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G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

Satoshi Okuda, Yutaka Yoshida

TL;DR

The paper extends the Gauge/Bethe correspondence by coupling G/G gauged WZW theory to matter, yielding a one-parameter G/G gauged WZW-matter model that maps to the q-boson integrable system via cohomological localization.It provides both the localized partition function as a sum over Bethe-like solutions and a detailed dictionary linking field-theoretic data to q-boson data, including a determinant expression for norms and a BRST-based localization framework.By connecting to Korff's commutative Frobenius algebra, the work shows the gauged WZW-matter model realizes a two-dimensional TQFT structure and clarifies when the Gauge/Bethe correspondence holds, additionally extending the correspondence to correlation functions.Numerical checks on partition functions for various genus, rank, and level support the proposed mappings and highlight the integrality and topological nature of the results, suggesting broader implications for 3D gauge theories and moduli-space indices.

Abstract

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.

G/G gauged WZW-matter model, Bethe Ansatz for q-boson model and Commutative Frobenius algebra

TL;DR

The paper extends the Gauge/Bethe correspondence by coupling G/G gauged WZW theory to matter, yielding a one-parameter G/G gauged WZW-matter model that maps to the q-boson integrable system via cohomological localization.It provides both the localized partition function as a sum over Bethe-like solutions and a detailed dictionary linking field-theoretic data to q-boson data, including a determinant expression for norms and a BRST-based localization framework.By connecting to Korff's commutative Frobenius algebra, the work shows the gauged WZW-matter model realizes a two-dimensional TQFT structure and clarifies when the Gauge/Bethe correspondence holds, additionally extending the correspondence to correlation functions.Numerical checks on partition functions for various genus, rank, and level support the proposed mappings and highlight the integrality and topological nature of the results, suggesting broader implications for 3D gauge theories and moduli-space indices.

Abstract

We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investigate this correspondence from a viewpoint of the commutative Frobenius algebra, the axiom of the two dimensional topological quantum field theory.

Paper Structure

This paper contains 12 sections, 1 theorem, 130 equations, 6 figures, 3 tables.

Table of Contents

  1. Introduction
  2. $q$-boson model
  3. $q$-boson model
  4. Algebraic Bethe Ansatz for the $q$-boson model
  5. $G/G$ gauged Wess-Zumino-Witten-matter model
  6. $G/G$ gauged Wess-Zumino-Witten-matter model
  7. Localization
  8. Numerical simulation
  9. Gauge/Bethe correspondence
  10. Partition function from the commutative Frobenius algebra
  11. Correlation functions of the gauged WZW-matter model
  12. Summary and Discussion

Key Result

Theorem 3.1

Let $\mathbbm{k}$ be the algebraically closed field of the Puiseux series and $\mathfrak{F}_{k,N} := {\cal F}_N^{\otimes k} \otimes_{\mathbb{C} (t)} \mathbbm{k}$. Define for $\mu, \nu \in {\cal A}_{N,k}^+$ the product where the structure constant of the commutative Frobenius algebra $N_{\mu\nu}^{\lambda}(t)$ is defined as Here, the transition matrix $S_{\mu\nu}(t)$ is defined in (deformedS). Mor

Figures (6)

  • Figure 1: Unit
  • Figure 3: Structure constant
  • Figure 4: Partition function on a sphere $S^2$
  • Figure 5: Partition function on a torus $T^2$
  • Figure 6: Handle operator
  • ...and 1 more figures

Theorems & Definitions (1)

  • Theorem 3.1: Commutative Frobenius algebra Korff:2013rsa