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 supersymmetric gauge theories and topological string models by focusing on the statistical model of a melting crystal, which is mathematically equivalent to a random plane partition. Starting from the conventional weight exp(−|π|) that penalises the volume |π| of a plane partition π, the authors introduce an infinite family of additional potentials Vₙ(π) with corresponding coupling constants tₙ. These potentials are linear combinations of row‑ or column‑height sums of the partition and can be interpreted physically as non‑local observables analogous to Wilson loops. The modified partition function

 Z({t}) = ∑_π exp