BV and BFV for the H-Twisted Poisson Sigma Model

Abstract

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using an auxiliary connection $\nabla$ on the target manifold M, we obtain alternative, purely geometrical expressions without the use of superfields, which are new also for the ordinary PSM and promise adaptations to other Lie algebroid-based gauge theories: The BV functional, in particular, is the sum of the classical action, the Hamiltonian lift of the (only on-shell nilpotent) BRST differential, and a term quadratic in the antifields which is essentially the basic curvature and measures the compatibility of $\nabla$ with the Lie algebroid structure on $T^*M$. We finally construct a $\hbox {Diff}(M)$-equivariant isomorphism between the two BV formulations.

Appendices

BV and BFV for the Poisson Sigma Model

The classical action of the Poisson sigma model [2, 3] results from putting to zero the 3-form H in the twisted generalization (2.1) studied in the present paper:

$$\begin{aligned} S = \int _{\Sigma } \left( A_i \wedge \mathrm {d}X{}^i + \tfrac{1}{2} \pi ^{ij}(X{}) A_i \wedge A_j \right) . \end{aligned}$$

(A.1)

It agrees with (the classical part of) the AKSZ model [8] in two dimensions [10]. In this appendix, we recall the BV formulation [9] of the action (A.1)—which is the starting point in the AKSZ approach—as well as the BFV formulation of the PSM.

1.1 Recollection of BV for the PSM

The BV symplectic form is

$$\begin{aligned} \omega _{\mathrm{BV}}= & {} \int _{T[1]\Sigma } d^2r\sigma d^2\theta \, \delta \varvec{X}^i \delta \varvec{A}_i, \end{aligned}$$

(A.2)

where $\varvec{X}^i$ and $\varvec{A}_i$ can be viewed as unconstrained superfields on $T[1]\Sigma$ of total degree zero and one, respectively, see (5.9) and (5.10) in the main text.

The BV action takes the form of the classical action (A.1), written in terms of the superfields:

$$\begin{aligned} S_{\mathrm{BV}} = \int _{T[1]\Sigma } d^2\sigma d^2\theta \, \left( \varvec{A}_i \varvec{\mathrm {d}}\varvec{X}^i + \tfrac{1}{2} \pi ^{ij}(\varvec{X}) \varvec{A}_i \varvec{A}_j \right) . \end{aligned}$$

(A.3)

Expanded in terms of the original fields, the ghost fields, and their antifields, this becomes:¹⁸

$$\begin{aligned} S_{\mathrm{BV}}= & {} \!\!\! \int _{\Sigma } \biggl (A_i \wedge \mathrm {d}X{}^i + \tfrac{1}{2} \pi ^{ij}(X{}) A_i \wedge A_j - \pi ^{ij}(X{}) X{}^{+}_i c_j + A^{+ i} \wedge \left( \mathrm {d}c_{i} + \pi ^{jk},_i(X{}) A_j c_k \right) \nonumber \\&+ \tfrac{1}{2} \pi ^{ij},_k(X{}) c^{+k} c_i c_j - \tfrac{1}{4} \pi ^{ij},_{kl}(X{}) A^{+k} \wedge A^{+l} c_i c_j \biggr ). \end{aligned}$$

(A.4)

In the AKSZ formalism, it is evident—by its very construction—that this action satisfies the classical master equation $(S_{\mathrm{BV}},S_{\mathrm{BV}})=0$: Indeed, the AKSZ sigma model can be seen as the Hamiltonian lift of the sum of two commuting differentials to the mapping space $\underline{\mathrm {Hom}}(T[1] \Sigma ,T^*[1]M)$, one being the de Rham differential on the source space $T[1]\Sigma$ and the other one the Lie algebroid differential on target space $T^*[1]M$. Note that this construction cannot be applied as such to the H-twisted Lie algebroid structure on $T^*[1]M$ since, for $H\ne 0$, the corresponding nilpotent vector field Q is not compatible with the symplectic form on $T^*[1]M$; this, however, is needed for the Hamiltonian lift in the AKSZ formalism, see [8, 10,11,12].

