# Kaluza-Klein Reduction of the 6 Dimensional \\ Dirac Equation on $\mathbb{S}^3 \cong SU(2)$ and \\ Non-abelian Topological Insulators

## TL;DR

This paper explores the Kaluza-Klein reduction of the Dirac equation on a 6D spacetime with an A^3
structure, deriving non-minimal SU(2) couplings that could shed light on non-abelian topological insulators.

## Contribution

It introduces a novel dimensional reduction approach for the Dirac equation on A^3
, revealing non-minimal SU(2) interactions relevant to topological insulators.

## Key findings

- Derived non-minimal SU(2) couplings from the reduction
- Compared non-minimal and minimal SU(2) Dirac equations
- Provided insights into non-abelian interactions in topological phases

## Abstract

In this work, the Kaluza-Klein reduction of the Dirac equation on a 6 dimensional spacetime $\mathbb{M}^{1+5} := \mathbb{M}^{1+2} \times \mathbb{S}^3$ is studied. Because of the group structure on $\mathbb{S}^3$, $\mathbb{M}^{1+5}$ can be seen as a principal $SU(2)$ bundle over the model Lorentzian spacetime $\mathbb{M}^{1+2}$. The dimensional reduction induces non-minimal $SU(2)$ couplings to the theory on $\mathbb{M}^{1+2}$. These interaction terms will be investigated by comparing with a minimally $SU(2)$ coupled Dirac equation on $\mathbb{M}^{1+2}$. We hope that these results may help us to understand non-abelian interactions of topological insulators.