Kac-Schwarz operators of type B, quantum spectral curves, and spin Hurwitz numbers

Abstract

Given a tau-function τ(t) of the BKP hierarchy satisfying τ(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 τ(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.