Topological Dirac Sigma Models and the Classical Master Equation

Abstract

We present the construction of the classical Batalin-Vilkovisky action for topological Dirac sigma models. The latter are two-dimensional topological field theories that simultaneously generalise the completely gauged Wess-Zumino-Novikov-Witten model and the Poisson sigma model. Their underlying structure is that of Dirac manifolds associated to maximal isotropic and integrable subbundles of an exact Courant algebroid twisted by a 3-form. In contrast to the Poisson sigma model, the AKSZ construction is not applicable for the general Dirac sigma model. We therefore follow a direct approach for determining a suitable BV extension of the classical action functional with ghosts and antifields satisfying the classical master equation. Special attention is paid on target space covariance, which requires the introduction of two connections with torsion on the Dirac structure.

Basic curvature and the Atiyah cocycle in gauge theory

“…Even though for the latter model the classical master equation cannot be determined from the vanilla AKSZ construction due to an obstruction to the existence of a QP structure on the target space, nevertheless using direct methods one finds that the 4-fermion term is still controlled by the basic curvature [28] (of a different connection though, this time with torsion.) A similar result holds for topological Dirac sigma models in 2D whose target spaces are Dirac manifolds; the 4-fermion terms in the BV action appear together with the basic curvature of two Lie algebroid connections on a Dirac structure [29].…”

“…We are not going to repeat here the technical analysis of [29]. To make our point, we will essentially reconstruct via deformation the BV operator of the general topological Dirac sigma model and emphasize its relation to the basic curvature tensor(s) of suitable E-connection(s).…”

“…The advantage now is that this is still true for the Wess-Zumino-Poisson sigma model in presence of the 3-form H, even though in that case there is no known AKSZ construction. What is more this addresses the BV operator for the general topological Dirac sigma model, whose BV action was found in [29].…”