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