Tropical Geometry and Five Dimensional Higgs Branches at Infinite Coupling

TL;DR

This paper uses tropical geometry to analyze the Higgs branches of five-dimensional SQCD theories at infinite coupling, revealing complex moduli space structures with multiple components and intersections.

Contribution

It introduces the application of tropical geometry's stable intersection to compute Higgs branches for a broad class of 5D SQCD theories, extending beyond previously known cases.

Findings

Discovery of multiple Higgs branch components depending on parameters

Identification of intersections as closures of nilpotent orbits

Extension of analysis to almost all theories in the family

Abstract

Superconformal five dimensional theories have a rich structure of phases and brane webs play a crucial role in studying their properties. This paper is devoted to the study of a three parameter family of SQCD theories, given by the number of colors $N_c$ for an $SU(N_c)$ gauge theory, number of fundamental flavors $N_f$, and the Chern Simons level $k$. The study of their infinite coupling Higgs branch is a long standing problem and reveals a rich pattern of moduli spaces, depending on the 3 values in a critical way. For a generic choice of the parameters we find a surprising number of 3 different components, with intersections that are closures of height 2 nilpotent orbits of the flavor symmetry. This is in contrast to previous studies where except for one case ($N_c=2, N_f=2$), the parameters were restricted to the cases of Higgs branches that have only one component. The new feature is achieved thanks to a concept in tropical geometry which is called stable intersection and allows for a computation of the Higgs branch to almost all the cases which were previously unknown for this three parameter family apart form certain small number of exceptional theories with low rank gauge group. A crucial feature in the construction of the Higgs branch is the notion of dressed monopole operators.