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

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 $\{Φ_j(x)\}_{j=0}^\infty$ of $W$. $q$-difference analogues $A$, $B$ of Kac-Schwarz operators are defined with the aid of $G$. $Φ_j(x)$'s satisfy the linear equations $AΦ_j(x) = q^jΦ_j(x)$, $BΦ_j(x) = Φ_{j+1}(x)$. The lowest equation $AΦ_0(x) = Φ_0(x)$ reproduces the quantum mirror curve in the authors' previous work.