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On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism

Abstract

We construct a formal global quantization of the Poisson Sigma Model in the BV-BFV formalism using the perturbative quantization of AKSZ theories on manifolds with boundary and analyze the properties of the boundary BFV operator. Moreover, we consider mixed boundary conditions and show that they lead to quantum anomalies, i.e. to a failure of the (modified differential) Quantum Master Equation. We show that it can be restored by adding boundary terms to the action, at the price of introducing corner terms in the boundary operator. We also show that the quantum Grothendieck BFV operator on the total space of states is a differential, i.e. squares to zero, which is necessary for a well-defined BV cohomology.

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Notes

  1. There is also an immersed Lagrangian submanifold \({\mathcal {L}}_2\in ({\mathcal {G}},\omega )\times ({\mathcal {G}},-\omega )\) representing the identity. The composition of \({\mathcal {L}}_3\) with \({\mathcal {L}}_2\) also defines an immersed Lagrangian submanifold of \(({\mathcal {G}},\omega )\times ({\mathcal {G}},\omega )\times ({\mathcal {G}},-\omega )\) that induces an equivalence relation and a quotient space which is precisely the symplectic reduction, so the symplectic groupoid G in case M is integrable.

  2. Here the letters BFV stand for Batalin, Fradkin and Vilkovisky, who introduced what is now known as BV [5, 6] and BFV [3, 4, 34, 35] formalisms for gauge fixing.

  3. We consider the Poisson Sigma Model as a perturbation around the trivial Poisson structure, so the moduli space of classical solutions on which \(\varvec{\widetilde{\psi }}\) is defined is identified with the target \({\mathscr {P}}\).

  4. This is related to the curvature that appears in the globalization of Kontsevich’s star product, see e.g. [19].

  5. This in particular will also allow us to consider 2D Yang–Mills theory in this formalism.

  6. \(\text {EL}_M\) is the term that depends only on the variations of the fields but not on higher jets.

  7. This means that the Hessian of the Lagrangian is weakly non degenerate.

  8. Only in special situations, i.e. \(\dim M = 1\), and some examples discussed in [38].

  9. In particular, \({\mathcal {V}}\) does not depend on derivatives of the fields.

  10. We have to assume certain conditions which are in particular satisfied for BF-like theories.

  11. We slightly changed the definition of quantum BV-BFV theory so that in principle it does not depend on a classical BV-BFV theory.

  12. Typically, \(\psi \) will have degree 0. This is the case when the gauge-fixing Lagrangian (see below) has degree zero, in the sense that its Berezinian bundle has degree zero. This is the case in all examples we consider.

  13. Note that this is a particular state induced by the bulk and not some choice of vacuum state.

  14. We have only considered the case of real polarizations so far.

  15. This assumption forces one to choose singular extensions of boundary fields.

  16. This assumption is rather strong but can be slightly relaxed to the notion of hedgehog fibration.

  17. This is the infinite-dimensional graded manifold adjoint to the Cartesian product (internal morphisms).

  18. Note that in BF-like AKSZ theories we have a target \(T^*[d-1](V[1])=V[1]\oplus V^*[d-2]\), where V is some graded vector space, whereas in the “split” case the target is of the form \(T^*[d-1]M\) for a graded manifold M.

  19. This connection can be extended to a differential on the complex of \(\widehat{S}T^*M\)-valued differential forms on \(\Gamma (\bigwedge ^\bullet T^*M\otimes \widehat{S}T^*M)\). Since \(\Gamma (\bigwedge ^\bullet T^*M\otimes \widehat{S}T^*M)\) is the algebra of functions on the formal graded manifold \({\mathcal {M}}:=T[1]M\oplus T[0]M\), the extended differential, which we also denote by \(D_{\mathsf {G}}\), gives \({\mathcal {M}}\) the structure of a differential graded manifold. In particular since \(D_{\mathsf {G}}\) vanishes on the body, we may linearize at each \(x\in M\) and get an \(L_\infty \)-algebra structure on \(T_xM[1]\oplus T_xM\oplus T_xM\).

  20. This is automatic for theories which admit an AKSZ formulation.

  21. We call the \(\frac{\delta }{\delta \mathbb {E}}\)-polarization the \(\mathbb {X}\)-representation and vice versa.

  22. Choosing a leaf \(b\in {\mathcal {B}}^{\mathcal {P}}_{\partial \Sigma }\) one considers it’s conjugated momentum \(-\mathrm{i}\hbar \frac{\delta }{\delta b}\).

  23. The reason why such counter terms always exists is due to the fact that the cohomology which would provide obstructions is trivial [16] (see also Remark 3.10).

  24. Note that we are not performing “extended” quantization of a manifold with corners in the sense of extended TQFTs, but simply apply BV-BFV quantization where we allow boundary conditions to change along connected components of the boundary.

  25. For any two \({\mathcal {E}}\)-valued 1-forms \(\gamma =\gamma _i{\mathrm {d}}x^{i},\sigma =\sigma _j{\mathrm {d}}x^{j}\in \Omega ^1({\mathscr {P}},{\mathcal {E}})\) one defines their star product by \(\gamma \star \sigma :=(\gamma _i\star \sigma _j){\mathrm {d}}x^{i}\wedge {\mathrm {d}}x^{j}\)

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Acknowledgements

We thank I. Contreras for helpful comments. Moreover, we want to thank the two referees for providing us with important and helpful comments.

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Correspondence to Alberto S. Cattaneo.

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This research was (partly) supported by the NCCR SwissMAP, funded by the Swiss National Science Foundation, and by the COST Action MP1405 QSPACE, supported by COST (European Cooperation in Science and Technology). We acknowledge partial support of SNF Grant No. 200020_172498/1. K. W. acknowledges partial support by the Forschungskredit of the University of Zurich, Grant No. FK-16-093.

Communicated by C. Schweigert

Appendices

Appendix A. Configuration Spaces and Their Compactifications

To define the quantum state, we need to recall the notion of configuration spaces and their compactification as in [2, 36] due to Fulton–MacPherson and Axelrod–Singer.

A.1. FMAS-compactification

We start with the definition of the configuration space.

Definition A.1

Let M be a manifold and S a finite set. The open configuration space of S in M is defined as

$$\begin{aligned} \mathsf {Conf}_S(M) := \{\iota :S \hookrightarrow M |\iota \,\, \text {injection}\}. \end{aligned}$$
(169)

Elements of \(\textsf {Conf}_{S}(M)\) are called S-configurations. To give an explicit definition of the compactification that can be extended to manifolds with boundaries and corners, we introduce the concept of collapsed configurations. Intuitively, a collapsed S-configuration is the result of a collapse of a subset of the points in the S-configuration. However, we remember the relative configuration of the points before the collapse by directions in the tangent space. This is a configuration in the tangent space that is well-defined only up to translations and scaling. The difficulty is that one can imagine a limiting configuration where two points collapse first together and then with a third (see Fig. 19). This explains the recursive nature of the following definition. Recall that if X is a vector space, then \(X\times {\mathbb {R}}_{>0}\) acts on X by translations and scaling.

