Mahler Measure for a Quiver Symphony

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.

Reflexions on Mahler: Dessins, Modularity and Gauge Theories

“…We find that the dessins obtained in such way are invariant under specular duality (Proposition 3.3). On the other hand, how Mahler measures behave under specular duality has been studied in [15]. Here, with the special maximally tempered coefficients, we find that Mahler measures are invariant for specular duals (Proposition 3.5 and Corollary 3.5.1).…”

“…Physically, this is called the Mahler flow equation [15]. It reveals the monotonic behaviour of the Mahler measure 5 :…”

“…For detailed steps, the readers are referred to, for example, the appendix in [15]. We plot the Mahler flow as In particular, we see that u 0 diverges when k approaches to 4.…”

“…Recently, Mahler measure has been more deeply explored in quiver gauge theories [11][12][13][14][15], which has been proposed as a "universal measure" for toric quivers. In particular, Mahler measure has monotonic behaviour along the so-called Mahler flow between different "limits" of the quiver gauge theory.…”

“…Along the Mahler flow, it encodes a phase transition that has nice interpretations in both mathematics and physics. Furthermore, the Mahler measure has also been found to have intimate relation with a-maximization, and also enjoys salient properties under dualities [15].…”