Mahler Measure for a Quiver Symphony

TL;DR

This paper introduces the Mahler measure from number theory into toric quiver gauge theories, revealing its role as a universal dynamical measure, its geometric interpretation via amoebas, and its invariance under Seiberg duality.

Contribution

It proposes the Mahler measure as a universal quiver measure, linking it to physical dynamics, geometric structures, and dualities, with novel insights into phase transitions and tropical limits.

Findings

Mahler measure encodes quiver dynamics and is monotonic along a flow.

Maximization of Mahler measure corresponds to $a$-maximization.

Mahler measure remains invariant under Seiberg duality.

Abstract

Adopting the Mahler measure from number theory, we introduce it to toric quiver gauge theories, and study some of its salient features and physical implications. We propose that the Mahler measure is a universal measure for the quiver, encoding its dynamics with the monotonic behaviour along a so-called Mahler flow including two special points at isoradial and tropical limits. Along the flow, the amoeba, from tropical geometry, provides geometric interpretations for the dynamics of the quiver. In the isoradial limit, the maximization of Mahler measure is shown to be equivalent to $a$-maximization. The Mahler measure and its derivative are closely related to the master space, leading to the property that the specular duals have the same functions as coefficients in their expansions, hinting the emergence of a free theory in the tropical limit. Moreover, they indicate the existence of phase transition. We also find that the Mahler measure should be invariant under Seiberg duality.