Definition A.2

(Collapsed configuration inM). Let M be a manifold, S a finite set and \({\mathfrak {P}} = \{S_1,\ldots ,S_k\}\) be a partition of S. A \({\mathfrak {P}}\)-collapsed configuration inM is a k-tuple \((p_{\sigma },c_{\sigma })\) such that \(((p_{\sigma },c_{\sigma }))_{\sigma = 1}^k\) satisfies

  1. (1)

    \(p_{\sigma } \in M\) and \(p_{\sigma } \ne p_{\sigma '}\), for \(\sigma \ne \sigma '\),

  2. (2)

    \(c_{\sigma } \in \widetilde{{\mathsf {C}}}_{S_{\sigma }}(T_{p_{\sigma }}M)\), where for \(|S| = 1\), \(\widetilde{{\mathsf {C}}}_S(X) := \{pt\}\) and for \(|S| \ge 2\)

    $$\begin{aligned} \widetilde{{\mathsf {C}}}_S(X):= & {} \coprod _{\begin{array}{c} {\mathfrak {P}}=\{S_1,\ldots ,S_k\} \\ S = \sqcup _\sigma S_\sigma , k\ge 2 \end{array}}\nonumber \\&\left\{ \left( x_\sigma , c_\sigma \right) _{1\le \sigma \le k}\ \bigg |\ (x_\sigma , c_\sigma ) {\mathfrak {P}}\text {-collapsed } S\text {-configuration in } X\right\} \bigg /(X \times {\mathbb {R}}_{>0}).\nonumber \\ \end{aligned}$$
    (170)

Here, \(\varphi \in X \times {\mathbb {R}}_{>0}\) acts on \((x_\sigma ,c_\sigma )\) by \((x_\sigma ,c_\sigma ) \mapsto (\varphi (x_\sigma ), {{\,\mathrm{d}\,}}\varphi _{x_\sigma }c_\sigma )\).

Intuitively, given a partition \({\mathfrak {P}}= \{S_1,\ldots ,S_k\}\), a k-tuple \((p_\sigma ,c_\sigma )\) describes the collapse of the points in \(S_\sigma \) to \(p_\sigma \). \(c_\sigma \) remembers the relative configuration of the collapsing points. This relative configuration can itself be the result of a collapse of some points.

Definition A.3

(FMAS compactification). The compactified configuration space\({\mathsf {C}}_S(M)\) of S in M is given by

$$\begin{aligned} {\mathsf {C}}_S(M) := \coprod _{\begin{array}{c} S_1,\ldots ,S_k \\ S = \sqcup _\sigma S_\sigma \end{array}}\left\{ (p_\sigma , c_\sigma )_{1\le \sigma \le k}\ \bigg |\ (p_\sigma , c_\sigma ) {\mathfrak {P}}\text {-collapsed } S\text {-configuration in } M\right\} .\nonumber \\ \end{aligned}$$
(171)

A.2. Boundary strata

A precise description of the combinatorics of the stratification can be found in [36], where it is also shown that \({\mathsf {C}}_S(M)\) is a manifold with corners and is compact if M is compact. For us, only strata in low codimensions are interesting. Let \(S=\{s_1,\ldots ,s_k\}\). The stratum of codimension 0 corresponds to the partition \({\mathfrak {P}}= \{\{s_1\},\ldots ,\{s_k\}\}\). For \(\ell >1\), strata of codimension 1 correspond to the collapse of exactly one subset \(S'=\{s_1,\ldots ,s_\ell \} \subset S\) with no further collapses, i.e a partition \({\mathfrak {P}}=\{\{s_1,\ldots ,s_\ell \},\{s_{\ell +1}\},\ldots ,\{s_k\}\}\) and configuration \((p_\sigma ,c_\sigma )\) with \(c_\sigma \) in the component of \(\widetilde{{\mathsf {C}}}_{S'}(X)\) given by the partition \({\mathfrak {P}} = \{\{s_1\},\ldots ,\{s_\ell \}\}\). This boundary stratum will be denoted by \(\partial _{S'}{\mathsf {C}}_S(M)\), in particular, we have

$$\begin{aligned} \partial {\mathsf {C}}_S(M) = \coprod _{S' \subset S}\partial _{S'}{\mathsf {C}}_S(M). \end{aligned}$$
(172)

There is a natural fibration \(\partial _{S'}{\mathsf {C}}_S(M) \rightarrow {\mathsf {C}}_{S {\setminus } S'\cup \{pt\}}(M)\) whose fiber is \(\widetilde{{\mathsf {C}}}_S({\mathbb {R}}^{\dim M})\). Finally, we note that if \(|S| = 2\), then \({\mathsf {C}}_{S}(M) \cong Bl_{{\overline{\Delta }}}(M \times M)\), the differential-geometric blow-up of the diagonal \({\overline{\Delta }} \subset M \times M\), and \(\widetilde{{\mathsf {C}}}_S(X) \cong S^{\dim X-1}\). See Fig. 19 for an example of a configuration of points and coresponding boundary strata.

Fig. 19
figure 19

An element of \({\mathsf {C}}_S(M)\)

A.3. Configuration spaces for manifolds with boundary

We proceed to recall the definition of a compactified configuration space for manifolds with boundary. Let M be a compact manifold with boundary \(\partial M\). Recall that for a manifold M with boundary \(\partial M\), at points \(p \in \partial M\) there is a well-defined notion of inward and outward half-space in \(T_pM\). If \(H \subset X\) is a half-space, then \(\partial H \subset X\) is a hyperplane. \(\partial H \times {\mathbb {R}}_{>0}\) acts on H by translations and scaling.

Definition A.4

(Configuration spaces for manifolds with boundary). Let M be a manifold with boundary \(\partial M\). For ST finite sets, we define the open configuration space by

$$\begin{aligned} \mathsf {Conf}_{S,T}(M,\partial M) := \{(\iota ,\iota '):S \times T \hookrightarrow M \times \partial M\}. \end{aligned}$$
(173)

Definition A.5

(Collapsed configuration on manifolds with boundary). Let \((M, \partial M)\) be a manifold with boundary. Let ST be finite sets and \({\mathfrak {P}}=\{S_1, \ldots , S_k\}\) a partition of \(S \sqcup T\). Then, a \({\mathfrak {P}}\)-collapsed (ST)-configuration in M is a k-tuple of pairs \((p_\sigma ,c_\sigma )\) such that

  1. (1)

    \(p_\sigma \in M\) and \(p_\sigma \ne p_{\sigma '}\), for all \(\sigma \ne \sigma '\),

  2. (2)

    \(S_\sigma \cap T \ne \varnothing \Rightarrow p_\sigma \in \partial M\),

  3. (3)
    $$\begin{aligned} c_\sigma \in {\left\{ \begin{array}{ll} \widetilde{{\mathsf {C}}}_{S_\sigma }(T_{p_\sigma }M) &{} p_\sigma \in M {\setminus } \partial M \\ \widetilde{{\mathsf {C}}}_{S \cap S_\sigma , T \cap S_\sigma }(\mathbb {H}(T_{p_\sigma }M)) &{} p_\sigma \in \partial M\end{array}\right. } \end{aligned}$$

where \(\mathbb {H}(T_{p_\sigma }M) \subset T_{p_\sigma }M\) denotes the inward half-space in \(T_{p_\sigma }M\). Here, for a vector space X and a half-space \(H \subset X\), \(\widetilde{{\mathsf {C}}}_{\varnothing ,\{pt\}}(H) := \widetilde{{\mathsf {C}}}_{\{pt\},\varnothing }(H) := \{pt\}\), and for \(\vert S \sqcup T\vert \ge 2\),

$$\begin{aligned} \widetilde{{\mathsf {C}}}_{S,T}(H):= & {} \coprod _{\begin{array}{c} {\mathfrak {P}}=\{S_1,\ldots ,S_k\}\\ S\sqcup T = \sqcup _\sigma S_\sigma , k \ge 2 \end{array}}\nonumber \\&\times \left\{ (v_\sigma ,c_\sigma )\ \bigg |\ (v_\sigma , c_\sigma ) {\mathfrak {P}}\text {-collapsed } (S,T)\text {-configuration in } H \right\} \bigg / (\partial H \times {\mathbb {R}}_{>0}).\nonumber \\ \end{aligned}$$
(174)

