The moduli space of non-Abelian vortices in Yang-Mills-Chern-Simons-Higgs theory

TL;DR

This paper calculates the dimension of the moduli space of non-Abelian vortices in a specific 2+1 dimensional gauge theory using index theorems, moduli matrix methods, and string theory techniques.

Contribution

It extends the moduli matrix approach and applies index theorems to determine vortex moduli in Yang-Mills-Chern-Simons-Higgs theory, linking field theory and string theory.

Findings

Derived the moduli space dimension using index theorem.

Extended the moduli matrix approach to this theory.

Connected vortex dynamics with string theory effective Lagrangian.

Abstract

We determine the dimension of the moduli space of non-Abelian vortices in Yang-Mills-Chern-Simons-Higgs theory in 2+1 dimensions for gauge groups $G=({\rm U}(1)\times G')/\mathbb{Z}_{n_0}$ with $G'$ being an arbitrary semi-simple group and $n_0$ the greatest common divisor of the Abelian charges of the $G'$ invariants. The calculation is carried out using a Callias-type index theorem, the moduli matrix approach and a D-brane setup in Type IIB string theory. We prove that the index theorem gives the number of zeromodes or moduli of the non-Abelian vortices, extend the moduli matrix approach to the Yang-Mills-Chern-Simons-Higgs theory and finally derive the effective Lagrangian of Collie and Tong using string theory.