Towards geometric Satake correspondence for Kac-Moody algebras -- Cherkis bow varieties and affine Lie algebras of type $A$

TL;DR

This paper proposes a geometric construction of Kac-Moody algebra modules via Coulomb branches of quiver gauge theories, refining the conjectural geometric Satake correspondence for affine type A.

Contribution

It provides a provisional, geometric realization of Kac-Moody modules using Coulomb branch geometry, verified for affine type A through Cherkis bow varieties.

Findings

Identified Coulomb branches with Cherkis bow varieties in affine type A.

Constructed Kac-Moody modules from hyperbolic restrictions of intersection cohomology.

Checked geometric properties necessary for the construction in affine type A.

Abstract

We give a provisional construction of the Kac-Moody Lie algebra module structure on the hyperbolic restriction of the intersection cohomology complex of the Coulomb branch of a framed quiver gauge theory, as a refinement of the conjectural geometric Satake correspondence for Kac-Moody algebras proposed in an earlier paper with Braverman, Finkelberg in 2019. This construction assumes several geometric properties of the Coulomb branch under the torus action. These properties are checked in affine type A, via the identification of the Coulomb branch with a Cherkis bow variety established in a joint work with Takayama.