Definition A.6

(FMAS compactification for manifolds with boundary). We define the compactification\({\mathsf {C}}_{S,T}(M,\partial M)\) of \(\mathsf {Conf}_{S,T}(M,\partial M)\) by

$$\begin{aligned}&{\mathsf {C}}_{S,T}(M,\partial M) \nonumber \\&\quad = \coprod _{\begin{array}{c} {\mathfrak {P}} =\{S_1,\ldots ,S_k\} \\ S\sqcup T= \sqcup _\sigma S_\sigma \end{array}}\left\{ \left( p_\sigma , c_\sigma \right) _{1\le \sigma \le k}\ \bigg |\ (p_\sigma ,c_\sigma ){\mathfrak {P}}\text {-collapsed } (S,T)\text {-configuration}\right\} .\nonumber \\ \end{aligned}$$
(175)

Again, this is a manifold with corners and is compact if M is compact. We proceed to describe the strata of low codimension. Let \(U= \{u_1,\ldots ,u_k\}, V = \{v_1,\ldots ,v_k\}.\) The codimension 0 stratum again is given by the partition \({\mathfrak {P}} = \{\{u_1\},\ldots ,\{u_k\},\{v_1\},\ldots , \{v_\ell \}\}.\) Let us describe the strata of codimension 1. We denote by \(\partial ^\text {I}_S{\mathsf {C}}_{U,V}(M,\partial M)\) a boundary stratum where a subset \(S \subset U\) collapses in the bulk, described in the same way as above. On manifolds with boundary, there are new boundary strata in the compactified configuration space given by the collapse of a subset of points to a point in the boundary. Concretely, given a subset \(S=\{u_1,\ldots ,u_{k'},v_1,\ldots ,v_{\ell '}\} \subset U \sqcup V\), there is a boundary stratum \(\partial ^{\text {II}}_S{\mathsf {C}}_{U,V}(M,\partial M)\) corresponding to the partition \({\mathfrak {P}}=\{S,\{u_{k'+1}\},\ldots , \{u_k\},\{v_{\ell '+1}\},\ldots ,\{v_\ell \}\}\) and collapsed configurations \((p_\sigma ,c_\sigma )\) with \(p_{\sigma } \in \partial M\) and \(c_\sigma \) corresponding to the partition \(\mathfrak {P'} = \{\{u_1\},\ldots ,\{u_k\},\{v_1\},\ldots ,\{v_\ell \}\}\). The boundary decomposes as

$$\begin{aligned} \partial {\mathsf {C}}_{U,V}(M,\partial M) = \coprod _{S \subseteq U} \partial ^\text {I}_S{\mathsf {C}}_{U,V}(M,\partial M) \amalg \coprod _{S \subseteq U \sqcup V} \partial ^{\text {II}}_S{\mathsf {C}}_{U,V}(M,\partial M). \end{aligned}$$
(176)

A.4. Configuration spaces for manifolds with corners

Finally, we consider a manifold M with boundary \(\partial M\) and corners \(\partial \partial M\). Note that around points in corners \(p \in \partial \partial M\) there is a notion of inward quadrant \({\mathbb {Q}}(T_pM) \subset T_pM\). It can be defined e.g. in coordinates, since the transition functions have to preserve both boundaries and corners. If \(Q \subset X\) is any quadrant, its boundary is the union of two half-hyperplanes whose intersection is a \((\dim X - 2)\)-dimensional subspace W. This subspace acts on Q by translations. Again, \({\mathbb {R}}_{>0}\) acts on Q by scaling. Note that in this case, \(\widetilde{{\mathsf {C}}}_{\{pt\}}(Q) \cong I\), where I is an interval. Hence the definition of collapsed configurations should be adapted to this case. We want to compactify the open configuration spaces

$$\begin{aligned} \mathsf {Conf}_{S,T,U}^{\mathscr {C}}(M,\partial M,\partial \partial M) \end{aligned}$$
(177)

where M is a manifold with corners. We proceed to define collapsed configurations as above:

Definition A.7

(Collapsed configurations for manifolds with corners). Let \((M,\partial M, \partial \partial M)\) be a manifold with corners. Let STU be finite sets and \({\mathfrak {P}} = (S_1,\ldots ,S_k)\) be a partition of \(S \sqcup T\sqcup U\). Then a \({\mathfrak {P}}\)-collapsed (STU)-configuration in M is a k-tuple of pairs \((p_\sigma ,c_\sigma )\) such that

  1. (1)

    \(p_\sigma \in M\) and \(p_\sigma \ne p_{\sigma '}\), for all \(\sigma \ne \sigma '\),

  2. (2)

    \(S_\sigma \cap T \ne \varnothing \Rightarrow p_\sigma \in \partial M\),

  3. (3)

    \(S_\sigma \cap U\ne \varnothing \Rightarrow p_\sigma \in \partial \partial M\),

  4. (4)
    $$\begin{aligned} c_\sigma \in {\left\{ \begin{array}{ll} \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{S_\sigma }(T_{p_\sigma }M) &{} p_\sigma \in M {\setminus } \partial M \\ \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{S \cap S_\sigma , T \cap S_\sigma }(\mathbb {H}(T_{p_\sigma }M)) &{} p_\sigma \in \partial M{\setminus } \partial \partial M\\ \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{S \cap S_\sigma , T \cap S_\sigma ,U\cap S_\sigma }({\mathbb {Q}}(T_{p_\sigma }M))&{} p_\sigma \in \partial \partial M\end{array}\right. } \end{aligned}$$

where, for Y a quadrant of X, we have \(\widetilde{{\mathsf {C}}}^{\mathscr {C}}_{S,\varnothing ,\varnothing }(Y) = \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{\varnothing ,T, \varnothing }(Y) = \{pt\}\), \( \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{\varnothing ,\varnothing ,\{pt\}}(Y)\cong I\), and for \(|S \sqcup T \sqcup U| \ge 2\) we define

$$\begin{aligned} \widetilde{{\mathsf {C}}}^{\mathscr {C}}_{S,T,U}(Y)&:= \coprod _{\begin{array}{c} {\mathfrak {P}}=\{S_1,\ldots ,S_k\}\\ S\sqcup T\sqcup U = \sqcup _\sigma S_\sigma , k \ge 2 \end{array}}\nonumber \\&\quad \times \left\{ (y_\sigma ,c_\sigma )\ \bigg |\ (y_\sigma , c_\sigma ) {\mathfrak {P}}\text {-collapsed } (S,T,U)\text {-configuration in } Y \right\} \nonumber \\&\bigg / (\partial Y \times {\mathbb {R}}_{>0}). \end{aligned}$$
(178)

