Coherent sheaves and quantum Coulomb branches II: quiver gauge theories and knot homology

TL;DR

This paper develops an algebraic framework for Coulomb branch resolutions in quiver gauge theories, leading to a new algebraic knot homology theory that categorifies Reshetikhin-Turaev invariants and aligns with Khovanov-Rozansky homology.

Contribution

It introduces a purely algebraic approach to Coulomb branch resolutions and constructs a new algebraic knot homology theory compatible with existing categorifications.

Findings

Algebraic description of Coulomb branches and their resolutions.

Construction of an algebraic knot homology theory.

Equivalence with existing categorifications and Khovanov-Rozansky homology.

Abstract

We continue our study of noncommutative resolutions of Coulomb branches in the case of quiver gauge theories. These include the Slodowy slices in type A and symmetric powers in $\mathbb{C}^2$ as special cases. These resolutions are based on vortex line defects in quantum field theory, but have a precise mathematical description, which in the quiver case is a modification of the formalism of KLRW algebras. While best understood in a context which depends on the geometry of the affine Grassmannian and representation theory in characteristic $p$, we give a description of the Coulomb branches and their commutative and non-commutative resolutions which can be understood purely in terms of algebra. This allows us to construct a purely algebraic version of the knot homology theory defined using string theory by Aganagi\'c, categorifying the Reshetikhin-Turaev invariants for minuscule representations of type ADE Lie algebras. We show that this homological invariant agrees with the categorification of these invariants previously defined by the author, and thus with Khovanov-Rozansky homology in type A.