Extended $5d$ Seiberg-Witten Theory and Melting Crystal

Abstract

We study an extension of the Seiberg-Witten theory of $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills on ${\mathbb{R}}^4\times S^1$. We investigate correlation functions among loop operators. These are the operators analogous to the Wilson loops encircling the fifth-dimensional circle and give rise to physical observables of topological-twisted $5d$ $\mathcal{N}=1$ supersymmetric Yang-Mills in the $\Omega$ background. The correlation functions are computed by using the localization technique. Generating function of the correlation functions of U(1) theory is expressed as a statistical sum over partitions and reproduces the partition function of the melting crystal model with external potentials. The generating function becomes a $\tau$ function of 1-Toda hierarchy, where the coupling constants of the loop operators are interpreted as time variables of 1-Toda hierarchy. The thermodynamic limit of the partition function of this model is studied. We solve a Riemann-Hilbert problem that determines the limit shape of the main diagonal slice of random plane partitions in the presence of external potentials, and identify a relevant complex curve and the associated Seiberg-Witten differential.

Final version to be published in Nucl. Phys. B. Typos are corrected. 38 pages, 4 figures

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• Small Instantons, Little Strings and Free Fermions
• Melting Crystal, Quantum Torus and Toda Hierarchy
• On certain aspects of string theory/gauge theory correspondence
• On Microscopic Origin of Integrability in Seiberg-Witten Theory
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Cited by in corpus (1)

• On two-dimensional quantum gravity and quasiclassical integrable hierarchies