This compactified configuration space has three types of boundary strata: Strata where a set of bulk points collapses in the bulk (called Type I strata), strata where a subset of bulk and boundary points collapses at the boundary (called Type II strata), and strata where a subset of all points collapses to a corner point (called Type III strata):

$$\begin{aligned}&\partial {\mathsf {C}}_{S,T,U}(M,\partial M, \partial \partial M) = \coprod _{S' \subseteq S} \partial ^\text {I}_{S'} {\mathsf {C}}_{S,T,U}(M,\partial M, \partial \partial M) \nonumber \\&\qquad \amalg \coprod _{S' \subseteq S \sqcup T} \partial _{S'}^{\text {II}} {\mathsf {C}}_{S,T,U}(M,\partial M, \partial \partial M) \amalg \coprod _{S' \subseteq S \sqcup T \sqcup U} \partial _{S'}^{\text {III}}{\mathsf {C}}_{S,T,U}(M,\partial M, \partial \partial M).\nonumber \\ \end{aligned}$$
(179)

Remark A.8

At this point, one can generalize the definitions above to that of compactifications of configuration spaces on stratified manifolds, with strata of any codimension. This is required for the extension of perturbative quantization to fully extended theories.

Notation A.9

For a manifold M without boundary, we also denote the compactified configuration space of n points \({\mathsf {C}}_{[n]}(M)\) on M by \({\mathsf {C}}_n(M)\) (here \([n] = \{1,\ldots , n\}\)). Moreover, for a manifold M with boundary, we denote the compactified configuration space \({\mathsf {C}}_{[n],[m]}(M)\) of n points on the bulk of M and m points on the boundary \(\partial M\) of M by \({\mathsf {C}}_{[n],[m]}(M,\partial M)\). We will also write \({\mathsf {C}}_\Gamma (M)\) for \({\mathsf {C}}_{[n],[m]}(M,\partial M)\), if \(\Gamma \) is a graph with \(n+m\) vertices, n vertices in the bulk of M and m vertices on \(\partial M\). Moreover, we will write \({\mathsf {C}}^{\mathscr {C}}_{n,m}(M)\) (or \({\mathsf {C}}^{\mathscr {C}}_\Gamma (M)\)) for \({\mathsf {C}}^{\mathscr {C}}_{[n],[m],\varnothing }(M, \partial M, \partial \partial M)\), if M is a manifold with corners.

Appendix B. Deformation Quantization and the Poisson Sigma Model

In this section we recollect some aspects of Kontsevich’s star product [10, 27, 45], its globalization construction [7, 16, 19, 30], and recall the relation with the Poisson Sigma Model [13, 17].

B.1. Kontsevich’s formality map on \({\mathbb {R}}^d\)

Kontsevich’s formality map is an \(L_{\infty }\) (quasi-iso)morphism from multivector fields \(T_{poly}{\mathbb {R}}^d:=\Gamma \left( \bigwedge ^\bullet T{\mathbb {R}}^d\right) \) to multidifferential operators \(D_{poly}^\bullet {\mathbb {R}}^d\) on \({\mathbb {R}}^d\). As such it consists of a family of maps

$$\begin{aligned} \begin{aligned}&{\mathcal {U}}_n:\Gamma \left( \bigwedge ^{k_1}T{\mathbb {R}}^d\right) \oplus \cdots \oplus \Gamma \left( \bigwedge ^{k_n} T{\mathbb {R}}^d\right) \rightarrow D^\bullet _{poly}{\mathbb {R}}^d\\&\qquad (\xi _1,\ldots ,\xi _n) \mapsto {\mathcal {U}}_n(\xi _1,\ldots ,\xi _n):=\sum _{\Gamma \in {\mathcal {G}}_{n,\ell }}w_\Gamma B_{\Gamma ,\xi _1,\ldots ,\xi _n}, \end{aligned} \end{aligned}$$
(180)

where \({\mathcal {G}}_{n,\ell }\) is the set of graphs with \(n+\ell \) numbered vertices, with \(\ell :=2-2n+\sum _{i=1}^nk_i\), such that the jth vertex for \(1\le j\le n\) emanates exactly \(k_j\) arrows (without short loops). Here \(k_i\) represents the degree of the multivector field \(\xi _i\). Note that \({\mathcal {U}}_n(\xi _1,\ldots ,\xi _n)\) acts on \(\ell \) functions. Here \(B_{\Gamma ,\xi _1,\ldots ,\xi _n}\) are multidifferential operators, depending a graph \(\Gamma \) and also on the vector fields \(\xi _1,\ldots ,\xi _n\), and the \(w_\gamma \) are weights corresponding to a graph \(\Gamma \) as in [45]. For a vector field \(\xi \) (i.e. \(\xi \) is of degree 1) and a bivector field \(\Pi \) (i.e. \(\Pi \) is of degree 2) we can define

$$\begin{aligned} P(\Pi )&:= \sum _{j=0}^{\infty } \frac{\varepsilon ^j}{j!}{\mathcal {U}}_j(\Pi ,\ldots ,\Pi ), \end{aligned}$$
(181)
$$\begin{aligned} A(\xi ,\Pi )&:=\sum _{j=0}^{\infty }\frac{\varepsilon ^j}{j!} {\mathcal {U}}_{j+1}(\xi ,\Pi ,\ldots ,\Pi ), \end{aligned}$$
(182)
$$\begin{aligned} F(\xi _1,\xi _2,\Pi )&:=\sum _{j=0}^\infty \frac{\varepsilon ^j}{j!}{\mathcal {U}}_{j+2}(\xi _1,\xi _2,\Pi ,\ldots ,\Pi ). \end{aligned}$$
(183)

We have chosen the letters in this way, because later we will think of P to be Kontsevich’s star product for \(\Pi \) a given Poisson tensor, A as a connection 1-form and F as its curvature. Let us take a look at some of the graphs appearing for some chosen multivector fields. For example, for a bivector field \(\Pi \), we get that the term \(U_{1}(\Pi )\) corresponds to the first graph of Fig. 20, whereas for a multivector field \({\mathcal {V}}\) of degree r we get for \({\mathcal {U}}_1({\mathcal {V}})\) the second graph of Fig. 20. Let now \(\xi \) be a vector field. Note that the number \(\ell \) for \({\mathcal {U}}_n(\xi ,\Pi ,\ldots ,\Pi )\) will always be 1 for every n, which implies that \(A(\xi ,\Pi )\) takes a smooth map f as an argument.

Fig. 20
figure 20

The graphs \({\mathcal {U}}_1(\Pi )\) and \({\mathcal {U}}_1({\mathcal {V}})\)

We want to look at graphs appearing for higher terms in A. We can, e.g., consider the \(n=3\) term, i.e. \({\mathcal {U}}_3(\xi ,\Pi ,\Pi )\). Some example of graphs in \({\mathcal {G}}_{3,1}\), which are taken in account for the sum, are given in Fig. 21.

