Dirac Operators on Coset Spaces

TL;DR

This paper develops a comprehensive theory of Dirac operators on coset spaces G/H, exploring geometric structures, obstructions to spin structures, and implications for gauge fields and string theory, with explicit spectra on spheres.

Contribution

It systematically analyzes Dirac operators on coset spaces, revealing obstructions to spin structures and their impact on gauge fields, with explicit spectral results on spheres.

Findings

Obstructions to spin and spin_c structures are identified.

Gauge fields required for defining spinors on non-spin manifolds.

Explicit spectra of Dirac operators on spheres S^n are computed.

Abstract

The Dirac operator for a manifold Q, and its chirality operator when Q is even dimensional, have a central role in noncommutative geometry. We systematically develop the theory of this operator when Q=G/H, where G and H are compact connected Lie groups and G is simple. An elementary discussion of the differential geometric and bundle theoretic aspects of G/H, including its projective modules and complex, Kaehler and Riemannian structures, is presented for this purpose. An attractive feature of our approach is that it transparently shows obstructions to spin- and spin_c-structures. When a manifold is spin_c and not spin, U(1) gauge fields have to be introduced in a particular way to define spinors. Likewise, for manifolds like SU(3)/SO(3), which are not even spin_c, we show that SU(2) and higher rank gauge fields have to be introduced to define spinors. This result has potential consequences for string theories if such manifolds occur as D-branes. The spectra and eigenstates of the Dirac operator on spheres S^n=SO(n+1)/SO(n), invariant under SO(n+1), are explicitly found. Aspects of our work overlap with the earlier research of Cahen et al..