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Coulomb Branch Quantization and Abelianized Monopole Bubbling

Mykola Dedushenko, Yale Fan, Silviu Pufu, Ran Yacoby

TL;DR

This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions by developing an approach to the study of Coulomb branch operators in 3D N.

Abstract

We develop an approach to the study of Coulomb branch operators in 3D $\mathcal{N}=4$ gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For `good' and `ugly' theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

Coulomb Branch Quantization and Abelianized Monopole Bubbling

TL;DR

This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions by developing an approach to the study of Coulomb branch operators in 3D N.

Abstract

We develop an approach to the study of Coulomb branch operators in 3D gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For `good' and `ugly' theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.

Paper Structure

This paper contains 66 sections, 2 theorems, 343 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Shift Operators
  3. Setup
  4. Theories
  5. Observables
  6. Gluing Formula
  7. Input from Localization
  8. Monopole Bubbling
  9. Shift Operators
  10. Reduction of Schur Index
  11. Dressing and Abelianized Bubbling
  12. Dressed Monopoles and Invariant Theory
  13. Abelianized Monopole Bubbling
  14. Relation to the Abelianization Map
  15. Bubbling from Polynomiality
  16. ...and 51 more sections

Key Result

Proposition 1

Let $G$ be a simple Lie group, $\mathcal{W}$ its Weyl group, and $b$ a dominant coweight (a magnetic charge). Then there exists a set of primitive monopoles (of magnetic charge $b$) $[P_1(\Phi)\mathcal{M}^b], [P_2(\Phi)\mathcal{M}^b], \ldots, [P_p(\Phi)\mathcal{M}^b]$, where $p=|\mathcal{W} b|$ is t

Figures (1)

  • Figure 1: A schematic depiction of $S^3$, obtained by gluing two hemispheres $HS^3_{\pm}\cong B^3$. The 1D TQFT lives on the $S^1$ parametrized by the angle $\varphi$ (thick orange line). The 1D TQFT circle intersects the equatorial $S^2=\partial HS^3_{\pm}$ at two points identified with its North ($N$) and South ($S$) poles.

Theorems & Definitions (5)

  • Definition 1
  • Example
  • Proposition 1
  • proof
  • Theorem 1