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The role of Coulomb branches in 2D gauge theory

Constantin Teleman

Abstract

I give a simple construction of certain Coulomb branches $C_{3,4}(G;E)$ of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group $G$ and a polarisable quaternionic representation $E$. The manifolds $C(G; 0)$ are abelian group schemes (over the bases of regular adjoint $G_c$-orbits, respectively conjugacy classes), and $C(G;E)$ is glued together from two copies of $C(G;0)$ shifted by a rational Lagrangian section $\varepsilon_V$, the Euler class of the index bundle of a polarisation $V$ of $E$. Extending the interpretation of $C_3(G;0)$ as "classifying space" for topological 2D gauge theories, I characterise functions on $C_3(G;E)$ as operators on the equivariant quantum cohomologies of $M\times V$, for all compact symplectic $G$-manifolds $M$. The non-commutative version has an analogous description in terms of the $Γ$-function of $V$, appearing to play the role of Fourier transformed J-function of the gauged linear Sigma-model $V/G$.

The role of Coulomb branches in 2D gauge theory

Abstract

I give a simple construction of certain Coulomb branches of gauge theory in 3 and 4 dimensions defined by Nakajima et al. for a compact Lie group and a polarisable quaternionic representation . The manifolds are abelian group schemes (over the bases of regular adjoint -orbits, respectively conjugacy classes), and is glued together from two copies of shifted by a rational Lagrangian section , the Euler class of the index bundle of a polarisation of . Extending the interpretation of as "classifying space" for topological 2D gauge theories, I characterise functions on as operators on the equivariant quantum cohomologies of , for all compact symplectic -manifolds . The non-commutative version has an analogous description in terms of the -function of , appearing to play the role of Fourier transformed J-function of the gauged linear Sigma-model .

Paper Structure

This paper contains 35 sections, 17 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Overview and key examples
  3. Background.
  4. Relation with quantum cohomology.
  5. Coulomb branches with matter.
  6. Example I: $G=\mathrm{U}_1$, with the standard representation $L$.
  7. Generalisation.
  8. Example II: $\mathrm{U}_1$ with a general representation $V$.
  9. Background on Coulomb branches
  10. The basic Coulomb branches bfm.
  11. Group scheme structure.
  12. Coulomb branches for $E=V\oplus V^\vee$.
  13. Massive versions.
  14. Main results
  15. The Euler Lagrangians.
  16. ...and 20 more sections

Key Result

Proposition 2.3

$\mathbb{C}[x,y]$ is the subring of regular functions $f(\tau,z)$ on $T^\vee\mathbb{C}^\times$ with the property that $f(\tau, z\tau)$ is also regular. ∎

Theorems & Definitions (38)

  • Remark 2.1
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 3.1: Sphere topology
  • Remark 3.3: Adjoint and Whittaker descriptions
  • Remark 3.6: $E_3$ Hecke property
  • Remark 4.2: Broader picture
  • ...and 28 more