The gravity dual of supersymmetric gauge theories on a two-parameter deformed three-sphere

We present rigid supersymmetric backgrounds for three-dimensional N=2N=2 supersymmetric gauge theories, comprising a two-parameter U(1)×U(1)U(1)×U(1)-invariant deformed three-sphere, and their gravity duals. These are described by supersymmetric solutions of four-dimensional N=2N=2 gauged supergravity with a self-dual metric on the ball and different instantons for the graviphoton field. We find two types of solutions, distinguished by their holographic free energies. In one type the holographic free energy is constant, whereas in another type it depends in a simple way on the parameters and is generically complex. This leads to a conjecture for the localized partition function of a class of N=2N=2 supersymmetric gauge theories on these backgrounds.