J. Bao

Y. He

A. Zahabi

Mahler measure from number theory is used for the first time in physics, yielding “Mahler flow” which extrapolates different phases in QFT.

In string theory, three complex structures, one from dessins d’enfants in number theory, another from QFT and the third from algebraic geometry, describe the torus of a brane tiling. For brane tilings described by reflexive polygons, lattice polygons with a single interior point, we show these three different structures are actually different limits of the Mahler measure, an indicator of a polynomial’s complexity.

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.

Mahler measure for a quiver symphony