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Analytical Traces on Coulomb Branches of Quiver Gauge Theories

Keke Zhang

TL;DR

The paper provides an explicit operator realization of the quantized Coulomb branch algebra $\mathcal{A}_\hbar(G, \mathbf N)$ for conical 3d $\mathcal{N}=4$ theories and derives a concrete integral formula for twisted traces, connecting to the correlation functions of the Beem–Peelaers–Rastelli conformal theory. By building holomorphic and antiholomorphic representations and employing localization to the maximal torus, it reduces nonabelian cases to Weyl-invariant abelian data and constructs a ground-state–based pairing to define the trace. The main result is a convergent, Weyl-invariant trace formula $\operatorname{Tr}(R(\sigma)) = \int_{\mathfrak{t}_G} e^{2\pi\zeta(\sigma)} R(\sigma) \mathrm{w}(\sigma+\underline{\mathbf{m}}) \, d\sigma$, with a weight $\mathrm{w}$ determined by roots and weights; this is verified in Ab- and GL$_2$-type examples. The work lays groundwork for extending to K-theoretic Coulomb branches and for advancing the quantum Hikita program, providing a practical computational toolkit for twisted traces on Coulomb branches.

Abstract

In this paper, we present an explicit construction of twisted traces for quantum Coulomb branches of conical theories. We develop an operator representation of the Coulomb branch algebra and use it to derive integral formulas for the twisted trace. Our construction provides a concrete realization of twisted traces that arise as the correlation functions of a conformal field theory, particularly in the work of Beem, Peelaers, and Rastelli. This complements recent developments in the study of twisted traces on quantum Higgs branches and offers new mathematical insights into the structure of quantum Coulomb branches.

Analytical Traces on Coulomb Branches of Quiver Gauge Theories

TL;DR

The paper provides an explicit operator realization of the quantized Coulomb branch algebra for conical 3d theories and derives a concrete integral formula for twisted traces, connecting to the correlation functions of the Beem–Peelaers–Rastelli conformal theory. By building holomorphic and antiholomorphic representations and employing localization to the maximal torus, it reduces nonabelian cases to Weyl-invariant abelian data and constructs a ground-state–based pairing to define the trace. The main result is a convergent, Weyl-invariant trace formula , with a weight determined by roots and weights; this is verified in Ab- and GL-type examples. The work lays groundwork for extending to K-theoretic Coulomb branches and for advancing the quantum Hikita program, providing a practical computational toolkit for twisted traces on Coulomb branches.

Abstract

In this paper, we present an explicit construction of twisted traces for quantum Coulomb branches of conical theories. We develop an operator representation of the Coulomb branch algebra and use it to derive integral formulas for the twisted trace. Our construction provides a concrete realization of twisted traces that arise as the correlation functions of a conformal field theory, particularly in the work of Beem, Peelaers, and Rastelli. This complements recent developments in the study of twisted traces on quantum Higgs branches and offers new mathematical insights into the structure of quantum Coulomb branches.
Paper Structure (22 sections, 21 theorems, 130 equations)

This paper contains 22 sections, 21 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of Theorem
  4. Future Directions
  5. Context
  6. Nakajima Quiver and Higgs Branches
  7. Coulomb branch
  8. Abelian Coulomb Branch Algebras and Localization
  9. Operator Representation
  10. Analysis
  11. Abelian Pure Gauge Theory
  12. Representation of $G = \mathbb C^*, \mathbf N = \mathbb C$ case
  13. The Pontryagin Dual
  14. Twisted Trace
  15. Faithful Representations
  16. ...and 7 more sections

Key Result

Theorem 1.3

For a conical pair $(G, \mathbf{N})$ coming from a quiver gauge theory of type $A$ with $G = \prod_{i \in Q_0} \operatorname{GL}(V_i)$ ($V_i = \mathbb{C}^{a_i}$) (see def for definition), the quantized Coulomb branch algebra $\mathcal{A}_\hbar(G, \mathbf{N})$ for a choice of $\underline{\mathbf{m}},

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2: 2020
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 34 more