Superconformal algebras and holomorphic field theories
We compute the holomorphic twists of four-dimensional superconformal algebras, and argue that the resulting algebras act naturally by holomorphic vector fields on holomorphically twisted superconformal theories. We demonstrate that this symmetry enhances to the action of an infinite-dimensional local Lie algebra, the Dolbeault resolution of all holomorphic vector fields on a punctured superspace. Global symmetries also enhance to the Dolbeault resolution of holomorphic functions valued in the Lie algebra; at the classical level, both of these higher symmetry algebras act naturally on the holomorphic twist of any Lagrangian theory, whether superconformal or not. We show that these algebras are related to two-dimensional chiral algebras extracted from four-dimensional superconformal theories in recent work; further deforming the differential induces the Koszul resolution of a plane in $\mathbb{C}^2$, and the cohomology of the higher symmetry algebras are the usual Virasoro and Kac-Moody chiral algebras. We show that the central extensions of those chiral algebras arise from recently studied central extensions of our higher symmetry algebras. However, the higher algebras admit many further deformations not originating in the global superconformal algebra; these localize to any smooth complex curve in $\mathbb{C}^2$, resolving the holomorphic vector fields there, and expect that they will lead to even more exotic behavior in the case of singular or nonreduced curves. We consider explicit examples of $\mathcal{N}=2$ gauge theories, and demonstrate that an anomaly to realizing the higher symmetry algebra at the quantum level vanishes precisely when the theory is, in fact, superconformal; for such theories, we also give an explicit description of the chiral algebras that result after further deformation.