Modified melting crystal model and Ablowitz-Ladik hierarchy

TL;DR

This paper reveals the integrable structure of a modified melting crystal model in topological string theory, connecting it to the 2D Toda and Ablowitz-Ladik hierarchies through quantum algebra and matrix factorizations.

Contribution

It demonstrates that the partition function of the model is a tau function of the 2D Toda hierarchy and relates it to the Ablowitz-Ladik hierarchy using shift symmetries and matrix factorizations.

Findings

Partition function expressed as vacuum expectation value of fermion operators.

Partition function is a tau function of the 2D Toda hierarchy.

Solution also satisfies the Ablowitz-Ladik hierarchy.

Abstract

This paper addresses the issue of integrable structure in a modified melting crystal model of topological string theory on the resolved conifold. The partition function can be expressed as the vacuum expectation value of an operator on the Fock space of 2D complex free fermion fields. The quantum torus algebra of fermion bilinears behind this expression is shown to have an extended set of "shift symmetries". They are used to prove that the partition function (deformed by external potentials) is essentially a tau function of the 2D Toda hierarchy. This special solution of the 2D Toda hierarchy can be characterized by a factorization problem of $\ZZ\times\ZZ$ matrices as well. The associated Lax operators turn out to be quotients of first order difference operators. This implies that the solution of the 2D Toda hierarchy in question is actually a solution of the Ablowitz-Ladik (equivalently, relativistic Toda) hierarchy. As a byproduct, the shift symmetries are shown to be related to matrix-valued quantum dilogarithmic functions.