Melting Crystal, Quantum Torus and Toda Hierarchy

TL;DR

This paper reveals a deep connection between melting crystal models, quantum torus Lie algebra, and integrable systems, specifically showing that certain partition functions form tau functions of the Toda hierarchy, with implications for supersymmetric gauge theories and topological strings.

Contribution

It demonstrates that melting crystal partition functions can be expressed as tau functions of the Toda hierarchy and links them to quantum torus Lie algebra, providing new insights into integrable structures in gauge theories and topological strings.

Findings

Partition functions form tau functions of the Toda hierarchy.

Potentials correspond to gauge theory observables like Wilson loops.

Possible topology change in topological strings due to observable condensation.

Abstract

Searching for the integrable structures of supersymmetric gauge theories and topological strings, we study melting crystal, which is known as random plane partition, from the viewpoint of integrable systems. We show that a series of partition functions of melting crystals gives rise to a tau function of the one-dimensional Toda hierarchy, where the models are defined by adding suitable potentials, endowed with a series of coupling constants, to the standard statistical weight. These potentials can be converted to a commutative sub-algebra of quantum torus Lie algebra. This perspective reveals a remarkable connection between random plane partition and quantum torus Lie algebra, and substantially enables to prove the statement. Based on the result, we briefly argue the integrable structures of five-dimensional $\mathcal{N}=1$ supersymmetric gauge theories and $A$-model topological strings. The aforementioned potentials correspond to gauge theory observables analogous to the Wilson loops, and thereby the partition functions are translated in the gauge theory to generating functions of their correlators. In topological strings, we particularly comment on a possibility of topology change caused by condensation of these observables, giving a simple example.