$q$-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 $|W⟩$ in the fermionic Fock space that represents a point $W$ of the Sato Grassmannian. $|W⟩$ is generated from the vacuum vector $|0⟩$ 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.

References in corpus (37)

• Matrix Models, Topological Strings, and Supersymmetric Gauge Theories
• The Topological Vertex
• Topological Strings and Integrable Hierarchies
• Quantum Dilogarithm
• Vortex Counting and Lagrangian 3-manifolds
• Open string amplitudes and large order behavior in topological string theory
• Remodeling the B-model
• Abelian Current Algebra and the Virasoro Algebra on the Lattice
• Generalized Kazakov-Migdal-Kontsevich Model: group theory aspects
• A & B model approaches to surface operators and Toda theories
• Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
• On KP-integrable Hurwitz functions
• A-polynomial, B-model, and Quantization
• Vertices, Vortices & Interacting Surface Operators
• Surface Operator, Bubbling Calabi-Yau and AGT Relation
• Hypergeometric τ-functions, Hurwitz numbers and enumeration of paths
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Quantum Curves and D-Modules
• Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes
• The spectral curve and the Schroedinger equation of double Hurwitz numbers and higher spin structures
• Multivariate hypergeometric functions as tau functions of Toda lattice and Kadomtsev-Petviashvili equation
• The Wave Function Behavior of the Open Topological String Partition Function on the Conifold
• Wall-crossing, free fermions and crystal melting
• Emergent Geometry and Mirror Symmetry of A Point
• All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds
• Crystal Model for the Closed Topological Vertex Geometry
• Non-compact Topological Branes on Conifold
• Generalized string equations for double Hurwitz numbers
• Quantum Mirror Curves for ${\mathbb{C}}^3$ and the Resolved Confiold
• Quantum curves
• Laplacian growth in a channel and Hurwitz numbers
• Multispecies quantum Hurwitz numbers
• Open string amplitudes of closed topological vertex
• Remarks on partition functions of topological string theory on generalized conifolds
• Weighted Hurwitz numbers and hypergeometric $\tau$-functions: an overview
• Phase space polarization and the topological string: a case study
• Radionic Non-uniform Stable Black Strings

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• Integrable structures of specialized hypergeometric tau functions
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