C ∞ (M). In each fiber W x for x ∈ M, one can construct an algebra structure by considering the associative product 2 We will use the Einstein summation convention.
Here we denote by (ω i j ) the components of the inverse ω −1 of the symplectic form.
For any x ∈ M, the tuple (W x , ) is called the Weyl algebra and is called the Moyal product. One can check that where { , } is the Poisson bracket coming from the symplectic structure ω, which makes sure that is actually a deformation quantization of T * x M with constant Poisson structure Moreover, we define the operators δ and δ * according to [26] by where ι denotes the contraction. Define δ −1 := 1 p+q δ * for p+q > 0 and zero if p+q = 0.

Symplectic connection and curvature.
Consider now a symplectic connection ∇ T M on the tangent bundle T M, i.e. a torsion-free connection such that ∇ T M ω = 0. This induces directly a connection ∇ W on W which we will just denote by ∇. The curvature of this connection is given by Moreover, consider the tensor In [26] it was shown that the curvature of ∇ can be formulated as where [ , ] denotes the commutator with respect to the Moyal product .

Fedosov's main theorems. Consider a connection
on W, where γ ∈ 1 (M, W). One can check that∇ is compatible with the Moyal product, i.e.∇ Theorem 3.1 [Fedosov [26]]. Consider a sequence {ω k } k≥1 of closed 2-forms on M. Then there is a flat connection∇ (that is∇ 2 = 0) defined as in (20) such that γ = i, j ω i j y i dx j + r , where r ∈ 1 (M, W) satisfying δ −1 r = 0. Moreover, γ satisfies Consider the symbol map a(x, y, ) = k, which sends all the y i s to zero.
Theorem 3.2 [Fedosov [26]]. The symbol map induces an isomorphism where H 0 ∇ ( (W)) denotes the space of flat sections of the Weyl bundle with respect to ∇. Moreover, since for any flat connection Equation (21) holds, we can construct a global star product on C ∞ (M) [[ ]] by the formula which defines a deformation quantization on (M, ω). There is a similar approach to globalization for any Poisson manifold, where we start with Kontsevich's star product on the local picture using elements of formal geometry, such as the construction of the Grothendieck connection. A modification (deformed version) of this connection will replace the symplectic connection in Fedosov's picture. In fact, Fedosov's construction uses the exponential map of a symplectic connection, whereas the more general approach uses the notion of a formal exponential map as we will discuss in the next section.

Formal Geometry and Grothendieck Connection
In this section we want to recall the most important notions of formal geometry as in [9,29], the construction of the Grothendieck connection, its deformed version and the relation to Fedosov's quantization approach for the case of a symplectic manifold [8,15,[17][18][19]21]. . We say that ϕ is a generalized exponential map if for all x ∈ M we have that ϕ x (0) = x, and dϕ x | y=0 = id T x M . In local coordinates we can write where the x i are coordinates on the base and the y i are coordinates on the fibers. We identify two generalized exponential maps if their jets at y = 0 agree to all orders. A formal exponential map is an equivalence class of generalized exponential maps.
It is completely specified by the sequence of functions ϕ i . By abuse of notation, we will denote equivalence classes and their representatives by ϕ. From a formal exponential map ϕ and a function f ∈ C ∞ (M), we can produce a section σ ∈ ( Sym(T * M)) by defining σ x = Tϕ * x f , where T denotes the Taylor expansion in the fiber coordinates around y = 0 and we use any representative of ϕ to define the pullback. We denote this section by Tϕ * f ; it is independent of the choice of representative, since it only depends on the jets of the representative.

The Grothendieck connection.
As it was shown [8,9,15,17,29], one can define a flat connection D on Sym(T * M) with the property that Dσ = 0 if and only if σ = Tϕ * f for some f ∈ C ∞ (M). Namely, we identify with (T M ⊗ Sym(T * M)). We have denoted by d x the de Rham differential on M and by L the Lie derivative. In coordinates we have Define R(x, y) For σ ∈ ( Sym(T * M)), L R σ is given by the Taylor expansion (in the y coordinates) of where we denote by d y the de Rham differential on the fiber. This shows that R does not depend on the choice of coordinates. One can generalize this also for any fixed vector where Here ξ(x) would replace the 1-form part dx i . The connection D is called the Grothendieck connection. Note that its flatness is equivalent to the Maurer-Cartan equation Moreover, using the Poincaré lemma on T x M it can be shown that its cohomology is concentrated in degree 0 and is given by 4.3. Lifting formal exponential maps to cotangent bundles. We want to consider the case were our manifold is given by a cotangent bundle.
Let (q,p) ∈ T (q, p) T * M, and henceq ∈ T q M andp ∈ T * q M. Note that ϕ q : T q M −→ M, and thus dq (ϕ q ) * ,−1 : Then we can write the lift of the exponential map asφ Thus, we get Remark 4.3. Proposition 4.2 will simplify the graphs in the Feynman graph expansion of the formal global Poisson Sigma Model for cotangent targets as we will see later on.

