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On higher Brézin-Gross-Witten tau-functions

Alexander Alexandrov, Saswati Dhara

TL;DR

The paper builds a comprehensive framework for higher BGW tau-functions within the KP hierarchy by constructing canonical KS operators and quantum spectral curves and proving W^{(3)}-constraints for both the original and generalized models. It demonstrates that higher BGW tau-functions are KP tau-functions corresponding to the (m+1)-reduction and provides an algebraic topological recursion via cut-and-join operators, giving explicit recursion relations for the topological expansion. A one-parameter deformation with N leads to generalized higher BGW tau-functions τ^{(m,N)}, with deformed KS algebras, quantum curves, and W-constraints, enriching the geometric interpretation through irregular spectral curves. Collectively, the results connect matrix models, Sato Grassmannian data, and enumerative geometry through explicit operator algebras and recursive structures, applicable to non-semi-simple cohomological field theories and related topological recursion frameworks.

Abstract

In this paper, we consider the higher Brézin--Gross--Witten tau-functions, given by the matrix integrals. For these tau-functions we construct the canonical Kac--Schwarz operators, quantum spectral curves, and $W^{(3)}$-constraints. For the simplest representative we construct the cut-and-join operators, which describe the algebraic version of the topological recursion. We also investigate a one-parametric generalization of the higher Brézin--Gross--Witten tau-functions.

On higher Brézin-Gross-Witten tau-functions

TL;DR

The paper builds a comprehensive framework for higher BGW tau-functions within the KP hierarchy by constructing canonical KS operators and quantum spectral curves and proving W^{(3)}-constraints for both the original and generalized models. It demonstrates that higher BGW tau-functions are KP tau-functions corresponding to the (m+1)-reduction and provides an algebraic topological recursion via cut-and-join operators, giving explicit recursion relations for the topological expansion. A one-parameter deformation with N leads to generalized higher BGW tau-functions τ^{(m,N)}, with deformed KS algebras, quantum curves, and W-constraints, enriching the geometric interpretation through irregular spectral curves. Collectively, the results connect matrix models, Sato Grassmannian data, and enumerative geometry through explicit operator algebras and recursive structures, applicable to non-semi-simple cohomological field theories and related topological recursion frameworks.

Abstract

In this paper, we consider the higher Brézin--Gross--Witten tau-functions, given by the matrix integrals. For these tau-functions we construct the canonical Kac--Schwarz operators, quantum spectral curves, and -constraints. For the simplest representative we construct the cut-and-join operators, which describe the algebraic version of the topological recursion. We also investigate a one-parametric generalization of the higher Brézin--Gross--Witten tau-functions.
Paper Structure (27 sections, 10 theorems, 224 equations)

This paper contains 27 sections, 10 theorems, 224 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Acknowledgments
  4. KP hierarchy and its symmetries
  5. KP hierarchy and Sato Grassmannian
  6. Kac--Schwarz operators
  7. Symmetries of the KP hierarchy
  8. BGW tau-function
  9. BGW tau-function
  10. KS description of the BGW tau-function
  11. Canonical KS operators
  12. Higher BGW tau-functions
  13. Matrix models and tau-functions
  14. Higher BGW model as a tau-function of KP hierarchy
  15. KS algebra for higher BGW tau-functions
  16. ...and 12 more sections

Key Result

Lemma 2.1

If a point of the Sato Grassmannian $\mathcal{W}$ is stabilized by a KS operator where parameters $a_k$, $b_k$, and $c_k$ do not depend on ${\bf t}$, then the corresponding tau-function $\tau_{\mathcal{W}}$ satisfies a linear constraint for some eigenvalue $\mu$ independent of ${\bf t}$.

Theorems & Definitions (31)

  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2.1
  • Theorem 1: KS2
  • Remark 3.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Definition 4
  • ...and 21 more