Spin Hurwitz theory and Miwa transform for the Schur Q-functions

TL;DR

This paper develops a Miwa transform representation for spin Schur Q-functions involving fermionic matrices, advancing the understanding of their algebraic structure and applications in matrix models and spin Hurwitz theory.

Contribution

It introduces a novel Miwa parametrization for spin Schur Q-functions using supermatrices, addressing a key gap in their matrix model representation.

Findings

Derived a fermionic matrix-based Miwa representation for spin Schur Q-functions.

Connected the representation to the algebra of spin Hurwitz theory and Sergeev group.

Enhanced the matrix model techniques for spin-related functions.

Abstract

Schur functions are the common eigenfunctions of generalized cut-and-join operators which form a closed algebra. They can be expressed as differential operators in time-variables and also through the eigenvalues of auxiliary $N\times N$ matrices $X$, known as Miwa variables. Relevant for the cubic Kontsevich model and also for spin Hurwitz theory is an alternative set of Schur Q-functions. They appear in representation theory of the Sergeev group, which is a substitute of the symmetric group, related to the queer Lie superalgebras $\mathfrak{q}(N)$.. The corresponding spin $\hat{\cal W}$-operators were recently found in terms of time-derivatives, but a substitute of the Miwa parametrization remained unknown, which is an essential complication for the matrix model technique and further developments. We demonstrate that the Miwa representation, in this case, involves a fermionic matrix $\Psi$ in addition to $X$, but its realization using supermatrices is {\it not} quite naive.