Fig. 21
figure 21

Example of graphs in \({\mathcal {G}}_{3,1}\)

We can also explicitly say what the differential operator given by a graph will be. E.g. for the graph as in 21 (b) we get

$$\begin{aligned} \partial _{i_1}\partial _{i_3}\xi ^{i_5}\partial _{i_2} \partial _{i_2}\Pi ^{i_3i_4}\partial _{i_5}\Pi ^{i_1i_2}\partial _{i_4}(f). \end{aligned}$$
(184)

By definition of F, for every n we get that \(\ell =0\), i.e. the image of \({\mathcal {U}}_n\) will be a differential operator of degree zero, which is a smooth function. Some examples for graphs in \({\mathcal {G}}_{3,0}\) are given in Fig. 22.

Fig. 22
figure 22

Example of graphs in \({\mathcal {G}}_{3,0}\)

B.2. Notions of formal geometry

We recall the most important notions of formal geometry as in [8, 37] following the presentation as in [16] and [7]. For a smooth manifold \({\mathscr {P}}\) we can consider a formal exponential map \(\varphi \) on \({\mathscr {P}}\), such that for \(x\in {\mathscr {P}}\) we have \(\varphi _x:T_x{\mathscr {P}}\rightarrow {\mathscr {P}}\), and we define a vector field \(R\in \Gamma (T^*{\mathscr {P}}\otimes T{\mathscr {P}}\otimes {\widehat{S}}T^*{\mathscr {P}})\), which is a 1-form with values in derivations of \({\widehat{S}}T^*{\mathscr {P}}\). Here \({\widehat{S}}\) denotes the completed symmetric algebra. In local coordinates we have \(R=R_i{\mathrm {d}}x^{i}\) with

$$\begin{aligned} R_i(x;y) = \left( \left( \frac{\partial \varphi _x}{\partial y}\right) ^{-1}\right) ^k_j\frac{\partial \varphi _x^{j}}{\partial x^{i}}\frac{\partial }{\partial y^k} =: Y^k_i(x;y)\frac{\partial }{\partial y^k} \end{aligned}$$
(185)

Then we can define the classical Grothendieck connection \(D_{\mathsf {G}}:={\mathrm {d}}+R\), which is flat. For a vector field \(\xi =\xi \frac{\partial }{\partial x^{i}}\) we have \(D^\xi _{\mathsf {G}}=\xi +{\widehat{\xi }}\), where

$$\begin{aligned} {\widehat{\xi }}(x;y)=\iota _\xi R(x;y)=\xi ^{i}Y_i^k(x;y)\frac{\partial }{\partial y^k}. \end{aligned}$$
(186)

B.3. Globalization

Now let us describe how to generalize the above procedure to an arbitrary Poisson manifold \(({\mathscr {P}},\Pi )\). Namely, let \(x \in {\mathscr {P}}\), and \(\varphi \) a formal exponential map on \({\mathscr {P}}\). Then \({\mathsf {T}}\varphi _x^*\Pi \), the Taylor expansion of \(\Pi \) around x defined using \(\varphi \), is a Poisson tensor on \(\widehat{S}T^*_x{\mathscr {P}}\). Any choice of coordinates on \(T_xM\) now allows us to identify \(\widehat{S}T^*_x{\mathscr {P}} \cong {\mathbb {R}}[[y_1,\ldots ,y_d]]\) and define Kontsevich’s star product \(P({\mathsf {T}}\varphi ^*_x\Pi )\). See [19] for a discussion of the equivariance of this construction in the choice of coordinates. In this way we get a new bundle \({\mathcal {E}}:= \widehat{S}T^*{\mathscr {P}}[[\varepsilon ]]\) of \(\star \)-algebras. One can use the Grothendieck connection defined in B.2 to give a description of a subalgebra \({\mathcal {A}}\subset \Gamma ({\mathcal {E}})\) which is a deformation quantization of \(C^\infty ({\mathscr {P}})\) seen as a subalgebra of \(\Gamma ({\mathcal {E}})\). Formally we have

$$\begin{aligned} \Gamma ({\mathcal {E}})\supset C^\infty ({\mathscr {P}}) \xrightarrow {\textsf {Deformation Quantization}}&{\mathcal {A}}\subset \Gamma ({\mathcal {E}}). \end{aligned}$$
(187)

The algebra \({\mathcal {A}}\) is given by closed sections under a deformation of the Grothendieck connection, which is defined in two steps: For a tangent vector \(\xi \in T_x{\mathscr {P}}\), we let

$$\begin{aligned} {\mathcal {D}}_{\mathsf {G}}^\xi :=\xi +A\left( {\widehat{\xi }}, {\mathsf {T}}\varphi ^*_x\Pi \right) = D_{\mathsf {G}}^\xi +O(\varepsilon ), \end{aligned}$$
(188)

where again we denote by \({\mathsf {T}}\varphi ^*_x\Pi \) the Poisson tensor \(\Pi \) lifted to a formal neighborhood and \(\widehat{\xi }\) is defined as in (186). One can write

$$\begin{aligned} {\mathcal {D}}_{{\mathsf {G}}} = {\mathrm {d}}+ A(R,{\mathsf {T}}\varphi ^*\Pi ) \end{aligned}$$
(189)

interpreting \(A(R,{\mathsf {T}}\varphi ^*\Pi )\) as a one-form valued in differential operators on \({\mathcal {E}}\). At some point \(x \in {\mathscr {P}}\), in coordinates \(x^i\) around x, it is given by

$$\begin{aligned} A(R,{\mathsf {T}}\varphi _x^*\Pi ) = {\mathrm {d}}x^i A(R_i(x;y),{\mathsf {T}}\varphi ^*_x\Pi ) = {\mathrm {d}}x^i A\left( Y_i^k(x;y)\frac{\partial }{\partial y^k},{\mathsf {T}}\varphi ^*_x\Pi \right) . \end{aligned}$$
(190)

One can then show [19] that \({\mathcal {D}}_{\mathsf {G}}\) is a globally defined connection on \(\Gamma ({\mathcal {E}})\), a derivation, and that \(({\mathcal {D}}_{\mathsf {G}})^2\) is an inner derivation, i.e.

$$\begin{aligned} ({\mathcal {D}}_{\mathsf {G}})^2\sigma =[F^{\mathscr {P}},\sigma ]_\star :=F^{\mathscr {P}}\star \sigma -\sigma \star F^{\mathscr {P}}, \end{aligned}$$
(191)

for any \(\sigma \in \Gamma ({\mathcal {E}})\), where \({F^{\mathscr {P}}}\) is the Weyl curvature tensor of \({\mathcal {D}}_{\mathsf {G}}\) given by \(F^{\mathscr {P}}(\xi _1,\xi _2):=F({\widehat{\xi }}_1,{\widehat{\xi }}_2,{\mathsf {T}}\varphi ^*\Pi )\), where \(\xi _1,\xi _2\in T_x{\mathscr {P}}\) are two tangent vectors on \({\mathscr {P}}\). More, precisely, \(F^{\mathscr {P}}\) is a 2-form valued in sections of \({\mathcal {E}}\) which in local coordinates can be expressed as

$$\begin{aligned} F^{\mathscr {P}}_x = {\mathrm {d}}x^i \wedge {\mathrm {d}}x^j F(R_i(x;y),R_j(x;y),{\mathsf {T}}\varphi ^*_x\Pi ). \end{aligned}$$
(192)

