Quiver Asymptotics and Amoeba: Instantons on Toric Divisors of Calabi-Yau Threefolds

TL;DR

This paper explores the asymptotic behavior of quiver gauge theories related to Calabi-Yau threefolds using tropical geometry, revealing phase transitions and thermodynamic properties from crystal models and Amoeba boundaries.

Contribution

It introduces a novel approach connecting tropical geometry and Amoeba analysis to compute thermodynamic observables in quiver gauge theories on Calabi-Yau divisors.

Findings

Explicit formulas for free energy, entropy, and growth rate from crystal limit shapes.

Identification of a Hagedorn phase transition in the instanton sector.

Concrete examples demonstrating the theoretical framework.

Abstract

The BPS bound states of D4-D2-D0 branes on the non-compact divisors of Calabi-Yau threefolds and the instantons in the dual quiver gauge theories are previously studied using two-dimensional crystal melting model and dimer model. Using the tropical geometry associated with the toric quiver, we study the asymptotic of the quiver gauge theory to compute some of their thermodynamic observables and extract the phase structure. We obtain that the thermodynamic observables such as free energy, entropy and growth rate are explicitly obtained from the limit shape of the crystal model, the boundary of the Amoeba and its Harnack curve characterization. Furthermore, we observe that there is a Hagedorn phase transition in the instanton sector inferred from the Gumbel distribution of the fluctuations in the crystal model. We present explicit computations of the results in some concrete examples of $\mathbb{C}^3$, conifold, local $\mathbb{P}^1\times \mathbb{P}^1$ and local $\mathbb{P}^2$ quivers.