Categorifying the Schur index of Seiberg-Witten theory

The”Schur index” is typically defined as a protected operator count in 4d N=2 superconformal field theories. It turns out in fact that one can define it for a generic 4d N=2 theory, conformal or not, by using the holomorphic-topological twist. Its categorification, namely the space of holomorphic-topological local operators, is expected to be a Poisson vertex algebra. However, for a general non-conformal theory, not much is known about the shape of this PVA. For 4d N=2 gauge theories with matter, I will formulate this PVA as a (relative) Lie algebra cohomology problem and then for the case of pure SU(2) Seiberg-Witten theory propose an explicit answer for the cohomology.