For the Weyl tensor we get \({\mathcal {D}}_{\mathsf {G}}F^{\mathscr {P}}=0\). The task is to modify the globalized connection \({\mathcal {D}}_{\mathsf {G}}\) slightly more, so that it becomes flat but still remaining a derivation. One can setFootnote 25

$$\begin{aligned} \overline{{{\mathcal {D}}}}_{\mathsf {G}}:={\mathcal {D}}_{\mathsf {G}} +[\gamma ,]_\star , \end{aligned}$$
(193)

and observe that for any 1-form \(\gamma \in \Omega ^1({\mathscr {P}},{\mathcal {E}})\) this connection is a derivation. Moreover, its Weyl curvature tensor is then given by

$$\begin{aligned} {\overline{F}^{\mathscr {P}}}=F^{\mathscr {P}}+{\mathcal {D}}_{\mathsf {G}}\gamma +\gamma \star \gamma . \end{aligned}$$
(194)

We call (188) the deformed Grothendieck connection and (193) the modified deformed Grothendieck connection. One then needs to find \(\gamma \in \Omega ^1({\mathscr {P}},{\mathcal {E}})\) such that \(\overline{F}^{\mathscr {P}}=0\), which implies that \((\overline{{\mathcal {D}}}_{\mathsf {G}})^2=0\), so that \(\overline{{{\mathcal {D}}}}_{\mathsf {G}}\)-closed sections will form the algebra \({\mathcal {A}}\) as a deformation quantization of \(C^\infty ({\mathscr {P}})\). If we compute \((\overline{{{\mathcal {D}}}}_{\mathsf {G}})^2\) explicitly, by using (193) we get

$$\begin{aligned} (\overline{{{\mathcal {D}}}}_{\mathsf {G}})^2= \underbrace{({\mathcal {D}}_{\mathsf {G}})^2}_{:=[F^{\mathscr {P}},]_\star } +{\mathcal {D}}_{\mathsf {G}}[\gamma ,]_\star +[\gamma ,[\gamma ,]_\star ]_\star . \end{aligned}$$
(195)

More precisely, \(\gamma \) has to satisfy

$$\begin{aligned} F^{\mathscr {P}}+{\mathcal {D}}_{\mathsf {G}}\gamma +\gamma \star \gamma =0. \end{aligned}$$
(196)

The existence of such a \(\gamma \) was shown in [16, 19] by homological perturbation theory. One can actually construct \(\gamma \) to be a solution of the more general equation given by

$$\begin{aligned} \overline{F}^{\mathscr {P}}_\omega =F^{\mathscr {P}} +\varepsilon \omega +{\mathcal {D}}_{\mathsf {G}}\gamma +\gamma \star \gamma =0, \end{aligned}$$
(197)

where \(\omega \in \Omega ^2({\mathscr {P}},{\mathcal {E}})\) such that \({\mathcal {D}}_{{\mathsf {G}}}\omega =0\) and \([\omega ,]_\star =0\) [19].

Now we want to focus on some special cases. We want to look at two important examples of Poisson structures.

B.3.1. Constant Poisson structure

The situation of a constant Poisson structure is a first example to think about. Let \(({\mathscr {P}},\Pi )\) be a Poisson manifold with constant Poisson structure \(\Pi \) and \(\xi \in T_x{\mathscr {P}}\) for \(x\in {\mathscr {P}}\) be a fixed tangent vector. By the definition of A, and the fact that each vertex has only one outgoing and no incoming arrow, we get \(A({\widehat{\xi }},{\mathsf {T}}\varphi ^*\Pi )={\widehat{\xi }}\), which leads to the fact that

$$\begin{aligned} {\mathcal {D}}^\xi _{\mathsf {G}}=(\xi +{\widehat{\xi }}) = D_{\mathsf {G}}^{\xi }. \end{aligned}$$
(198)

Therefore we get \(({\mathcal {D}}_{\mathsf {G}})^2=0\) and thus \(F^{\mathscr {P}}=0\). We can then choose \(\gamma =0\).

B.3.2. Linear Poisson structure

Let now \(({\mathscr {P}}={\mathfrak {g}}^*,\Pi )\) be a Poisson manifold with linear Poisson structure \(\Pi (x)=\Pi _k^{ij}x^k\frac{\partial }{\partial x^{i}}\wedge \frac{\partial }{\partial x^j}\), where \(\Pi ^{ij}_k\) represent the structure constants of a Lie algebra \({\mathfrak {g}}\), and \(\xi \in T_x{\mathscr {P}}\) for \(x\in {\mathscr {P}}\) be a fixed tangent vector. As in the constant case, we observe that \(A({\widehat{\xi }},{\mathsf {T}}\varphi ^*\Pi )={\widehat{\xi }}\), which is the case since the integral of a bulk vertex with one incoming and one outgoing arrow is zero, and since there is at most one incoming arrow for each vertex. Again we may choose \(\gamma =0\).

B.4. Connection to the Poisson Sigma Model

In [13] and [17] it was shown that Kontsevich’s formality map on \({\mathbb {R}}^d\) can be intepreted as the perturbative computation of expectation values of observables of the Poisson Sigma Model on the upper half plane (or respectively the disk) with values in \({\mathbb {R}}^d\). The graphs which appear in the construction of Kontsevich’s star product on Poisson manifolds [45] are given on the upper half plane, where they can collapse, according to the boundary of the configuration space, on the boundary of the upper half plane. This means that the graphs that appear in the Poisson Sigma Model are exactly the graphs that appear for Kontsevich’s star product. More precisely, if one considers the disk D in \({\mathbb {R}}^2\) and the classical action of the Poisson Sigma Model on D given by \(S_D[(X,\eta )]=\int _D\left( \langle \eta ,{\mathrm {d}}X\rangle +\frac{1}{2}\langle \Pi (X),\eta \wedge \eta \rangle \right) \), we can asymptotically write Kontsevich’s star product for two smooth maps f and g as a perturbative expansion of the following path integral:

$$\begin{aligned} f\star g(x)=\int _{X(\infty )=x}f(X(0))g(X(1))\text {e}^{\frac{\mathrm{i}}{\hbar }S_D[(X,\eta )]}, \end{aligned}$$
(199)

where \(0,1,\infty \) represent some marked points on the boundary of D. Note that \(x\in {{\,\mathrm{Map}\,}}(D,{\mathbb {R}}^d)\) is a constant map, i.e. the we get a local representation of the star product. If one considers a general Poisson manifold \(({\mathscr {P}},\Pi )\), one can consider the constant map \(x\in {{\,\mathrm{Map}\,}}(D,{\mathscr {P}})\) as a point sitting in \({\mathscr {P}}\) giving a local product on each fiber. As already described in B.3, one can then algebraically construct the star product on all of \({\mathscr {P}}\).

Appendix C. On the Propagator

We have an explicit propagator for the Poisson Sigma Model, i.e. using the superfields of it, on a disk with alternating boundary conditions, which was computed in [14], in [26] and, in full generality, in [31].

