Melting Crystal, Quantum Torus and Toda Hierarchy

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.

💡 Research Summary

The paper investigates the integrable structures underlying the melting crystal model—equivalently, random plane partitions—by introducing a family of potentials that depend on an infinite set of coupling constants. Starting from the standard statistical weight $q^{|\pi|}$ for a plane partition $\pi$, the authors define functions $\Phi_k(\lambda,p)$ on charged partitions $(\lambda,p)$, which are essentially $q$‑deformed power‑sum type observables. By linearly combining these with couplings $t_k$, they construct a generalized weight $e^{\Phi(t;p)(\pi(0))}$ that depends only on the main diagonal partition $\pi(0)$. The resulting partition function \