Integrable structure of melting crystal model with two q-parameters

Abstract

This paper explores integrable structures of a generalized melting crystal model that has two $q$-parameters $q_1,q_2$. This model, like the ordinary one with a single $q$-parameter, is formulated as a model of random plane partitions (or, equivalently, random 3D Young diagrams). The Boltzmann weight contains an infinite number of external potentials that depend on the shape of the diagonal slice of plane partitions. The partition function is thereby a function of an infinite number of coupling constants $t_1,t_2,...$ and an extra one $Q$. There is a compact expression of this partition function in the language of a 2D complex free fermion system, from which one can see the presence of a quantum torus algebra behind this model. The partition function turns out to be a tau function (times a simple factor) of two integrable structures simultaneously. The first integrable structure is the bigraded Toda hierarchy, which determine the dependence on $t_1,t_2,...$. This integrable structure emerges when the $q$-parameters $q_1,q_2$ take special values. The second integrable structure is a $q$-difference analogue of the 1D Toda equation. The partition function satisfies this $q$-difference equation with respect to $Q$. Unlike the bigraded Toda hierarchy, this integrable structure exists for any values of $q_1,q_2$.

27 pages, no figure, latex2e(package amsmath,amssymb,amsthm); (v2) typos in eq. (62) and the next equation corrected, version accepted for publication

References in corpus (17)

• Integrable Hierarchies and Dispersionless Limit
• Instanton counting on blowup. I. 4-dimensional pure gauge theory
• Extended Seiberg-Witten Theory and Integrable Hierarchy
• The Extended Bigraded Toda hierarchy
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Phase transitions, double-scaling limit, and topological strings
• Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
• Five-Dimensional Supersymmetric Yang-Mills Theories and Random Plane Partitions
• The solution to the q-KdV equation
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
• Free Fermion and Seiberg-Witten Differential in Random Plane Partitions
• On a c-number quantum $\tau$-function
• $q$-analogue of modified KP hierarchy and its quasi-classical limit
• Differential Fay identities and auxiliary linear problem of integrable hiearchies
• Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit
• Yang-Mills Gauge Conditions from Witten's Open String Field Theory
• Integrable structure of melting crystal model with external potentials

Cited by in corpus (4)

• Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra
• Cluster Toda chains and Nekrasov functions
• Mirror of the refined topological vertex from a matrix model
• Orbifold melting crystal models and reductions of Toda hierarchy