C.1. Construction of the branes

Consider an n-sided polygon \(P_n=u(\mathbb {H}^+)\) where \(u:\mathbb {H}^+\rightarrow P_n\) is a suitable homeomorphism between the compactified complex upper half plane \(\mathbb {H}^+\) and \(P_n\), depending on the number of the branes considered. Let \(G_{S_i}\), be the relevant superpropagators for the Poisson Sigma Model with n branes defined by constraints \(C_j=\{x^{\mu _j}=0\mid \mu _j\in I_j\}\) (also called branes) and index sets \(S_1=I_1^C\cap I_2\cap I_3^C\cap \cdots \cap I_n\), \(S_2=I_1\cap I_2^C\cap I_3\cap \cdots \cap I_n^C\) for n even, and \(S_1=I_1^C\cap I_2\cap I_3^C\cap \cdots \cap I_n^C\), \(S_2=I_1\cap I_2^C\cap \cdots \cap I_n\) for n odd, which are called relevant. It turns out that the \(C_i\subset {\mathscr {P}}\) are coisotropic submanifolds of \({\mathscr {P}}\) [14].

C.2. Constructing integral kernels

The integral kernels \(\theta (Q,P)_{S_i}:=-\frac{\mathrm{i}}{\hbar }\langle \widehat{{\mathsf {X}}}^\bullet (Q)\widehat{\varvec{\eta }}_\bullet (P)\rangle \) for the two brane case are given by:

$$\begin{aligned} \theta (Q;P)_{S_i}&=\frac{1}{2\pi }{\mathrm {d}}\arg \frac{(u-v)({\bar{u}}-v)}{({\bar{u}}+v)(u+v)}, \end{aligned}$$
(200)
$$\begin{aligned} \theta (Q,P)_{S_2}&=\frac{1}{2\pi }{\mathrm {d}}\arg \frac{(u-v)({\bar{u}}+v)}{(\bar{u}-v)(u+v)}, \end{aligned}$$
(201)

where \(P_2:=u(\mathbb {H}^+)\) with \(u(z)=\sqrt{z}\), \(v:=u(w)\), \({\mathrm {d}}={\mathrm {d}}_u+{\mathrm {d}}_v\). We identify (PQ) with the couple (uv). Consider e.g. \(P_2\) to be the worldsheet disk \(\Sigma \) with boundary \(\partial \Sigma =\bigsqcup _{1\le j\le 6}J_j\) (we denote the intervals here by J instead of I such that there is no confusion with the index sets) and the branes \(C_1=\{x^{\mu _1}=0\mid \mu _1\in I_1=\{1,\ldots ,n\}\}\) and \(C_2=\{x^{\mu _2}=0\mid \mu _2\in I_2=\varnothing \}\), which correspond to the boundary conditions of \(\partial _1\Sigma \) and \(\partial _2^{\text {tot}}\Sigma \) respectively. The components \(\partial _1\Sigma \) and \(\partial _2^{\text {tot}}\Sigma \) are such that \(\partial \Sigma =\partial _1\Sigma \sqcup \partial _2^{\text {tot}}\Sigma \), where \(\partial _1\Sigma \) is chosen to be some \(J_1\) endowed with the \(\frac{\delta }{\delta \mathbb {E}}\)-polarization and \(\partial _2^\text {tot}\Sigma =\bigsqcup _{2\le j\le 6} J_j\) such that \(J_j\) is endowed with the \(\frac{\delta }{\delta \mathbb {X}}\)-polarization and with the boundary condition \(\widehat{\varvec{\eta }}\equiv 0\) for j odd and even respectively. Now we get \(S_1=I_1^C\cap I_2=\varnothing \) and \(S_2=I_1\cap I_2^C=\{1,\ldots ,n\}\). Now \(P_2\) is defined by \(P_2=u(\mathbb {H}^+)\), where u is the map \(z\mapsto \sqrt{z}\). Points \((P,Q)\in P_2\times P_2\) are represented respectively by a pair of complex numbers (uv) in the first quadrant, with \(u=u(z)\), \(v=u(w)\) for all \((z,w)\in \mathbb {H}^+\times \mathbb {H}^+\). The boundary \(\partial _1 P_2\) (corresponding to \(\partial _1\Sigma \)) is given by the positive imaginary axis, while \(\partial _2P_2\) (corresponding to \(\partial _2^{\text {tot}}\Sigma \)) is given by the positive real axis.

C.3. Construction of superpropagators

The boundary conditions imposed by the index sets \(S_i\) are \(\theta (v,u\in \partial _1P_2)_{S_1}=\theta (u\in \partial _2P_2,u)_{S_1}=0\), \(\theta (v,u\in \partial _2P_2)_{S_2}=\theta (v\in \partial _1P_2,u)_{S_2}=0\). Let

$$\begin{aligned} \psi (u,v)_{S_1}&=\arg \frac{(u-v)({\bar{u}}-v)}{({\bar{u}}+v)(u+v)}, \end{aligned}$$
(202)
$$\begin{aligned} \psi (u,v)_{S_2}&=\arg \frac{(u-v)({\bar{u}}+v)}{(\bar{u}-v)(u+v)}, \end{aligned}$$
(203)

which satisfy the same boundary conditions as \(\theta (v,u)_{S_i}\). Now for vanishing cohomology, we get the following Theorem.

Theorem C.1

The integral kernels for the superpropagators \(G_{S_i}\) in presence of two branes are given by

$$\begin{aligned} \theta (v,u)_{S_i}=\frac{1}{2\pi }{\mathrm {d}}\psi (u,v)_{S_i}, \end{aligned}$$
(204)

with angle maps (202) and (203). The integral kernels satisfy the additional boundary conditions \(\theta (v,u)_{S_1}=\theta (v,{\bar{u}})=\theta (-{\bar{v}},u)_{S_1}\), \(\theta (v,u)_{S_2}=\theta (v,-{\bar{u}})_{S_2}=\theta ({\bar{v}},u)_{S_2}\), i.e. every boundary component of \(P_2\) is labeled by a boundary condition for both the variables (uv). By construction \(\theta (v,u)_{S_1}=\theta (u,v)_{S_2}\), \(\theta (v,u)_{S_2}=\theta (u,v)_{S_1}\).

C.4. Relation to Kontsevich’s propagator

Let \(\phi \) be Kontsevich’s angle 1-form. Then, one can show that

$$\begin{aligned} \theta (v,u)_{{\mathcal {A}}_1}&=\frac{1}{2\pi }{\mathrm {d}}\arg \frac{(u-v)(u+v)}{(u+{\bar{v}})(u-{\bar{v}})}=\frac{1}{2\pi } {\mathrm {d}}\arg \frac{(z-w)}{(z-{\bar{w}})}=\frac{1}{2\pi }{\mathrm {d}}\phi (z,w), \end{aligned}$$
(205)
$$\begin{aligned} \theta (v,u)_{{\mathcal {A}}_2}&=\frac{1}{2\pi }{\mathrm {d}}\arg \frac{(u-v)(u+v)}{({\bar{u}}- v)({\bar{u}}+ v)}=\frac{1}{2\pi }{\mathrm {d}}\arg \frac{(z-w)}{({\bar{z}}- w)}=\frac{1}{2\pi }{\mathrm {d}}\phi (w,z), \end{aligned}$$
(206)

where \({\mathcal {A}}_1=I_2\cap I_2\) and \({\mathcal {A}}_2=I_1^C\cap I_2^C\).

