I will explain the construction of a new homology theory for three-manifolds, defined using perverse sheaves on the SL(2,C) character variety. Our invariant is a model for an SL(2,C) version of Floer’s instanton homology. I will present a few explicit computations for Brieskorn spheres, and discuss the connection to the Kapustin-Witten equations and Khovanov homology. This is joint work with Mohammed Abouzaid.