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Coulomb branch algebras via symplectic cohomology

Eduardo Gonzalez, Cheuk Yu Mak, Daniel Pomerleano

Abstract

Let $(\bar{M}, ω)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose that $(\bar{M}, ω)$ is equipped with a convex Hamiltonian $G$-action for some connected, compact Lie group $G$. We construct an action of the pure Coulomb branch of $G$ on the $G$-equivariant symplectic cohomology of $\bar{M}.$ Building on work of Teleman, we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima in terms of equivariant symplectic cohomology.

Coulomb branch algebras via symplectic cohomology

Abstract

Let be a compact symplectic manifold with convex boundary and . Suppose that is equipped with a convex Hamiltonian -action for some connected, compact Lie group . We construct an action of the pure Coulomb branch of on the -equivariant symplectic cohomology of Building on work of Teleman, we use this construction to characterize the Coulomb branches of Braverman-Finkelberg-Nakajima in terms of equivariant symplectic cohomology.
Paper Structure (35 sections, 68 theorems, 195 equations)

This paper contains 35 sections, 68 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Floer theory over a family, pull-back and push-forward
  3. Continuation and invariance
  4. Product structure on $SH^*(P)$
  5. Pull-back
  6. Push-forward
  7. Equivariant Floer theory
  8. Useful properties
  9. Free actions
  10. Classifying models
  11. Product structure
  12. Weyl group action
  13. Equivariant Seidel morphism
  14. Overview
  15. Tautological isomorphism
  16. ...and 20 more sections

Key Result

Theorem 1.1

Let $(\bar{M}, \omega)$ be a compact symplectic manifold with convex boundary and $c_1(T\bar{M})=0$. Suppose further that $(\bar{M}, \omega)$ is equipped with a convex Hamiltonian $G$-action. Then there is an algebra homomorphism:

Theorems & Definitions (163)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 153 more