Amoebas and Instantons

Abstract

We study a statistical model of random plane partitions. The statistical model has interpretations as five-dimensional $\mathcal{N}=1$ supersymmetric SU(N) Yang-Mills on ${\mathbb{R}}^4\times S^1$ and as Kähler gravity on local SU(N) geometry. At the thermodynamic limit a typical plane partition called the limit shape dominates in the statistical model. The limit shape is linked with a hyperelliptic curve, which is a five-dimensional version of the SU(N) Seiberg-Witten curve. Amoebas and the Ronkin functions play intermediary roles between the limit shape and the hyperelliptic curve. In particular, the Ronkin function realizes an integration of thermodynamical density of the main diagonal partitions, along one-dimensional slice of it and thereby is interpreted as the counting function of gauge instantons. The radius of $S^1$ can be identified with the inverse temperature of the statistical model. The large radius limit of the five-dimensional Yang-Mills is the low temperature limit of the statistical model, where the statistical model is frozen to a ground state that is associated with the local SU(N) geometry. We also show that the low temperature limit corresponds to a certain degeneration of amoebas and the Ronkin functions known as tropical geometry.

58 pages, 28 figures, references added

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Cited by in corpus (15)

• Brane Tilings and Their Applications
• Melting Crystal, Quantum Torus and Toda Hierarchy
• Intersecting Solitons, Amoeba and Tropical Geometry
• Emergent Calabi-Yau Geometry
• Crystal Melting and Wall Crossing Phenomena
• Quantum Torus symmetry of the KP, KdV and BKP hierarchies
• Two-dimensional crystal melting and D4-D2-D0 on toric Calabi-Yau singularities
• Instanton Counting and Matrix Model
• Extended $5d$ Seiberg-Witten Theory and Melting Crystal
• Instanton Counting and Dielectric Branes
• Thermodynamic limit of random partitions and dispersionless Toda hierarchy
• Signatures of partition functions and their complexity reduction through the KP II equation
• Integrable Structure of $5d$ $\mathcal{N}=1$ Supersymmetric Yang-Mills and Melting Crystal
• Topological string amplitudes and Seiberg-Witten prepotentials from the counting of dimers in transverse flux
• Supersymmetric Gauge Theories with Matters, Toric Geometries and Random Partitions