1.2 BFV for the PSM

The BFV formulation [3] of the PSM permits a superfield description similar to the one above, too. The BFV symplectic form can be written as

$$\begin{aligned} \omega _{\mathrm{BFV}}= & {} \int _{T[1]S^1} d \sigma d \theta \, \delta {\widetilde{X}}{}^i \wedge \delta {\widetilde{A}}{}_i. \end{aligned}$$

(A.5)

In terms of the expansion (3.8), this gives rise to the following Poisson brackets for the fields and ghosts:

$$\begin{aligned} \{X{}^i(\sigma ) p_{j}(\sigma ^{\prime })\}= & {} \delta ^i{}_j\delta (\sigma - \sigma ^{\prime }), \end{aligned}$$

(A.6)

$$\begin{aligned} \{c_j(\sigma ) b^i(\sigma ^{\prime })\}= & {} \delta ^i{}_j\delta (\sigma - \sigma ^{\prime }). \end{aligned}$$

(A.7)

Likewise, also the Hamiltonian BFV or BRST charge permits a compact description in terms of these superfields:

$$\begin{aligned} S_{\mathrm{BFV}} = \int _{T[1]S^1} d \sigma d \theta \, \left( {\widetilde{A}}{}_i {\mathrm {d}}{\widetilde{X}}{}^i + \tfrac{1}{2} \pi ^{ij}({\widetilde{X}}{}) {\widetilde{A}}{}_i {\widetilde{A}}{}_j \right) , \end{aligned}$$

(A.8)

which, when decomposed into its components takes the form:

$$\begin{aligned} S_{\mathrm{BFV}}= & {} \int _{S^1} d \sigma \left( c_i \partial X{}^i + \pi ^{ij}(X{}) c_i p_{j} + \tfrac{1}{2} \pi ^{ij},_k(X{}) b^k c_i c_j \right) . \end{aligned}$$

(A.9)

It satisfies $\{S_{\mathrm{BFV}}, S_{\mathrm{BFV}}\} =0$. The BFV symplectic form and the BFV-BRST charge are just a one-dimensional superextension of the classical symplectic form (on the cotangent space of loop space) and the classical action, respectively.

Naive Generalization of AKSZ Does not Work for the HPSM

We already explained at the end of Sect. A.1, why—at least in its direct, unmodified form—the AKSZ procedure cannot be applied to the HPSM for non-vanishing H; in fact, the PSM is the most general AKSZ sigma model for a two-dimensional choice of $\Sigma$, see, e.g., [11]. One may still hope that there may be some elegant superfield formalism that yields the BV theory of the HPSM without much explicit work. We want to show here that, at least in the most direct way, this is not the case.

Rewriting the classical symplectic form (2.27) of the HPSM and its classical action (2.1) in terms of superfields on $T[1]\Sigma$ as before, one would arrive at a BV symplectic form

$$\begin{aligned} \omega _{\mathrm{BV}}= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \, \left( \delta \varvec{X}^i \delta \varvec{A}_i + \tfrac{1}{2} H_{ijk}(\varvec{X}) \varvec{\mathrm {d}}\varvec{X}^k \delta \varvec{X}^i \delta \varvec{X}^j \right) , \end{aligned}$$

(B.1)

and a BV action

$$\begin{aligned} S_{\mathrm{BV}}= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \ \left( \varvec{A}_i \varvec{\mathrm {d}}\varvec{X}^i + \tfrac{1}{2} \pi ^{ij}(\varvec{X}) \varvec{A}_i \varvec{A}_j \right) \nonumber \\&+ \int _{T[1]N} d^3 \sigma d^3 \theta \ \tfrac{1}{3!} H_{ijk}(\varvec{X}) \varvec{\mathrm {d}}\varvec{X}^i \varvec{\mathrm {d}}\varvec{X}^j \varvec{\mathrm {d}}\varvec{X}^k. \end{aligned}$$

(B.2)

However, this naive candidate of the BV action does not satisfy the classical master equation. In fact, with the induced BV brackets

$$\begin{aligned} \left( \varvec{X}^i(\sigma , \theta ),\varvec{A}_j(\sigma ^{\prime }, \theta ^{\prime })\right)= & {} \delta ^i{}_j \delta ^2(\sigma - \sigma ^{\prime }) \delta ^2(\theta - \theta ^{\prime }), \end{aligned}$$

(B.3)

$$\begin{aligned} \left( \varvec{A}_i(\sigma , \theta ),\varvec{A}_j(\sigma ^{\prime }, \theta ^{\prime })\right)= & {} - H_{ijk}(\varvec{X}) \varvec{\mathrm {d}}\varvec{X}^k \delta ^2(\sigma - \sigma ^{\prime }) \delta ^2(\theta - \theta ^{\prime }), \end{aligned}$$

