New supersymmetric localizations from topological gravity

Supersymmetric field theories can be studied exactly on off-shell “localizing” supergravity backgrounds. We show that these supergravity configurations can be identified with BRST invariant configurations of background topological gravity coupled to background topological gauge multiplets. We apply this topological point of view to two-dimensional $$ \mathcal{N}=\left(2,2\right) $$ supersymmetric matter theories to obtain, in a simple and straightforward way, a complete classification of localizing supersymmetric backgrounds in two dimensions. We recover all known localizing backgrounds and (infinitely) many more that have not been explored so far. The newly found localizing backgrounds are characterized by quantized fluxes for both graviphotons of the $$ \mathcal{N}=\left(2,2\right) $$ supergravity multiplet. The BRST invariant topological backgrounds are parametrized by both Killing vectors and $$ {\mathbb{S}}^1 $$ -equivariant cohomology of the two-dimensional spacetime. We completely reconstruct the supergravity backgrounds from the topological data: some of the supergravity fields are twisted versions of the topological backgrounds, but others are composite, in that they are nonlinear functionals of topological fields. Moreover, we show that the supersymmetric Ω-deformation is nothing but the background value of the ghost-for-ghost of topological gravity, a result which holds for higher dimensions too.