The deformed Grothendieck connection.
Let M ⊂ R d be an open subset and consider a Poisson structure π on R d . We will denote its associated Weyl bundle 3 by similarly as in the symplectic case. Using Kontsevich's formality map, one can construct a global connection D on (W) as follows: For some vector field ξ , define a differential operator using the formality map U. Define the quantized version D of D by replacing ξ by (29), where we consider a fixed vector ξ ∈ T x M. Hence we have This connection can be extended to a well-defined global connection D on W. It is in fact given as a deformation of D, i.e. D = D + O( ). Moreover, D is not flat but one can check that it is an inner derivation as in (19). Let denote Kontsevich's star product. For any section σ ∈ (W) we have where F ∈ 2 (M, W) denotes the Weyl curvature tensor of D, which can be also expressed by Kontsevich's L ∞ -morphism. For two vector fields ξ, ζ , define a function in terms of the L ∞ -morphism U. Then we can define the Weyl curvature tensor of D to be given by Moreover, one can check that the Bianchi identity DF = 0 holds and that for any γ ∈ 1 (M, W) the mapD is a derivation, i.e.D(σ τ ) =D(σ ) τ + σ D (τ ) for all σ, τ ∈ (W). Computinḡ D 2 directly, one can see that the Weyl curvature tensorF ofD is given bȳ Proposition 4.4. There exists a γ ∈ 1 (M, W) such thatF = 0. More generally, for It is clear that (42) is the special case of (43) for ω = 0. Proposition 4.4 can be shown by using techniques of homological perturbation theory. Note that the Bianchi identity for F implies thatDω = Dω = 0 if ω is a central element of the Weyl algebra endowed with Kontsevich's star product. Equation (43) can be seen as a more general version of (22) for Poisson manifolds, where D takes the place of the symplectic connection ∇ andD the one of∇. We will say that a connection is compatible if its extension to differential forms • (M, W) is a derivation of degree +1 with respect to the star product on the Weyl algebra. A compatible connection on (W) is called a Fedosov connection if it is an inner derivation with respect to its Weyl curvature tensor and it satisfies the Bianchi identity. By the constructions above, the deformed Grothendieck connection D is a Fedosov connection as well as any symplectic connection ∇ on the tangent bundle of a symplectic manifold as in Fedosov's construction. Note that, as we have seen, if D is a Fedosov connection, thenD = D + [γ, ] is also a Fedosov connection.

Grothendieck connection on symplectic manifolds.
Let (M, ω) be a symplectic manifold which can be considered as a special case of a Poisson manifold with Poisson structure π coming from the symplectic form. By Darboux's theorem, we consider a constant symplectic form ω ϕ := ϕ * x ω lifted to the formal construction for any x ∈ M. Note that in this case R is a 1-form on M with values in formal Hamiltonian vector fields 4 for the corresponding Hamiltonian functions h x such that h x | y=0 = 0. For any x ∈ M, h x is a 1-form with values in Sym(T * M) and for any section σ ∈ (W) Consider a symplectic connection ∇ on T M, which induces a connection D on (W), which acts as a derivation on the Weyl algebra. Its curvature is then given by has a solution γ = − 1 2 ω i j y i dx j +γ 0 + γ 1 + 2 γ 2 +· · · ∈ 1 (M, W) such that γ | y=0 = 0. Moreover, we consider the formal exponential map coming from the symplectic connection ∇ Then the connection ∇ +[γ , ] is given byD = D+[γ, ] with γ = − 1 2 h x +γ , where γ is a solution of (43) with ω = ω 0 + ω 1 + 2 ω 2 + · · · . The star product constructed in this way, using a closed two form ω ∈ 2 (M, R)[[ ]], is equivalent to the one constructed by Fedosov associated to the class − 1 2 ω + ω . Note that the deformations of the symplectic form are in one-to-one correspondence with their characteristic classes, which are formal power series ω = ω 0 + ω 1 + 2 ω 2 + · · · , with ω i ∈ H 2 (M, R) such that −ω 0 is the class of the symplectic form ω. For more details on these constructions see [18].  More precisely, we can construct a cochain map which implies a quantization map This map induces an isomorphism , since there are no cohomological obstructions. Note that this is the analogue of the symbol map as in Fedosov's quantization. Moreover, there is a unique ρ for eachD such that ρ| y=0 = id. Using this map, one can define a global version of Kontsevich's star product, defined on the whole Poisson manifold M by Indeed, the map ρ sends D-flat sections toD-flat sections since ρ is a cochain map, i.e. we have ρ • D =D • ρ, and by compatibility with the star product, one can obtain . But since J isD-closed, we know that it has to lie in the image of ρ.
Hence there exists some j ∈ ( Sym(T * M)) such that ρ( j) = J . This implies that j is D-closed and thus of the form j = Tϕ * j for somej . Setting the formal variables y = 0 one finds a global construction for the star product. This approach generalizes Fedosov's construction for the Moyal product, to the globalization of Kontsevich's star product. It can be translated into field theoretic concepts using the Grothendieck connection together with the Poisson Sigma Model as we will also briefly recall in Sect. 5.4.

The Poisson Sigma Model and its Globalization
We consider the space of fields as vector bundle maps F = Map VecBun (T , T * M), i.e. we have the following diagram Consider now the case where ∂ = ∅ and let ι ∂ : ∂ → denote the inclusion of the boundary. Then we set the boundary conditions such that ι * ∂ η = 0. This is convenient to choose, since the Euler-Lagrange equations are given by Hence, it is easy to consider the solution where X = const. and η = 0. In [11] it was shown that this model is directly connected to Kontsevich's star product as formulating it by a quantum field theory where the space-time manifold is modelled by the disk D = {x ∈ R 2 | x ≤ 1}. If we choose three points 0, 1, ∞ on the boundary ∂D counterclockwise (i.e. if we move from 0 counterclockwise on the boundary, we will first meet 1 and then ∞, see Fig. 1), Kontsevich's star product is given by the semiclassical expansion of the path integral modelled by the Poisson Sigma Model as [4][5][6] is a way of dealing with gauge theories 5 , i.e. of theories where the action is invariant under certain symmetries. There we usually associate to a space-time manifold a BV space of fields F (in general, if one starts with the BRST formalism, we get F BV = T * [−1]F BRST ), which is a Z-graded supermanifold, endowed with a (−1)-shifted symplectic structure ω and an action functional Equation), where { , } denotes the BV bracket coming from the odd symplectic form ω . Here we denote by O(X ) functions on a space X . We would like our theory to be local, i.e. we require the action to be given as an integral over some Lagrangian density L depending on fields and higher derivatives

