Advances in Theoretical and Mathematical Physics

Volume 23 (2019)

Number 2

Ring objects in the equivariant derived Satake category arising from Coulomb branches

Pages: 253 – 344



Alexander Braverman (Department of Mathematics, University of Toronto, Ontario, Canada; Perimeter Institute of Theoretical Physics, Waterloo, Ontario, Canada; and Skolkovo Institute of Science and Technology, Moscow, Russia)

Michael Finkelberg (Department of Mathematics, National Research University Higher School of Economics, Moscow, Russia; Skolkovo Institute of Science and Technology, Moscow, Russia; and Institute for Information Transmission Problems, Moscow, Russia)

Hiraku Nakajima (Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan; and Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba, Japan)


This is the second companion paper of [Part II]. We consider the morphism from the variety of triples introduced in [Part II] 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 [Part II] 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 F. Benini, Y. Tachikawa, and D. Xie [“Mirrors of $3d$ Sicilian theories”, JHEP 1009 (2010), 63].

The author of Appendix B is Gus Lonergan.

Published 11 November 2019