Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 009, 28 pages      arXiv:1609.00882
Contribution to the Special Issue on Combinatorics of Moduli Spaces: Integrability, Cohomology, Quantisation, and Beyond

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

Kanehisa Takasaki a and Toshio Nakatsu b
a) Department of Mathematics, Kinki University, 3-4-1 Kowakae, Higashi-Osaka, Osaka 577-8502, Japan
b) Institute of Fundamental Sciences, Setsunan University, 17-8 Ikeda Nakamachi, Neyagawa, Osaka 572-8508, Japan

Received September 08, 2016, in final form February 17, 2017; Published online February 21, 2017

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.

Key words: topological vertex; mirror symmetry; quantum curve; $q$-difference equation; KP hierarchy; Kac-Schwarz operator.

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