BV formulation. The Batalin-Vilkovisky (BV) formalism
Moreover, we consider the BV Laplacian [35,39], acting on functions on F . We will denote by O loc (F ) the space of such local functions on F . One can check that 6 . Moreover, we can define a cohomological vector field (similarly as in the linear case, which would be the usual BRST charge) as the degree +1 Hamiltonian vector field where D = θ μ ∂ ∂ x μ is the superdifferential for even coordinates (x μ ) and odd coordinates (θ μ ) and , denotes the pairing of tangent and cotangent space of M. One can write out the components of the superfields in terms of fields, antifields and ghosts as follows where β denotes the ghost field. For a field φ we denote by φ + its antifield. Note that we have the relation gh(φ) + gh(φ + ) = −1 and deg(φ) + deg(φ + ) = 2, where "gh" denotes the ghost number which corresponds to the Z-grading on F , and "deg" denotes the form degree. Thus we get In local coordinates we have where now d denotes the de Rham differential on . Note that the BV action has the same form as the classical action (49) and thus it produces the same Euler-Lagrange equations, where the classical fields are replaced by the superfields and the de Rham differential d on is replaced by the superdifferential D.

Equivariant BV formulation
Consider a Lie algebra g acting on via a vector field v X for some X ∈ g. Note that the cohomological vector field is given by where d F and π are the Hamiltonian vector fields for the Hamiltonians for X ∈ g. Choosing a basis (e j ) of g, we get where S ι v j is the Hamiltonian of ι v j which is the vector field on F obtained from the vector field ι v j , such that where S L v j is the Hamiltonian of the vector field L v j which is the vector field on F defined by L v j . The equivariant Quantum Master Equation is then given by For the case where = D we have an S 1 -action and hence we can consider the S 1equivariant theory. For more details on the equivariant BV construction see [7].

Splitting of the space of fields.
We consider a symplectic splitting of the space of fields into residual fields (low energy fields) and fluctuations (high energy fields), which, for the examples considered in this paper, exists by techniques of Hodge theory (see e.g., [13]). We write where M 1 is the space of residual fields and M 2 the space of fluctuation fields. We want to assume that M 1 is finite-dimensional, which is the case for B F-like theories (such as the Poisson Sigma Model). In this case it is always possible to find a split = 1 + 2 , where j is a BV Laplacian on M j , j = 1, 2. Consider a half-density f on F . Then for any Lagrangian submanifold L ⊂ M 2 we get Here L denotes the BV pushforward, which is defined on half-densities by restricting the half-density to L which makes it a density and apply the Berezinian integral. Note that the choice of L is equivalent to gauge-fixing since, assuming the Quantum Master Equation we have an invariance of the BV pushforward L exp i S under continuous deformation of L up to 1 -exact terms. This is due to the following theorem.
Theorem 5.1 [Batalin-Vilkovisky]. The following holds: If we take f = exp i S , we get that the Quantum Master Equation has to hold for the second point of the theorem. For B F-like theories, F is given as the direct sum of two complexes C ⊕C endowed with the differentials δ andδ. We want them to be endowed with a nondegenerate pairing , of degree −1 such that the differentials are related by B, δ A = δ B, A for all A ∈ C and B ∈C. In that case M 1 is given by the cohomology H ⊕H and M 2 is just a complement in F . For the case of the Poisson Sigma Model with boundary (∂ = ∅) such that the boundary is given by the disjoint union of two boundary components ∂ 1 and ∂ 2 we have for a constant background field x : −→ M, and thus According to the splitting of the space of fields, we write X = x + X and η = e + E , where x, e ∈ M 1 and X , E ∈ M 2 .
Remark 5.2. Note that functions on the shifted tangent bundle T [1] are given by the algebra of differential forms • ( ), which indeed allows us to write the space of fields as in (66). Moreover, if we would have a manifold with boundary ∂ = ∂ 1 ∂ 2 as mentioned before, we can split the space of fields as F = B×M 1 ×M 2 , where B would denote the leaf space of the symplectic foliation induced by a chosen polarization on the boundary to perform geometric quantization, where we would choose the convenient δ δE -polarization on ∂ 1 and the opposite δ δX -polarization on ∂ 2 , where E and X denote the ηand X-boundary fields respectively (elements of the leaf space B). Moreover, one can always obtain a symplectic structure on the space of boundary fields by symplectic reduction. Hence, by techniques of geometric quantization, one would obtain a vector space for each boundary component and one can speak of "boundary states" as elements of these spaces. This construction is needed for treating the Poisson Sigma Model in the Hamiltonian approach of the BFV formalism (space of boundary fields) coupled together to the BV formalism, which is called the BV-BFV formalism [12][13][14]. We will not use the BV-BFV construction, since we will only deal with the disk D with one single boundary component together with the boundary condition ι * ∂D η = 0.

