Gauging the Poisson sigma model

Abstract

We show how to carry out the gauging of the Poisson sigma model in an AKSZ inspired formulation by coupling it to a generalization of the Weil model worked out in ref. [31]. We call the resulting gauged field theory, Poisson-Weil sigma model. We study the BV cohomology of the model and show its relation to Hamiltonian basic and equivariant Poisson cohomology. As an application, we carry out the gauge fixing of the pure Weil model and of the Poisson-Weil model. In the first case, we obtain the 2-dimensional version of DonaldsonWitten topological gauge theory, describing the moduli space of flat connections on a closed surface. In the second case, we recover the gauged A topological sigma model worked out by Baptista describing the moduli space of solutions of the so-called vortex equations.Keywords: quantum field theory in curved spacetime; geometry, differential geometry and topology.

Observables in the equivariant A-model

“…The model that we are going to discuss was considered in [15,4,11]. The graded geometric formulation of the equivariant formulation and its AKSZ theory that we are going to use was discussed in [4].…”

“…We consider here a different gauge fixing of the Lie algebra sector that recovers the so called topological Yang-Mills theory in two dimensions, considered by Witten in [14]. This connection was already established in [15]; here we use a slightly different gauge fixing and emphasize the relation between the residual gauge symmetry and the gauge multiplet of supersymmetry. The basic tool for introducing topological Yang-Mills theory is the gauge multiplet of 2d supersymmetry.…”

“…In this paper we extend the analysis to an equivariant version of the Poisson Sigma Model. This is an AKSZ theory that was studied in [4,11,15]. The geometrical data of the target encode a hamiltonian G space, i.e.…”

AKSZ construction from reduction data

“…The BFV-AKSZ model corresponding to (5.23) is the sigma model proposed by Zucchini in [35,36] as Poisson-Weil sigma models and by Signori in [33] under the name of JPSM. In [36] it is shown that a sector of the underlying BV-cohomology is related to the equivariant Poisson cohomology of (M, π M ). In Appendix D we clarify this relation, by relating the cohomology of the target QP-manifold to Poisson equivariant cohomology.…”

“…In a series of recent papers [35,36] the procedure of gauging and reduction for the Poisson Sigma model (PSM in short) has been considered. The model is constructed when there is an action of a Lie group G on a Poisson manifold M by Poisson diffeomorphisms and this action is hamiltonian with momemntum map µ : M → g * .…”