New Supersymmetric Localizations from Topological Gravity

TL;DR

This paper classifies all supersymmetric backgrounds in two-dimensional theories using topological gravity, revealing new backgrounds with non-trivial fluxes and connecting supersymmetry to topological data.

Contribution

It provides a complete classification of 2D supersymmetric backgrounds via topological gravity, including previously unexplored configurations with fluxes.

Findings

Recovered all known 2D supersymmetric backgrounds.

Identified new backgrounds with non-trivial fluxes.

Linked the Omega-deformation to topological ghost backgrounds.

Abstract

Supersymmetric field theories can be studied exactly on suitable off-shell supergravity backgrounds. We show that in two dimensions such backgrounds are identifiable with BRST invariant backgrounds of topological gravity coupled to an abelian topological gauge multiplet. This latter background is required for the consistent coupling of the topological `matter' YM theory to topological gravity. We make use of this topological point of view to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. The BRST invariant topological backgrounds are parametrized by both Killing vectors and $S^1$-equivariant cohomology of the 2-dimensional world-sheet. We reconstruct completely the supergravity backgrounds from the topological data: some of the supergravity fields are twisted versions of the topological backgrounds, but others are "composite", i.e. they are non-linear functionals of them. We recover all the known localizing 2-dimensional backgrounds and (infinitely) many more that have not been explored so far. We show that the supersymmetric $\Omega$-deformation is nothing but the background value of the ghost-for-ghost of topological gravity, a result which holds for other dimensions too. The new localizing backgrounds are characterized by non-trivial fluxes for both the graviphotons of the supergravity multiplet.