The formal global action.
Let us consider for a multivector field ξ k ∈ ( k T M) the local functional 7 Note that for any k ≥ 0 we have Q (S ξ ) = {S , S ξ } = 0. In [21] it was shown that the Poisson Sigma Model action can be formally globalized by adding another term to the action, which is given by where X and η are defined by the following equations Recall that x : −→ M denotes a constant background field. Denote by S 0 the free part of the action, i.e. S 0 := η i ∧ dX i . Lifting the Poisson Sigma Model action to the formal construction, we get the formal global action If we denote by π ϕ := Tϕ * π , we can observe Tϕ * S π = S π ϕ . Note that the de Rham differential in d X i is on and the de Rham differential in dx i is on the moduli space of constant solutions to the Euler-Lagrange equations Remark 5. 3. In general, one can consider any type of classical solution of the Poisson Sigma Model for the point of expansion. We choose the moduli space (72) since it makes things much easier.
One can show that (113) satisfies the differential Classical Master Equation For quantization, consider the partition function, given by (113) for some Lagrangian submanifold L ⊂ M 2 . The Quantum Master equation is not satisfied in general. It can be shown that if π is divergence free (unimodular), the Quantum Master Equation exp i (S 0 + S π ) = 0 holds. Another case would be if the Eulercharacteristic of is zero (e.g., the torus). The choice of a unimodular Poisson structure can be seen as a renormalization procedure. One form of renormalization is to impose that there are no tadpoles (short loops), which results in the fact that where is the BV Laplacian acting on the coefficients of the residual fields. If we choose a volume form on M, we can define a divergence operator div and thus a renormalized BV Laplacian by setting (see also Appendix 9.3) Note that S π = 0 if div π = 0. Since S ϕ x = 0, we get a differential version of the Quantum Master Equation Remark 5.4. As we will see, the formal global action for the Poisson Sigma Model (see Equation (113)) has to be extended to an equivariant version such that the S 1 -action on the disk is taken into account. This can be done by using the methods of [7] and formulate it as in Equation (59).

Traces and Algebraic Index Theorem
(hence the name "trace"). There is a canonical trace associated to any star product coming from a symplectic manifold (M, ω) which is described within the local picture. Locally, all deformations are equivalent to the Weyl algebra and on the Weyl algebra there is a canonical trace which is constructed as an integral with respect to the Liouville measure [26]. If we consider functions with support in neighborhoods of any point of M, we set the trace equal to this canonical trace restricted to these functions. Let A(T M) denote the A-genus of M, which is a characteristic class of the tangent bundle T M. One can express it by a de Rham representative as where R denotes the curvature of any connection on T M. Theorem 6.1 ). Let (M, ω) be a compact symplectic manifold and let be a star product with characteristic class ω = −ω + ω 1 + 2 ω 2 + · · · . Then the canonical trace associated to obeys If we assume that there is a volume form on M and that the Poisson structure π is unimodular and div π = 0, we can construct a map Now we can define an integration map on the zeroth periodic cyclic homology by composition where we have used Shoikhet's T π map for the isomorphism C H 0 (A ) ∼ = H P 0 (M) (we could have also used the trace density map of Dolgushev-Rubtsov [25]). Let R be a DG ring with differential d R . For a projective R-module M, one defines a connection to be a map The Atiyah class of a connection ∇ is then defined by In fact, [At(∇)] measures the the obstruction to find a d R -compatible connection. We define the Chern character of a connection ∇ by Moreover, one can then define more generally the A-genus of a connection ∇ on M in terms of these classes by where Td denotes the Todd class, defined by . (88)

Theorem 6.2 (Tamarkin-Tsygan [41]). Let M be a compact manifold with formal Poisson structure π ∈ ( 2 T M)[[ ]]
and a volume form on M with div π = 0 and c ∈ PC 0 (A ). Then Here where A j (T M) ∈ H 2 j (M) are the components of the A-genus.

A trace map for negative cyclic chains.
In [16] it was shown how one can obtain a trace map by constructing an L ∞ -morphism from negative cyclic chains to multivector fields with an adjunction of the formal parameter u of degree 2. Moreover, the relation to the BV formulation of the Poisson Sigma Model and how the former formula can be interpreted as an expectation value with respect to the corresponding quantum field theory was shown. However, this construction was only given for open subsets M of R d . We will extend this construction to a global one using notions of formal geometry as we have seen before.
The Taylor components of V are given by maps Note that an element of degree +1 in (T • poly (M) [u], δ ) has the formπ = π + uh, where π is a bivector field and h a function. The Maurer-Cartan equation δ π − 1 2 [π,π ] = 0 translates to [π, π] = 0 and div π − [h, π] = 0, and hence π is Poisson and h corresponds to the Hamiltonian function of the Hamiltonian vector field δ π . As we have seen, this is equivalent to the unimodularity condition.
Remark 6.4. The morphism V is in fact related to Shoikhets morphism [40] in the proof of Tsygan's formality theorem on chains [42] for M = R d . It is a morphism of L ∞ -modules over T •+1 poly (M) from C • (A) to the DG module of differential forms ( −• (M, R), d = 0) and extends to (11). The action of ξ ∈ T •+1 poly (M) on • (M, R) is given by Lie derivative L ξ = d • ι ξ ± ι ξ • d, where the internal multiplication of vector fields is extended to multivector fields by ι ξ ι ζ = ι ξ ∧ζ . This construction was globalized by Dolgushev to any manifold M. Moreover, recall that a volume form ∈ d (M, R) defines an isomorphism and thus we identify the differential d on • (M, R) by the divergence operator div on T • poly (M). By the fact that V is an L ∞ -morphism we get ι div ξ = dι ξ . Letπ

