Intersecting Solitons, Amoeba and Tropical Geometry

TL;DR

This paper explores the intersection of vortices and instantons in five-dimensional supersymmetric gauge theories, linking their geometric configurations to amoeba and tropical geometry, and extends these concepts to non-Abelian theories.

Contribution

It establishes a mathematical framework connecting solitons in gauge theories to amoeba and tropical geometry, providing new insights into vortex sheet configurations and moduli space metrics.

Findings

A dictionary relating gauge theory solitons to amoeba and tropical geometry.

Description of vortex sheets via amoeba shapes and charge distributions.

Extension of geometric methods to non-Abelian gauge theories.

Abstract

We study generic intersection (or web) of vortices with instantons inside, which is a 1/4 BPS state in the Higgs phase of five-dimensional N=1 supersymmetric U(Nc) gauge theory on R_t \times (C^\ast)^2 \simeq R^{2,1} \times T^2 with Nf=Nc Higgs scalars in the fundamental representation. In the case of the Abelian-Higgs model (Nf=Nc=1), the intersecting vortex sheets can be beautifully understood in a mathematical framework of amoeba and tropical geometry, and we propose a dictionary relating solitons and gauge theory to amoeba and tropical geometry. A projective shape of vortex sheets is described by the amoeba. Vortex charge density is uniformly distributed among vortex sheets, and negative contribution to instanton charge density is understood as the complex Monge-Ampere measure with respect to a plurisubharmonic function on (C^\ast)^2. The Wilson loops in T^2 are related with derivatives of the Ronkin function. The general form of the Kahler potential and the asymptotic metric of the moduli space of a vortex loop are obtained as a by-product. Our discussion works generally in non-Abelian gauge theories, which suggests a non-Abelian generalization of the amoeba and tropical geometry.