Ring objects in the equivariant derived Satake category arising from Coulomb branches (with an appendix by Gus Lonergan)

TL;DR

This paper explores the structure of ring objects in the equivariant derived Satake category derived from Coulomb branches, extending previous work to arbitrary rings and connecting to quiver gauge theories.

Contribution

It generalizes the construction of ring objects in the equivariant derived Satake category to arbitrary commutative rings and studies Coulomb branches related to star-shaped quivers.

Findings

Ring objects in the equivariant derived Satake category are constructed for arbitrary rings.

The morphism from the variety of triples to the affine Grassmannian induces a ring object.

Connections between Coulomb branches and Higgs branches of 3d Sicilian theories are established.

Abstract

This is the second companion paper of arXiv:1601.03586. We consider the morphism from the variety of triples introduced in arXiv:1601.03586 to the affine Grassmannian. The direct image of the dualizing complex is a ring object in the equivariant derived category on the affine Grassmannian (equivariant derived Satake category). We show that various constructions in arXiv:1601.03586 work for an arbitrary commutative ring object. The second purpose of this paper is to study Coulomb branches associated with star shaped quivers, which are expected to be conjectural Higgs branches of $3d$ Sicilian theories in type $A$ by arXiv:1007.0992.