Glossary

\(\mathrm{i}\) :

Imaginary unit

M :

A finite-dimensional manifold

\(F_M\) :

Space of fields

\(S_M\) :

Local action functional

\(\Delta \) :

Global BV Laplacian on half-densitites

\({\mathcal {F}}_M\) :

BV space of fields

\(\omega _M\) :

BV symplectic form

\({\mathcal {S}}_M\) :

BV action functional

\(\omega ^\partial \) :

BFV symplectic form

\({\mathcal {F}}^\partial _M\) :

BFV space of boundary fields

\({\mathcal {S}}^\partial \) :

BFV boundary action

\(\Omega _{\partial M}\) :

BFV boundary operator

\({\mathcal {H}}_{\partial M}\) :

Space of boundary states

\(\psi _M\) :

Quantum state

\(\widehat{{\mathcal {H}}}_M\) :

The space \({\text {Dens}^\frac{1}{2}({\mathcal {V}}_M)\otimes {\mathcal {H}}_{\partial M}}\)

\({\mathcal {V}}_M\) :

Space of residual fields

\(\text {Dens}^{\frac{1}{2}}(M)\) :

Half-densities on a manifold M

\({\mathcal {P}}\) :

Polarization of \({\mathcal {F}}^\partial _M\)

\({\mathcal {B}}^{\mathcal {P}}_{\partial M}\) :

Leaf space of the polarization \({\mathcal {P}}\)

\({\mathcal {H}}^{\mathcal {P}}_{\partial M}\) :

Full space of boundary states

\({\mathcal {H}}^{{\mathcal {P}},\text {princ}}_{\partial M}\) :

Principal space of boundary states

\(\mathsf {Conf}_\Gamma (M)\) :

Configuration space

\(\mathsf {C}_\Gamma (M)\) :

Compactified configuration space

\(\varvec{\psi }_M\) :

Full quantum state

\(\Omega ^{\mathrm{princ}}\) :

Principal part of the BFV boundary operator

\(\Omega ^\mathbb {X}_0\) :

\(\mathbb {X}\)-part of the free BFV boundary operator

\(\Omega ^\mathbb {E}_0\) :

\(\mathbb {E}\)-part of the free BFV boundary operator

\(\Omega ^\mathbb {X}_\text {pert}\) :

\({\mathbb {X}}\)-part of the perturbative BFV boundary operator

\(\Omega ^\mathbb {E}_\mathrm{pert}\) :

\({\mathbb {E}}\)-part of the perturbative BFV boundary operator

\(\varvec{\Omega }_{\partial M}\) :

Full BFV boundary operator

\(\varphi \) :

Formal exponential map

\(\mathsf {T}\) :

Taylor expansion

\(\widehat{S}\) :

Completed symmetric algebra

\(D_\mathsf {G}\) :

Classical Grothendieck connection

R :

Vector field in the definition of \({D_\mathsf {G}}\)

x :

Constant background field

\(\Sigma \) :

Worldsheet manifold with boundary in the Poisson Sigma Model

\(\widetilde{{\mathcal {S}}}_{\Sigma ,x}\) :

Formal globalized action

\(\widehat{{\mathsf {X}}}\) :

Lift of \({\mathsf {X}}\) by \({\varphi _x}\)

\(\widehat{\varvec{\eta }}\) :

Lift of \({\varvec{\eta }}\) by \({{\mathrm {d}}\varphi _x}\)

\(\widetilde{\psi }_{\Sigma ,x}\) :

Principal covariant quantum state

\(\varvec{\widetilde{\psi }}_{\Sigma ,x}\) :

Full covariant quantum state

\(\nabla _\mathsf {G}\) :

Quantum Grothendieck BFV operator

\(\mathscr {P}\) :

Poisson manifold

\(\Pi \) :

Poisson structure

X :

Base map of the Poisson Sigma Model

\(\eta \) :

Fiber map of the Poisson Sigma Model

\(\mathsf {X}\) :

Superfield version of X

\(\varvec{\eta }\) :

Superfield version of \({\eta }\)

\(\mathbb {X}\) :

Boundary field part of \({\mathsf {X}}\)

\(\mathbb {E}\) :

Boundary field part of \({\varvec{\eta }}\)

\(\mathsf {x}\) :

Residual field part of \({\mathsf {X}}\)

\(\mathsf {e}\) :

Residual field part of \({\varvec{\eta }}\)

\(\mathscr {X}\) :

Fluctuation field part of \({\mathsf {X}}\)

\(\mathscr {E}\) :

Fluctuation field part of \({\varvec{\eta }}\)

\(\star \) :

Kontsevich’s star product

\({[},{]}_\star \) :

Star commutator

\(\widetilde{\varvec{\Omega }}^{\mathbb {E}}_{\partial \Sigma }\) :

\({\mathbb {E}}\)-part of the globalized full BFV boundary operator

\(\mathcal {D}_\mathsf {G}\) :

Deformed Grothendieck connection

A :

Connection term coming from Kontsevich’s formality map

F :

Curvature term coming from Kontsevich’s formality map

\(\widetilde{\varvec{\Omega }}^{\mathbb {X}}_{\partial \Sigma }\) :

\({\mathbb {X}}\)-part of the globalized full BFV boundary operator

\(\gamma \) :

A solution to \({F^\mathscr {P}+{\mathcal {D}}_\mathsf {G}\gamma +\gamma \star \gamma =0}\)

\(\overline{\mathcal {D}}_\mathsf {G}\) :

The flat connection \({{\mathcal {D}}_\mathsf {G}+[\gamma ,]_\star }\)

\({\mathcal {E}}\) :

The deformed bundle \({\widehat{S}T^*\mathscr {P}[[\varepsilon ]]}\)

\(\varepsilon \) :

Formal deformation parameter

\(\omega \) :

A \({\star }\)-central \({{\mathcal {D}}_\mathsf {G}}\)-closed 2-form

\(\mathbb {S}_{\Sigma ,x}\) :

Modified formal globalized action

\({\mathcal {S}}_{\Sigma ,\gamma }\) :

\({\gamma }\)-action term

\({\mathcal {S}}_{\Sigma ,\omega }\) :

\({\omega }\)-action term

\(\widetilde{\varvec{\Omega }}^{\mathbb {E},\gamma }_{\partial \Sigma }\) :

Twisted \({\mathbb {E}}\)-part of the full BFV boundary operator

\(\mathscr {C}\) :

Set of all corner points of \({\Sigma }\)

\(\nabla ^\gamma _\mathsf {G}\) :

Twisted quantum Grothendieck BFV operator

\({\mathcal {H}}_C\) :

Space of corner states

\(\widehat{{\mathcal {H}}}^\mathscr {C}_{\partial \Sigma ,x}\) :

Extended state space

\(\widehat{{\mathcal {H}}}^\mathscr {C}_{\partial \Sigma ,\text {tot}}\) :

Total extended state space

\(\varvec{\Omega }_\mathscr {C}\) :

Corner contribution of the full BFV boundary operator

\(\widetilde{\nabla }^\gamma _\mathsf {G}\) :

Twisted quantum Grothendieck BFV operator for corners

\(F^\mathscr {P}\) :

Weyl curvature tensor of \({{\mathcal {D}}_\mathsf {G}}\)

\(\overline{F}^\mathscr {P}\) :

Weyl curvature tensor of \({\overline{{\mathcal {D}}}_\mathsf {G}}\)

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Cattaneo, A.S., Moshayedi, N. & Wernli, K. On the Globalization of the Poisson Sigma Model in the BV-BFV Formalism. Commun. Math. Phys. 375, 41–103 (2020). https://doi.org/10.1007/s00220-020-03726-z

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