Thermodynamics of Isoradial Quivers and Hyperbolic 3-Manifolds

TL;DR

This paper explores the thermodynamics and phase structure of isoradial quivers in 4d $ ext{N}=2$ theories, linking BPS observables to hyperbolic 3-manifolds and Mahler measure theory, providing new asymptotic analysis methods.

Contribution

It introduces a novel approach connecting Mahler measure, hyperbolic geometry, and BPS sector asymptotics for isoradial quivers, advancing understanding of their thermodynamics.

Findings

Derived BPS free energy and entropy density as functions of R-charges and hyperbolic volumes.

Identified critical R-charges where phase transitions occur.

Provided explicit examples illustrating the theoretical framework.

Abstract

The BPS sector of $\mathcal{N}=2$, $4d$ toric quiver gauge theories, and its corresponding D6-D2-D0 branes on Calabi-Yau threefolds, have been previously studied using integrable lattice models such as the crystal melting model and the dimer model. The asymptotics of the BPS sector, in the large N limit, can be studied using the Mahler measure theory, \cite{Zah}. In this work, we consider the class of isoradial quivers and study their thermodynamical observables and phase structure. Building on our previous results, and using the relation between the Mahler measure and hyperbolic 3-manifolds, we propose a new approach in the asymptotic analysis of the isoradial quivers. As a result, we obtain the observables such as the BPS free energy, the BPS entropy density and growth rate of the isoradial quivers, as a function of the $R$-charges of the quiver and in terms of the hyperbolic volumes and the dilogarithm functions. The phase structure of the isoradial quivers is studied via the analysis of the BPS entropy density at critical $R$-charges and universal results for the phase structure in this class are obtained. Explicit results for the observables are obtained in some concrete examples of the isoradial quivers.