Amoebas and Instantons

TL;DR

This paper explores the connection between random plane partitions, gauge theories, and algebraic geometry, revealing how limit shapes relate to hyperelliptic curves, amoebas, and Ronkin functions, with implications for instanton counting and low-temperature limits.

Contribution

It introduces a novel link between the limit shape of plane partitions and hyperelliptic curves, amoebas, and Ronkin functions, providing new insights into gauge instantons and tropical geometry.

Findings

The limit shape is connected to a hyperelliptic curve resembling the Seiberg-Witten curve.

Ronkin functions interpret the counting of gauge instantons.

Low temperature limit corresponds to tropical geometry degeneration.

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 K\"ahler 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.