W$W$‐algebras associated to surfaces

Abstract

We define an integral form of the deformed W $W$ -algebra of type gl r ${\mathfrak {gl}}_r$ , 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.