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

Alexander Braverman, Michael Finkelberg, Hiraku Nakajima

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.

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

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 Sicilian theories in type by arXiv:1007.0992.

Paper Structure

This paper contains 48 sections, 33 theorems, 244 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Complexes on the affine Grassmannian
  3. Categorical generalities
  4. Pushforward to the affine Grassmannian
  5. Commutativity
  6. A complex on the affine Grassmannian of the flavor symmetry group
  7. An alternative construction of a regular sheaf
  8. Line bundles via homology groups of fibers
  9. Wakimoto sheaves
  10. Gluing construction
  11. Proof of commutativity
  12. Commutativity constraint
  13. Factorization version of $\mathcal{R}$
  14. Definition of another multiplication
  15. The case N=0
  16. ...and 33 more sections

Key Result

Proposition 2.1

Let $\pi\colon \mathcal{R}\to \mathrm{Gr}_G$ be the projection and $\mathscr A \overset{\operatorname{ def.}}{=} \pi_* \boldsymbol\omega_\mathcal{R}[-2\dim\mathbf N_\mathcal{O}]\in D_{G}(\mathrm{Gr}_G)$. (1) There exists a natural multiplication homomorphism where the left hand side is the convolution product of $\mathscr A$ with itself given by the diagram eq:1 . (2) Let $\mathbf 1_{\mathrm{Gr}_

Figures (1)

  • Figure 1: A star shaped quiver gauge theory

Theorems & Definitions (85)

  • Proposition 2.1
  • proof
  • Theorem 2.11
  • Theorem 2.13
  • Lemma 2.17
  • proof
  • Remark 2.23
  • Theorem 2.24
  • Remark 2.25
  • Claim
  • ...and 75 more