W$W$ ‐algebras associated to surfaces

Abstract

We define an integral form of the deformed W$W$ ‐algebra of type glr${\mathfrak {gl}}_r$ , and construct its action on the K$K$ ‐theory groups of moduli spaces of rank r$r$ stable sheaves on a smooth projective surface S$S$ , under certain assumptions. Our construction generalizes the action studied by Nakajima, Grojnowski and Baranovsky in cohomology, although the appearance of deformed W$W$ ‐algebras by generators and relations is a new feature. Physically, this action encodes the Alday–Gaiotto–Tachikawa correspondence for 5‐dimensional supersymmetric gauge theory on S×$S \times$ circle.