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BKP-Affine Coordinates and Emergent Geometry of Generalized Brézin-Gross-Witten Tau-Functions

Zhiyuan Wang, Chenglang Yang, Qingsheng Zhang

TL;DR

The paper develops an emergent-geometry framework for generalized Brézin-Gross-Witten BKP tau-functions by deriving a spectral curve and its special deformation from Virasoro constraints. It provides explicit BKP-affine coordinates and a generating-series description that yield a closed formula for connected n-point functions, and it demonstrates that the Eynard-Orantin topological recursion on the emergent spectral curve reproduces these correlators. A quantum spectral curve of type B is constructed via Kac-Schwarz operators, linking the BKP data to a Schrödinger-type equation whose semiclassical limit matches the classical curve $x^2y^2=x^2+S^2$. Together, these results connect BKP tau-function techniques, topological recursion, and quantum curve concepts, offering concrete computational tools for BGW-related enumerative geometries and highlighting the role of BKP-affine coordinates in emergent geometry.

Abstract

Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected $n$-point functions. Finally, we show that the quantum spectral curve of type $B$ in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.

BKP-Affine Coordinates and Emergent Geometry of Generalized Brézin-Gross-Witten Tau-Functions

TL;DR

The paper develops an emergent-geometry framework for generalized Brézin-Gross-Witten BKP tau-functions by deriving a spectral curve and its special deformation from Virasoro constraints. It provides explicit BKP-affine coordinates and a generating-series description that yield a closed formula for connected n-point functions, and it demonstrates that the Eynard-Orantin topological recursion on the emergent spectral curve reproduces these correlators. A quantum spectral curve of type B is constructed via Kac-Schwarz operators, linking the BKP data to a Schrödinger-type equation whose semiclassical limit matches the classical curve . Together, these results connect BKP tau-function techniques, topological recursion, and quantum curve concepts, offering concrete computational tools for BGW-related enumerative geometries and highlighting the role of BKP-affine coordinates in emergent geometry.

Abstract

Following Zhou's framework, we consider the emergent geometry of the generalized Brézin-Gross-Witten models whose partition functions are known to be a family of tau-functions of the BKP hierarchy. More precisely, we construct a spectral curve together with its special deformation, and show that the Eynard-Orantin topological recursion on this spectral curve emerges naturally from the Virasoro constraints for the generalized BGW tau-functions. Moreover, we give the explicit expressions for the BKP-affine coordinates of these tau-functions and their generating series. The BKP-affine coordinates and the topological recursion provide two different approaches towards the concrete computations of the connected -point functions. Finally, we show that the quantum spectral curve of type in the sense of Gukov-Sułkowski emerges from the BKP-affine coordinates and Eynard-Orantin topological recursion.
Paper Structure (24 sections, 15 theorems, 171 equations)

This paper contains 24 sections, 15 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Backgrounds and overview
  3. Description of main results
  4. Preliminaries of Generalized Brézin-Gross-Witten Tau-Functions
  5. Generalized BGW tau-functions and Virasoro constraints
  6. Recursion for connected $n$-point functions
  7. Schur $Q$-function expansion and BKP-affine coordinates
  8. BKP-Affine Coordinates of Generalized BGW Tau-Functions
  9. BKP-affine coordinates of $\tau_{\text{BGW}}^{(N)}({\bf t}/2)$
  10. A formula for connected $n$-point functions
  11. Generating series for the BKP-affine coordinates
  12. Emergence of Spectral Curve and Its Special Deformation
  13. A brief review of Zhou's emergent geometry
  14. Special deformation and spectral curve for $\tau_{\text{BGW}}^{(N)}$
  15. Quantum deformation theory of the spectral curve
  16. ...and 9 more sections

Key Result

Theorem 1.1

Denote by $B(x_1,x_2)$ the following symmetric bi-differential on the spectral curve eq-intro-curve: where $W_{g,n}^{\text{gBGW}}$ is the connected $n$-point function of genus $g$: Let $\omega_{g,n}$ (where $g\geq 0,n\geq 1$) be the Eynard-Orantin invariants for the spectral curve eq-intro-curve and Bergman kernel $B$, then for $(g,n)$ with $2g-2+n>0$ we have:

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Example 2.1
  • Remark 2.3
  • Proposition 3.1
  • proof
  • Remark 3.1
  • ...and 32 more