Poisson Vertex Algebra of Seiberg-Witten Theory

The space of local operators in the $Q$-cohomology of the holomorphic-topological supercharge in a four-dimensional $\mathcal{N}=2$ theory carries the structure of a Poisson vertex algebra. This note studies the Poisson vertex algebra associated to the pure $\mathcal{N}=2$ gauge theory with gauge group $SU(2)$. We propose an explicit Poisson vertex algebra $A$, claimed to be isomorphic to the algebra of holomorphic-topological observables to all orders in perturbation theory. We compute the Hilbert-Poincar\'e series of $A$ and show that it refines the Schur index of the pure $SU(2)$ theory. We show that $A$ admits a further differential $Q_{\text{inst}}$ which we hypothesize captures non-perturbative corrections, and compute the cohomology of this differential. We thus present an explicit candidate for the space of non-perturbative holomorphic-topological observables of Seiberg-Witten theory.