Melting Crystal, Quantum Torus and Toda Hierarchy

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.

Final version to be published in Commun. Math. Phys. . A new section is added and devoted to Conclusion and discussion, where, in particular, a possible relation with the generating function of the absolute Gromov-Witten invariants on CP^1 is commented. Two references are added. Typos are corrected. 32 pages. 4 figures

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• q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement
• Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities
• Generalized string equations for double Hurwitz numbers
• Quantum torus symmetries of multicomponent modified KP hierarchy and reductions
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries
• Quantum Torus symmetries of the CKP and multi-component CKP hierarchies
• Slavnov determinants, Yang-Mills structure constants, and discrete KP
• Thermodynamic Limit and Dispersive Regularisation in Matrix Models
• Integrable structure of melting crystal model with two q-parameters
• Open string amplitudes of closed topological vertex
• Matrix models and stochastic growth in Donaldson-Thomas theory
• Extended $5d$ Seiberg-Witten Theory and Melting Crystal
• Instanton Counting and Dielectric Branes
• Remarks on partition functions of topological string theory on generalized conifolds
• Instanton Calculus and Loop Operator in Supersymmetric Gauge Theory
• Orbifold melting crystal models and reductions of Toda hierarchy
• Generalized Ablowitz-Ladik hierarchy in topological string theory
• Modified melting crystal model and Ablowitz-Ladik hierarchy
• Thermodynamic limit of random partitions and dispersionless Toda hierarchy
• Integrable structure of melting crystal model with external potentials
• Integrable structure of modified melting crystal model
• Three-partition Hodge integrals and the topological vertex
• Heisenberg XX chain, non-homogeneously parameterised generating exponential, and diagonally restricted plane partitions
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• $q$-Difference Kac-Schwarz Operators in Topological String Theory
• 4D limit of melting crystal model and its integrable structure
• Coadjoint Orbits and Wilson Loops in Five Dimensional Topological Gauge Theories
• Intertwining operator and integrable hierarchies from topological strings
• Toda tau functions with quantum torus symmetries