Atiyah-Singer Index Theorem in an SO(3) Yang-Mills-Higgs system and derivation of a charge quantization condition

TL;DR

This paper extends the Atiyah-Singer index theorem to a two-dimensional SO(3) Yang-Mills-Higgs system, deriving a charge quantization condition for non-Abelian monopoles that aligns with previous findings.

Contribution

It generalizes the index theorem to a new gauge system and derives a consistent charge quantization condition using heat kernel methods.

Findings

Derived a charge quantization condition eg=n for non-Abelian monopoles.

Proved the generalized index theorem using heat kernel and nonlinear SU(2) realization.

Connected the quantization integer to the Dirac operator index.

Abstract

The Atiyah-Singer index theorem is generalized to a two-dimensional SO(3) Yang-Mills-Higgs (YMH) system. The generalized theorem is proven by using the heat kernel method and a nonlinear realization of SU(2) gauge symmetry. This theorem is applied to the problem of deriving a charge quantization condition in the four-dimensional SO(3) YMH system with non-Abelian monopoles. The resulting quantization condition, eg=n (n: integer), for an electric charge e and a magnetic charge g is consistent with that found by Arafune, Freund and Goebel. It is shown that the integer n is half of the index of a Dirac operator.