AKSZ construction from reduction data

Abstract

We discuss a general procedure to encode the reduction of the target space geometry into AKSZ sigma models. This is done by considering the AKSZ construction with target the BFV model for constrained graded symplectic manifolds. We investigate the relation between this sigma model and the one with the reduced structure. We also discuss several examples in dimension two and three when the symmetries come from Lie group actions and systematically recover models already proposed in the literature.Comment: 42 page

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 briefly recall it.…”

“…(24) so that (W (M, π, g), D) coincides with the Kalkman model for equivariant cohomology (see [8]). If we look at the target manifold T * [1](M × T [1]g [1]) again as a tangent bundle T [1](M × g [1] × g * [−1]) so that the de Rham differential is defined as dx µ = b µ , dc a = φ a and d ξ a = ξ a we immediately recognize that [4], the antighost degree ag = −gh + deg, where gh is the natural degree of the target graded manifold, gives the target manifold the structure of BF V manifold, a model for the symplectic reduction of T * [1]M with respect to the constraints µ = 0 and v ν a b ν = 0. We recall that the BFV (Batalin-Fradkin-Vilkovisky) manifolds in general give an homological resolution of constrained system and can be seen as a mathematical formulation of BRST in the hamiltonian setting (see [3,12]).…”

“…We consider two different gauge fixings which are compatible with the A-model observables. The first one is relevant when the symplectic reduction of the target space is smooth; we conjecture that the theory computes the A-model correlator of the reduced symplectic manifold in the spirit of [4]. In the second one, we recover for the Lie algebra sector the supersymmetric Yang Mills action and the residual BV symmetry is the supersymmetry generator.…”

“…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.…”

“…The target homological vector field encodes the Weil model for equivariant geometry so that the AKSZ observables are associated to equivariant cohomology. In [4] this theory was considered as a model for the PSM with target the symplectic reduction µ −1 (0)/G. We introduce the analogue of A-model observables that depend on a minimal set of fields and introduce an explicit homotopy with the AKSZ observables.…”