Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and Spin Hurwitz Numbers

TL;DR

This paper explores the relationship between BKP tau-functions, affine coordinates, and Kac-Schwarz operators, leading to the derivation of quantum spectral curves for spin Hurwitz numbers, advancing the understanding of integrable systems and algebraic geometry.

Contribution

It introduces a new formulation of Kac-Schwarz operators for BKP tau-functions and computes the quantum spectral curve for spin single Hurwitz numbers.

Findings

Derived Kac-Schwarz operators satisfying [P,Q]=1

Computed affine coordinates for spin Hurwitz tau-functions

Established the quantum spectral curve for spin Hurwitz numbers

Abstract

Given a tau-function $\tau(t)$ of the BKP hierarchy satisfying $\tau(0)=1$, we discuss the relation between its BKP-affine coordinates on the isotropic Sato Grassmannian and its BKP-wave function. Using this result, we formulate a type of Kac-Schwarz operators for $\tau(t)$ in terms of BKP-affine coordinates. As an example, we compute the affine coordinates of the BKP tau-function for spin single Hurwitz numbers with completed cycles, and find a pair of Kac-Schwarz operators $(P,Q)$ satisfying $[P,Q]=1$. By doing this, we obtain the quantum spectral curve for spin single Hurwitz numbers.