which is a Maurer-Cartan element if π + uh is a Maurer-Cartan element in (T • poly (M)[u][[ ]]
, δ ). We will denote the twist of V byπ by Vπ , which is defined through its Taylor components V n (π , . . . ,π | ). (98) Then we can define a trace map [16] Tr since Vπ : , u div ) is a chain map. We will elaborate on this fact a bit more in Sect. 8.
where ξ is a •-vector field and α is a •-form. Here denotes again a chosen volume form on M. We have denoted by • c (M, R) differential forms with compact support. It is obvious that this map can be extended u-bilinearly. Moreover, there is an isomorphism

Construction via the Poisson sigma model. Consider now the Poisson Sigma Model on the disk . Let
and define the vacuum expectation value of an observable by the map The map V n can be expressed as the vacuum expectation of an observable S ξ 1 · · · S ξ j O a 0 ,...,a m , where O a 0 ,...,a m := a 0 (X(t 0 )) t 1 <t 2 <···<t m ∈∂D\{t 0 } a 1 (X(t 1 )) · · · a m (X(t m )).
For m points t 1 , . . . , t m ∈ ∂D we consider the ordering t 0 < . . . < t m , which means that if we start at t 1 and move counterclockwise on ∂D, we wil first meet t 2 , then t 3 , and so on. If we embed the disk into the complex plane, i.e. we have D = {z ∈ C | |z| ≤ 1} and set t 0 = 1, we can express the counterclockwise condition on ∂D by 0 < arg(t 1 ) < arg(t 2 ) < · · · < arg(t m ) < 2π .
given by the expectation value of the corresponding observable. Recall thatπ is a Maurer-Cartan element for a unimodular Poisson structure and that we work with the boundary condition ι * ∂D η = 0. For two functions f, g ∈ C ∞ (M), we define O f (X, η) := f (X(1)) for 1 ∈ ∂D and O g (X, η) := g(X(0)) for 0 ∈ ∂D. Moreover, define 10 Then we observe where δ was the differential on the complex C in the definition of the space of fields in Sect. 5.3. This follows from the Ward identity which is true by (65), the fact that Z 0 is constant on M 1 and the Leibniz rule for the BV Laplacian (see also Appendix 9.3, Equation (189)). Hence by (51) the two functions f, g can move under the trace map from both sides 11 to each other on ∂D. Thus we get vertices in the bulk, the ones representing the formally lifted Poisson structure π ϕ := Tϕ * π = Tϕ * π − uTϕ * h and the ones representing the R vector field coming from the definition of the Grothendick connection. We will also consider additional vertices on the boundary where we place solutions γ of (42). Then we can consider the vacuum expectation value where S ϕ π,R := S π ϕ + ϕ * S R .
Remark 6.6. The additional vertices labeled by a solution γ of (42) give rise to another additional term in the formal global action [21]. In particular, we have to consider the actionS We will call the Poisson Sigma Model with actionS ϕ the Fedosov-type formal global Poisson Sigma Model and we callS ϕ the Fedosov-type formal global action.

Proposition 6.7. The map
coincides with where we consider V π ϕ n (R · · · R | ) to be defined on the negative cyclic complex for sections of the Weyl algebra W.
This can be seen by constructing the maps V n in terms of graphs. We will do this in Sect. 7.1.
Remark 6.8. In fact, one can construct Kontsevich's star product directly by using a path integral quantization with respect to the formal global action S ϕ as in (113), using a similar approach as in [11], with the difference that the observables on the boundary are given byD-closed sections of the form O ρ(Tϕ * f ) (see Fig. 2). Hence we can write it down as a path integral

Construction via graphs.
We want to describe how the Taylor components of V are given in terms of graphs. In fact we have where ξ = ξ 1 · · · ξ n , with ξ i ∈ ( k i T M) [u], k = (k 1 , . . . , k n ) and a = [a 0 ⊗ · · · ⊗ a m ] ∈ C m (A). Here w ∈ R denotes the weight of a graph according to the given Feynman rules, which can be computed as integrals over configuration spaces of points on the the interior of the disk and on the boundary. We want to recall the definition of the finite set G k,m of oriented graphs as in [16]. For each graph 12 ∈ G k,m with n + m vertices (n vertices in the bulk and m vertices on the boundary), we assign a vertex set V ( ) = V 1 ( ) V 2 ( ) V w ( ). We will distinguish between two different types of vertices which we call the black vertices V b ( ) = V 1 ( ) V 2 ( ) and the white vertices V w ( ). Within the black vertices we will also distinguish between vertices of type 1 and of type 2 according to the following rules.
• There are n vertices in V 1 ( ). There are exactly k i edges originating at the ith vertex of V 1 ( ). • There are m vertices in V 2 ( ). There are no edges originating at these vertices.
• There is exactly one edge pointing at each vertex in V w ( ) and no edge originating from it. • There are no edges starting and ending at the same vertex.
• For each pair of vertices (i, j) there is at most one edge from i to j.
Each multivector field ξ i can be endowed with a power of the formal parameter v , which represent the residual field assigned to a black vertex.
where we sum over all indices and where we set To compute the configuration integrals, we want to make a degree count, i.e. we want the form degree to be equal to the dimension of the configuration space. Let be a manifold with boundary and define the configuration space of n points in the bulk and m points on the boundary by The FMAS-compactification, C X ( ), is then defined as the closure of Conf X ( ) in this embedding.

