Kac-Schwarz Operators of Type $B$, Quantum Spectral Curves, and 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.

References in corpus (3)

• Superintegrability of Kontsevich matrix model
• Enumerative geometry, tau-functions and Heisenberg-Virasoro algebra
• BKP Hierarchy, Affine Coordinates, and a Formula for Connected Bosonic $N$-Point Functions

Cited by in corpus (1)

• Diagonal Tau-Functions of 2D Toda Lattice Hierarchy, Connected $(n,m)$-Point Functions, and Double Hurwitz Numbers