Coulomb branches of $3d$ $\mathcal N=4$ quiver gauge theories and slices in the affine Grassmannian (with appendices by Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes)

TL;DR

This paper explores Coulomb branches of 3d N=4 quiver gauge theories of type ADE, revealing their geometric structures as moduli spaces and slices in the affine Grassmannian, and connecting quantized Coulomb branches to shifted Yangians.

Contribution

It establishes isomorphisms between Coulomb branches and geometric objects like rational maps and affine Grassmannian slices, and identifies their quantizations with shifted Yangians.

Findings

Coulomb branches correspond to moduli spaces of rational maps and affine Grassmannian slices.

Quantized Coulomb branches are identified with truncated shifted Yangians.

Provides geometric and algebraic descriptions of Coulomb branches for ADE quiver theories.

Abstract

This is a companion paper of arXiv:1601.03586. We study Coulomb branches of unframed and framed quiver gauge theories of type $ADE$. In the unframed case they are isomorphic to the moduli space of based rational maps from ${\mathbb C}P^1$ to the flag variety. In the framed case they are slices in the affine Grassmannian and their generalization. In the appendix, written jointly with Joel Kamnitzer, Ryosuke Kodera, Ben Webster, and Alex Weekes, we identify the quantized Coulomb branch with the truncated shifted Yangian.