Let now
= D and fix the point 1 on ∂D. Then we have to work on the section space The space (120) has dimension 2n + m. Moreover, the number m represents the amount of points on the boundary distinct from the fixed point 1, i.e. the total amount of points on the boundary is m + 1. In fact, (120) is equal to the set {(z, t) ∈ C n,m+1 (D) | t 0 = 1} for m ≥ 1.
As already mentioned, we have an S 1 -action on the disk. Instead of working with the quotient of the configuration space by P SL 2 (R), we will work with equivariant differential forms, which arise from the equivariant BV construction of the Poisson Sigma Model within the Feynman graph expansion.

Equivariant differential forms and equivariant Stokes' theorem.
We want to work with equivariant differential forms with respect to the S 1 -action on the disk. We define them as where the differential is given by Here v ∈ (T D) denotes the image of the infinitesimal vector field d dt , which is the generator of the infinitesimal action R d dt −→ (T D). Now consider a differential form ω on the configuration space C 0 n,m (D). We want to describe the boundary of the configuration space. Let S be a subset ofn ≥ 2 points in the bulk which collapse at a point in the bulk of the disk. Then the stratum of type I is given by The stratum of type II is constructed as follows. Let S be the subset ofn points in the bulk and T the subset ofm points on the boundary which collapse at a point on the boundary of the disk. Hence we get the stratum where H denotes the upper half plane. [16]). Let ω ∈ • S 1 (C n,m+1 (D)). Denote also by ω its restriction on C 0 n,m (D) ⊂ C n,m+1 (D). Denote by ω ∂ its restriction to the coboundary 1 strata ∂ i C 0 n,m (D). Then 3. Weights of graphs. We will consider a propagator P on D × D \ diag, where diag := {(z, z) | z ∈ D} ⊂ D × D denotes the diagonal on the disk. The propagator will be a 1-form on the configuration space of the disk. In particular, we have

Theorem 7.2 (Equivariant Stokes
Note that this propagator is equivariant under the S 1 -action.
3. An important fact [10,13] of the propagator is where π 1 , π 2 are the projections to the first and second factor respectively. Here χ j , χ j are representatives of the cohomology classes and their duals respectively, such that Computing this directly, we get The first term of (129) is a volume form on the disk and hence a representative of the cohomology class, hence the whole is a representative of the equivariant cohomology class. Graphically, this corresponds to the fact that if the de Rham differential acts on an edge of a graph between two (black) vertices (which represents a propagator), it will split into residual fields (see Fig. 4). This can be extended to the equivariant differential d S 1 . The white vertices mentioned in the graph construction before are actually represented by zero modes on D. More precisely, we have the following Lemma. Lemma 7.4 (e.g., [13,16]). Let ∂ e be the graph which is obtained from the graph by adding a white vertex • and replacing the edge e ∈ E b ( ) connecting two black vertices by an edge originating at the same vertex as e but ending at the white vertex •. Then (130) Fig. 5. The first picture corresponds to the integrand of (134) and the second picture corresponds to the one of (135). Graphs with such a vertex w vanish The represented zero modes are parametrized by the formal variable u attached to each vertex. The weight of a graph ∈ G (k 1 ,...,k n ),m is then computed by The equivariant cohomology H • S 1 (D) is generated by the constant function 1. Moreover, the relative equivariant cohomology H • S 1 (D, ∂D) is generated by the class of Remark 7.5. Note that with this notation we have d S 1 P = −π * 1 φ.
The differential form ω ∈ 2n+m S 1 (C 0 n,m (D)) is given by where the number r i is given by the degree of the vertex i plus the amount of white vertices attached to it. Moreover, we have the following lemma. Proof. This follows by the fact that is a chain map, which follows from Theorem 7.2 and Lemma 7.4. Using the construction with the Poisson Sigma Model, the trace property follows from (51) and the constructions in Sect. 6.3.
. Note that the configuration integrals are considered on the section space where the point 1 is fixed on the boundary, labeled by the observable O ρ(Tϕ * f ) . We consider another point 0 on the boundary, which is not fixed, labeled by the observable O ρ(Tϕ * g) . Moreover, we have some additional m−1 boundary points labeled by γ . Note that there are boundary strata of the configuration space where g collides to f from the left and one where it collides from the right. Recall that the dimension of the configuration space C 0 n,m (D) ⊂ C n,m+1 (D) is given by 2n + m. Without the point 0 we would have that the dimension is equal to 2n + m − 1, which has to be the same as the form degree of the differential form ω within the configuration integral for any graph ∈ G (k 1 ,...,k n ),m . Hence, we look at its equivariant differential d S 1 ω and apply the equivariant Stokes' theorem (Theorem 7.2). Using (117) and (131), we can write where k = (k 1 , . . . , k n ) and < is a subgraph of , where n < n points collapse in the bulk and m < m collapse on the boundary. Moreover, is a graph whose vertex set satisfies |V ( )| = |V ( )| + 1 with the same amount of vertices in the bulk and on the boundary plus an additional vertex on the boundary. Note that by setting u = 0, Theorem 7.2 reduces to the usual Stokes' theorem for corners. The dimension of the configuration space C n,m (H) modulo scaling and translation is given by 2n + m − 2. This has to be equal to the form degree of the differential form we want to integrate. Let p be the amount of vertices labeled by π ϕ and r the amount of vertices labeled by R.
Then we have This implies three different cases (see Fig. 6); Summing over all these graphs, the third picture will exactly correspond to F = F(R, R), the curvature of the deformed Grothendieck connection D, the second picture to Dγ and the first picture is exactly the star product γ γ . Thus, summing them together we get a contribution (third picture). Note that in each picture p can be arbitrary. The thick arrows denote the fact that there can be arbitrarily many incoming arrows, depending on the combinatorics and hence these terms vanish. Hence the only strata that survive within the boundary of the configuration space are the ones where g approaches f from the left and from the right, so by [11] we get the boundary contribution g f − f g. In fact, for any ξ = ξ 1 · · · ξ n ∈ (Sym n T •+1 poly (M) [u], δ ) and a ∈ CC − m (A), we have whereξ i denotes the projection of ξ i to T •+1 poly (M), S p,q ⊂ S p+q is the set of ( p, q)shuffles and the signs ε(σ, ξ ), ε i j are the Koszul signs coming from the permutation of the ξ i , and |ξ | = i |ξ i |. Note that δ is extended to a degree +1 derivation on SymT •+1 poly (M) [u]. The maps U k : Indeed, one can show that for any a = [a 0 ⊗ · · · ⊗ a m ] ∈ C −m (A), ∈ G k,m and where j k is defined as follows: Define a map Moreover, note that and the second term on the right-hand side of (142) is given by V n+1 (ξ | Ba). Let us look at the boundary integral in the first term of the right hand side of (142). As argued in [16], one can show that treating the boundary strata of type I, the only remaining term will be the sum in (140) containing the Schouten-Nijenhuis bracket. The strata of type II will give a contribution as the sum in (140) containing Kontsevich's L ∞ -morphism and a term V n−1 (ξ | ba).
Using Equation (140), we get that the twist of V by π ϕ is indeed a chain map.
Recall from Sect. 6 that the zeroth cyclic homology C H 0 (A ) is isomorphic to the zeroth Hochschild homology H H 0 (A ), which is again isomorphic to the zeroth Poisson homology H P 0 (M). Hence the chain map V π ϕ induces a map given by integration as in (81). Tamarkin Proof. Note that since the Lie derivative with respect to the Poisson tensor π is defined by L π := d • ι π − ι π • d, we get an isomorphism of complexes