(B.4)

one obtains

$$\begin{aligned} \left( S_{\mathrm{BV}},S_{\mathrm{BV}}\right)= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \ \left( \pi ^{kl} H_{ijl} \varvec{\mathrm {d}}\varvec{X}^i \varvec{\mathrm {d}}\varvec{X}^j \varvec{A}_k - \pi ^{jm} \pi ^{kn} H_{imn} \varvec{\mathrm {d}}\varvec{X}^i \varvec{A}_j \varvec{A}_k \right. \nonumber \\&\left. + \tfrac{1}{3} \pi ^{il} \pi ^{jm} \pi ^{kn} H_{lmn} \varvec{A}_i \varvec{A}_j \varvec{A}_k \right) \end{aligned}$$

(B.5)

$$\begin{aligned}= & {} - \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \ \tfrac{1}{3} H_{ijk} ( \varvec{\mathrm {d}}\varvec{X}^i + \pi ^{il} \varvec{A}_l)( \varvec{\mathrm {d}}\varvec{X}^j + \pi ^{jm} \varvec{A}_m) ( \varvec{\mathrm {d}}\varvec{X}^k + \pi ^{kn} \varvec{A}_n).\nonumber \\ \end{aligned}$$

(B.6)

This is zero only for $H = 0$. Note that the second expression shows that the master equation holds true on-shell (in some sense), i.e., upon usage of the obvious superfield extension of the field Eq. (2.2). A similar statement also follows if one drops the H contribution to (B.1), taking $\omega _{\mathrm{BV}}$ to have Darboux form, the choice made in [14]. The on-shell validity of the master equation is, however, not sufficient for the BV formulation. In fact, this is precisely the reason imposing the use of the more elaborate BV formalism over its simpler BRST version (see also the discussion around (4.10)).

Let us try a more general ansatz. Suppose that one is not sure of how to distribute the 3-form H between the BV symplectic form and the BV action (see also “Appendix C” for a related question). We thus introduce two parameters $\alpha ,\beta \in {{\mathbb {R}}}$ and attempt to use

$$\begin{aligned} \omega _{\mathrm{BV}}= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \, \left( \delta \varvec{X}^i \delta \varvec{A}_i + \tfrac{\alpha }{2} H_{ijk}(\varvec{X}) \varvec{\mathrm {d}}\varvec{X}^k \delta \varvec{X}^i \delta \varvec{X}^j \right) , \end{aligned}$$

(B.7)

as the BV symplectic form and

$$\begin{aligned} S_{\mathrm{BV}}&= \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \ \left( \varvec{A}_i \varvec{\mathrm {d}}\varvec{X}^i + \tfrac{1}{2} \pi ^{ij}(\varvec{X}) \varvec{A}_i \varvec{A}_j + \tfrac{\beta }{2} B_{ij}(\varvec{X}) \varvec{\mathrm {d}}\varvec{X}^i \varvec{\mathrm {d}}\varvec{X}^j \right) , \end{aligned}$$

(B.8)

as the BV action. Here, we assumed, for simplicity, that $H = \mathrm {d}B$. The previous ansatz is reproduced for $\alpha =\beta =1$, the considerations of [14] correspond to $\alpha =0$ and $\beta =1$, but one might still hope that for some other choice of parameters there is some decisive cancellation. However, a direct calculation yields

$$\begin{aligned} \left( S_{\mathrm{BV}},S_{BV}\right)= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \ \left( \pi ^{ij},_l \pi ^{lk} \varvec{A}_i \varvec{A}_j \varvec{A}_k \right. \nonumber \\&\left. - \alpha \pi ^{jm} \pi ^{kn} H_{imn} \varvec{\mathrm {d}}\varvec{X}^i \varvec{A}_j \varvec{A}_k + (2 \alpha - \beta ) \pi ^{kl} H_{ijl} \varvec{\mathrm {d}}\varvec{X}^i \varvec{\mathrm {d}}\varvec{X}^j \varvec{A}_k \right) .\nonumber \\ \end{aligned}$$

(B.9)

This calculation holds true for every choice of the bivector field $\pi$ and the 3-form $H=\mathrm {d}B$, not necessarily restricted by means of the twisted Poisson condition (1.1)—which was used to obtain (B.5B.6), to which it then reduces for the initial choice of constants.

We see that, for $H \ne 0$, (B.9) vanishes if and only if $\alpha = 0$, $\beta = 0$, and $\pi$ Poisson. But in this case, H drops out from both (B.7) and (B.8) and we only reproduce the BV formulation of the untwisted Poisson sigma model (see “Appendix A.1”).

In terms of superfields as above and for the case that one writes the BV symplectic form in Darboux coordinates as in (A.2), the correct BV extension of the HPSM is found to be rather of the involved form (5.20) in the main text. A qualitatively new feature of this functional, which is at least not easy to guess, is the appearance of the Euler vector field $\varvec{\varepsilon }$ inside $S_{\mathrm{BV}}$—a fact that holds true also for the simpler BFV functional $S_{\mathrm{BFV}}$ of the HPSM, see (3.10).

