Rational, trigonometric, elliptic algebras and moduli spaces of sheaves on surfaces

Abstract

We survey the well-known Yangian of ${\widehat{\mathfrak{gl}}}_1$ /quantum toroidal ${\mathfrak{gl}}_1$ action on the cohomology / $K$-theory of moduli spaces of stable sheaves on surfaces, and give the generalization of this construction to elliptic cohomology.