Relation to the
Let A t be a 1-parameter family of algebras given by A [[t]] as an R[t]-module. Denote by V π := ∞ n=0 1 n! V π n . Consider the Gauss-Manin connection on the periodic cyclic cohomology viewed as a vector bundle over the parameter space (see e.g [20,32,43]). In [20] it was shown that for a 1-parameter family π t of solutions to the Maurer-Cartan equation e.g., with polynomial dependence π t = tπ for a Poisson tensor π , and a cyclic cycle c t ∈ PC −m (A t ), which is horizontal with respect to the Gauss-Manin connection, the class of ((exp u j is independent of t. Denote by V π := exp ι π t /u V π t the image of V π t under the isomorphism (148) with formal Poisson structure π t . Since [ V π t (c t )] is independent of t, we can set the Poisson tensor to be zero. In our case we have t = and this will allow us to put = 0 in π ϕ and to merge all the π vertices at zero (see Fig. 7). This will produce wheel graphs as considered e.g., in [44,45]. Let us denote the weight for a wheel graph with j vertices by w j .
Note that the curvature of the Grothendieck connection is contained in the R-vertices (see Sect. 4.5). Hence, considering the Propagator P on the disk we can compute the weight of a wheel diagram with j black vertices. We will get where z 1 , . . . , z j are the vertices labeled by R. Moreover, if we recall that R (x, y) = R k (x, y) ∂ ∂ y k and R k (x, y) := R k (x, y)dx , we get a differential form where we sum over all indices. Thus permuting everything into the right place we get Tr(R j ), where by abuse of notation we also denote by R the appearing curvature. The permutation will give a sign To get the correct form degree and be consistent with the isomorphism (148), we will need a factor of u j . Indeed, note that φ(0, u) = u, which in fact appears for any π ϕ -vertex and hence merging this j times we get a factor u j .
One can easily see that the propagator P will reduce to Kontsevich's angle propagator P(0, z i ) = 1 2π d arg(z i ) and hence w j vanishes if j is odd. Note that if j is even, we have Therefore, as it was computed in [44,45], we get where B j are the Bernoulli numbers 14 . Hence we have Thus we get Recall from Sect. 2.2 that the Connes isomorphism Co is given by Then for any c ∈ PC • (A) we can see that one gets