BV Symplectic form with H

The graded symplectic form $\omega _{\mathrm{BFV}}$ of the HPSM necessarily contains the 3-form H, see (3.4) or (3.9). This is not the case for the BV symplectic form $\omega _{\mathrm{BV}}$, see (4.21) or (5.26) (in fact, for degree reasons, $\omega _{\mathrm{BV}}$ is always exact [11]). One may be curious, however, how the BV functional of the HPSM would look like if one adds an H contribution to the BV symplectic form similar to the one in $\omega _{\mathrm{BFV}}$. Indeed, in terms of the new field¹⁹

$$\begin{aligned} \varvec{A}_i= & {} \varvec{\check{A}}_{i} + H_{ijk}(\varvec{X}) (\varepsilon \varvec{X}^j) {\mathrm {d}}\varvec{X}^k, \end{aligned}$$

(C.1)

the BV symplectic form (5.26) changes to

$$\begin{aligned} \omega _{\mathrm{BV}}= & {} \int _{T[1]\Sigma } d^2 \sigma d^2 \theta \, \left( \delta \varvec{X}^{i} \delta \varvec{\check{A}}_i - \tfrac{1}{2} H_{ijk} {\mathrm {d}}\varvec{X}^i \delta \varvec{X}^j \delta \varvec{X}^k \right) . \end{aligned}$$

(C.2)

and the BV action (5.20), becomes

$$\begin{aligned} S_{\mathrm{BV}}= & {} \int _{T[1]\Sigma } \!\!\!\! d^2 \sigma d^2 \theta \, \left[ \varvec{\check{A}}_i \, \varvec{\mathrm {d}}\varvec{X}^i + \tfrac{1}{2} \pi ^{ij}(\varvec{X})\, \varvec{\check{A}}_i \varvec{\check{A}}_{j} \right] + \int _{T[1]N} \!\!\!\! d^3 \sigma d^3 \theta \, H(\varvec{X}) \nonumber \\&+ \int _{T[1]\Sigma } \!\!\!\! d^2 \sigma d^2 \theta \, \left[ \tfrac{1}{4} (\pi ^{il} \pi ^{jm} H_{lmk})(\varvec{X}) \, \varvec{\check{A}}_i \varvec{\check{A}}_j \varvec{\varepsilon } \varvec{X}^k - \tfrac{1}{2} (\pi ^{il} H_{jkl}) (\varvec{X})\, \varvec{\check{A}}_i (\varvec{\mathrm {d}}\varvec{X}^j) \varvec{\varepsilon } \varvec{X}^k \right] \nonumber \\&+ \int _{T[1]\Sigma } \!\!\!\! d^2 \sigma d^2 \theta \, \left[ \tfrac{1}{8} (\pi ^{im} \pi ^{jn} \pi ^{pq} H_{mql}H_{npk})(\varvec{X})\, \varvec{\check{A}}_i \varvec{\check{A}}_j (\varvec{\varepsilon } \varvec{X}^k) \varvec{\varepsilon } \varvec{X}^l \right] . \nonumber \\ \end{aligned}$$

(C.3)

This answers the question posed in this appendix.

Bianchi Identity for the Basic Curvature S

In this appendix, we want to provide a non-sophisticated derivation of the Bianchi identity (4.35) for the tensor S, defined in (4.13).²⁰ We do this in terms of a component calculation. In a local coordinate system, the curvature R and the torsion $\Theta$ of the affine TM connection $\Gamma _{ij}^k$ as well as the E torsion T and the basic curvature S take the following form:

$$\begin{aligned} R_{lij}^k= & {} \partial _i \Gamma _{lj}^k - \partial _j \Gamma _{li}^k + \Gamma _{lj}^m \Gamma _{mi}^k - \Gamma _{li}^m \Gamma _{mj}^k, \end{aligned}$$

(D.1)

$$\begin{aligned} \Theta ^k_{ij}= & {} - \Gamma _{ij}^k + \Gamma _{ji}^k = \pi ^{kl} H_{ijl}, \end{aligned}$$

(D.2)

$$\begin{aligned} T_k^{ij}= & {} - f_k^{ij} - \pi ^{il} \Gamma _{kl}^j - \pi ^{lj} {\Gamma _{kl}^i}, \end{aligned}$$

