$q$-Difference Kac-Schwarz Operators in Topological String Theory

TL;DR

This paper introduces $q$-difference Kac-Schwarz operators in topological string theory, linking quantum mirror curves, tau functions, and fermionic representations to deepen understanding of the mathematical structures underlying the theory.

Contribution

The authors develop $q$-difference analogues of Kac-Schwarz operators and connect them to quantum mirror curves and tau functions in topological string theory.

Findings

Defined $q$-difference Kac-Schwarz operators $A$, $B$

Established linear equations satisfied by basis functions $\

Reproduced quantum mirror curve from the lowest equation

Abstract

The perspective of Kac-Schwarz operators is introduced to the authors' previous work on the quantum mirror curves of topological string theory in strip geometry and closed topological vertex. Open string amplitudes on each leg of the web diagram of such geometry can be packed into a multi-variate generating function. This generating function turns out to be a tau function of the KP hierarchy. The tau function has a fermionic expression, from which one finds a vector $|W\rangle$ in the fermionic Fock space that represents a point $W$ of the Sato Grassmannian. $|W\rangle$ is generated from the vacuum vector $|0\rangle$ by an operator $g$ on the Fock space. $g$ determines an operator $G$ on the space $V = \mathbb{C}((x))$ of Laurent series in which $W$ is realized as a linear subspace. $G$ generates an admissible basis $\{\Phi_j(x)\}_{j=0}^\infty$ of $W$. $q$-difference analogues $A$, $B$ of Kac-Schwarz operators are defined with the aid of $G$. $\Phi_j(x)$'s satisfy the linear equations $A\Phi_j(x) = q^j\Phi_j(x)$, $B\Phi_j(x) = \Phi_{j+1}(x)$. The lowest equation $A\Phi_0(x) = \Phi_0(x)$ reproduces the quantum mirror curve in the authors' previous work.