Relation to the Nest-Tsygan theorem.
Let M = T * N be the cotangent bundle for a manifold N endowed with its canonical symplectic form ω and consider the constant function 1 on the boundary of the disk. In this setting we get the following theorem. Proof. One can easily check that by Proposition 4.2 and degree reasons the only diagrams contributing within the trace formula are given by wheel-like loops as in Fig. 8, and residual graphs as in Fig. 9. Using the same construction as in Sect. 8.2, we can merge the gray vertices to the center, and obtain wheel graphs which again will give rise to . Note that we choose to be the symplectic volume form ω d d! and, using (89), we can see that if c = 1, the u's will all cancel each other and thus it will not depend on u. Indeed, we have and therefore By degree reasons, the only surviving terms in A u (T M) exp(ι π /u) ω d d! are From the field theoretical construction, it is easy to check that the sum over all residual graphs will exactly give a contribution exp (ω / ). Indeed, the integral and for s = 1, we get D φ = 1. Hence summing over all such graphs we get exp(π ϕ ) = exp(ω / ). Putting everything together, we have where where d T M is the de Rham differential on T M, := L π = [d T M , ι π ] with π the Poisson structure induced by ω (here L denotes the Lie derivative), and F is the Weyl curvature tensor given as in (18). In fact ( −• (T M), ) is a BV algebra like as in Appendix 9.3, which is why in [33]  : Fix a solution γ of (22). Then one can construct a nilpotent 16 solution γ ∞ of (167) as an effective action 16 i.e. there is some N 0 such that γ N ∞ = 0, which is in fact true since the exponential map will terminate for some power.
where G 0 denotes the set of all connected graphs, ( ) denotes the number of loops of , Aut( ) denotes the automorphism group of , and ω (γ , P S 1 ) a differential form depending on a chosen propagator P S 1 on S 1 and γ .
Define a map which represents a factorization map from local observables on the interval to global observables on S 1 . The trace map in this setting is defined by where σ is the symbol map (23). For the equivariant formulation, extend the map σ to the BV bundle by sending y i , dy i → 0, and define the S 1 -equivariantly extended complexes where t is the coordinate on S 1 , and and show that it still remains a cochain map for the equivariant differentials. Furthermore, one also defines an equivariant twisted integration map Remark 9.1. In fact one can show that (178) extends (168) as Fig. 10. The interaction vertices appearing in the cotangent case. The straight arrows represent aq-derivative and the wavy arrows represent ap-derivative. There are no incoming arrows at the π ϕ -vertices and exactly two emanating arrows. There are arbitrarily many incoming arrows representing theq-derviatives for an R-vertex, but at most one arrow representing ap-derivative and exactly one arrow emanating. For the γ -vertices we have arbitrarily many incomingq-andp-derivatives and no emanating arrows Again, one can show that (178) remains a cochain map with respect to the extended complexes, and in particular the composition is a cochain map. The S 1 -equivariant trace map is then defined by Moreover, the relation to (173) is

Feynman graphs for cotangent targets. Consider the case of the Poisson Sigma
Model with target a cotangent bundle M = T * N for some manifold N . Then by Proposition 4.2 and Lemma 7.6 the graphs will reduce to a certain class of graphs. We have two different bulk vertices. There are vertices labeled by π ϕ and vertices labeled by R.
The π ϕ -vertices emanate two arrows, representingq-andp-derivatives as in Sect. 4.3, and there are no arrows arriving at them, since the Poisson structure is constant. The R-vertices emanate one arrow and there can be an arbitrary amount of arrows representingq-derivatives arriving at them, but by Proposition 4.2 we can only have at most one arrow representing ap-derivative arriving. We also consider vertices on the boundary representing solutions γ of (42). For each of them there are no arrows emanating and arbitrarily many arriving.
where h was the Hamiltonian function for π such that div π − [h, π] = 0. Indeed, by considering the Feynman graph expansion of V π ϕ n , we get that Tr V ( f ) = M ∞ n=0 n n! P n (Tϕ * π, Tϕ * h, ρ(Tϕ * f )) exp(Tϕ * h) , where P n are differential polynomials in Tϕ * π, Tϕ * h, and ρ(Tϕ * f ). Now, considering Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
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Appendix A. BV Algebras and Relation to Field Theory
We want to recall some notions on BV algebras as in [31], and how it is related to the original gauge formalism developed by Batalin and Vilkovisky within quantum field theory.
An identity element in B is an element 1 of degree 0 such that it is an identity for the product and [1, ] = 0.

A.2. BV algebras. A BV algebra
An identity in A is an element 1 of degree 0 such that it is an identity for the product and (1) = 0. One can show that a BV algebra is in fact a special type of a braid algebra. More precisely, a BV algebra is a braid algebra endowed with an operator : A • −→ A •+1 such that 2 = 0 and such that the bracket and are related by A.3. Connection to field theory. We would like to explain the name "BV" algebra. This comes from the approach to deal with gauge theories in quantum field theory developed by Batalin-Vilkovisky in the setting of odd symplectic (super)manifolds. Let (F, ω) be an odd symplectic (super)manifold. In physics, F is called the space of fields. Let f ∈ C ∞ (F) and consider its Hamiltonian vector field X f . One can check that C ∞ (F) endowed with the Poisson bracket is a braid algebra. Let μ ∈ (Ber(F)) be a nowhere-vanishing section of the Berezinian bundle of F. This represents a density which is characterized by the integration map : c (F, Ber(F)) −→ R. Hence μ induces an integration map on functions with compact support for some Lagrangian submanifold L ⊂ F, where the integral exists. Then one can define a divergence operator div μ X by Lemma A.1. For a vector field X let X * = −X − div μ X . Then Moreover, div μ ( f X) = f div μ X − (−1) | f ||X | X ( f ) and if S ∈ C ∞ (F) is an even function, then div exp(S)μ X = div μ X + X (S).
One can then define to be the odd operator on C ∞ (F) given by A BV (super)manifold (F, ω, μ) is then an odd symplectic (super)manifold with Berezinian μ such that 2 = 0.
we get that 2 S = 0, which ensures a BV algebra structure. In physics, the function S is called the action and Equation (197) is usually called the Quantum Master Equation 17 . 17 Here we have set i = 1, whereas in quantum field theory we want dependence on as a formal variable and consider formal power series as Taylor expansions (cf. perturbative expansion of path integrals)