(D.3)

$$\begin{aligned} S_{ij}{}^{kl}= & {} \nabla _j T_i^{kl} - \pi ^{mk} R^l_{ijm} + \pi ^{ml} R^k_{ijm} \nonumber \\&\equiv - f_{(i}^{kl}{},_{j)} + \Gamma _{(ij)}^m f_m^{kl} + 2 \Gamma _{m(j}^{[k} f_{i)}^{l]m} + 2 \pi ^{m[k} \Gamma ^{l]}_{ij},_m + 2 \pi ^{m[k},_j \Gamma ^{l]}_{im} - 2 \Gamma ^{[k}_{im}\Gamma ^{l]}_{nj} \pi ^{mn}. \nonumber \\ \end{aligned}$$

(D.4)

The Bianchi identities for $R_{ijk}^l$ and $\Theta _{ij}^k$ are

$$\begin{aligned}&R_{lij}^k = - R_{lji}^k, \end{aligned}$$

(D.5)

$$\begin{aligned}&R_{[ijk]}^l = \nabla _{[i} \Theta ^l_{jk]} - \Theta _{[ij}^m \Theta _{k]m}^l, \end{aligned}$$

(D.6)

$$\begin{aligned}&\nabla _{[i} R_{l|jk]}^m + \Theta _{[ij}^n R_{k]ln}^m =0. \end{aligned}$$

(D.7)

A twisted Poisson structure on M gives rise to a Lie algebroid structure on $T^*M$, which, by means of a connection with the prescribed torsion as in (2.21), can be expressed as follows:

$$\begin{aligned}&[\pi ^{\sharp }(\alpha ), \pi ^{\sharp }(\beta )]^{\nabla } = \pi ^{\sharp }([\alpha , \beta ]_{\pi }^{\nabla }), \end{aligned}$$

(D.8)

$$\begin{aligned}&[[\alpha , \beta ]_{\pi }^{\nabla }, \gamma ]_{\pi }^{\nabla } + \text{ cyclic }(\alpha \beta \gamma ) = 0, \end{aligned}$$

(D.9)

where $\alpha , \beta , \gamma \in \Omega ^1(M)$. Here, $[X, Y]^{\nabla }$ is the covariantized Lie bracket for vector fields $X, Y \in \mathfrak {X}(M)$, which is defined by replacing the derivative $\partial _i$ by the covariant derivative $\nabla _i$. Similarly, $[\alpha , \beta ]_{\pi }^{\nabla }$ is the covariantized Koszul bracket [56]

$$\begin{aligned}&[\alpha , \beta ]_{\pi }^{\nabla } = L^{\nabla }_{\pi ^{\sharp } (\alpha )}\beta - L^{\nabla }_{\pi ^{\sharp } (\beta )} \alpha - \nabla (\pi (\alpha , \beta )), \end{aligned}$$

(D.10)

where $L_X = \mathrm{D}\circ \iota _X + \iota _X \circ \mathrm{D}$ is the covariantized Lie derivative (as before, $\mathrm{D}$ denotes the exterior covariant derivative induced by the connection $\nabla$). In local coordinates, (D.8) and (D.9) take the form

$$\begin{aligned}&\pi ^{l[i} \nabla _l \pi ^{jk]} =0, \end{aligned}$$

(D.11)

$$\begin{aligned}&\pi ^{m[i} \nabla _m T_l^{jk]} - T_{l}^{m[i} T_m^{jk]} - \pi ^{[i|m} \pi ^{|j|n} R_{lmn}^{k]} = 0. \end{aligned}$$

(D.12)

In addition, one may verify the following formulas:

$$\begin{aligned}&[\nabla _m, \nabla _k]T_n^{ij} = \Theta _{km}^l \nabla _l T_n^{ij} - R_{nmk}^l T_l^{ij} + R_{lmk}^i T_n^{lj} + R_{lmk}^j T_m^{il}, \end{aligned}$$

(D.13)

and (cf. Eq. (2.26))

$$\begin{aligned}&- \nabla _k \pi ^{ij} + \pi ^{il} \Theta _{kl}^j = T_k^{ij}. \end{aligned}$$

(D.14)

Using (D.11), (D.12), (D.13), and (D.14), one now verifies

$$\begin{aligned}&\pi ^{m[i} \nabla _m S_{nk}{}^{jl]} + T_m^{[ij} S_{nk}{}^{l]m} - T_n^{m[i} S_{mk}{}^{jl]} - T_k^{m[i} S_{nm}{}^{jl]} = 0, \end{aligned}$$

(D.15)

which is the searched-for Bianchi identity of the basic curvature $S_{nk}{}^{ij}$.