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

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.

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• Quantum Spin Hall Insulator State in HgTe Quantum Wells
• Non-Abelian gauge fields and topological insulators in shaken optical lattices
• Topological Insulators and Superconductors from String Theory