Solitons in Supersymmetric Gauge Theories: Moduli Matrix Approach

TL;DR

This paper reviews the moduli matrix approach to classifying and analyzing BPS solitons in supersymmetric U(Nc) gauge theories with multiple Higgs fields, revealing the structure of their moduli spaces and interactions.

Contribution

It introduces the moduli matrix as a comprehensive tool to characterize all BPS solitons and their moduli spaces in supersymmetric gauge theories with eight supercharges.

Findings

The total moduli space of walls is a complex Grassmann manifold.

Composite solitons have charges that can be positive or negative.

The moduli matrix simplifies the construction of effective Lagrangians.

Abstract

We review our recent works on solitons in U(Nc) gauge theories with Nf (>Nc) Higgs fields in the fundamental representation, which possess eight supercharges. The moduli matrix is proposed as a crucial tool to exhaust all BPS solutions, and to characterize all possible moduli parameters. Since vacua are in the Higgs phase, we find domain walls (kinks) and vortices as the only elementary solitons. Stable monopoles and instantons can exist as composite solitons with vortices attached. Webs of walls are also found as another composite soliton. The moduli space of all these elementary as well as composite solitons are found in terms of the moduli matrix. The total moduli space of walls is given by the complex Grassmann manifold SU(Nf)/[SU(Nc)x SU(Nf-Nc) x U(1)] and is decomposed into various topological sectors corresponding to boundary conditions specified by particular vacua. We found charges characterizing composite solitons contribute negatively (either positively or negatively) in Abelian (non-Abelian) gauge theories. Effective Lagrangians are constructed on walls and vortices in a compact form. The power of the moduli matrix is illustrated by an interaction rule of monopoles, vortices, and walls, which is difficult to obtain in other methods. More thorough description of the moduli matrix approach can be found in our review article (hep-th/0602170).