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

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.