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

TL;DR

This paper surveys the Yangian and quantum toroidal actions on the cohomology and K-theory of moduli spaces of sheaves on surfaces, extending to elliptic cohomology, highlighting algebraic structures and geometric applications.

Contribution

It generalizes known algebraic actions on moduli spaces to elliptic cohomology, enriching the mathematical framework for sheaves on surfaces.

Findings

Yangian and quantum toroidal actions on cohomology and K-theory

Extension of algebraic actions to elliptic cohomology

Connections between algebraic structures and moduli spaces of sheaves

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.