Dirac Operators on Coset Spaces

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..

section on Riemannian structure improved, references added

References in corpus (9)

• Topologically nontrivial field configurations in noncommutative geometry
• Noncommutative Geometry and Gauge Theory on Fuzzy Sphere
• Fuzzy CP2
• Chirality and Dirac Operator on Noncommutative Sphere
• The Weyl Character Formula, the half-spin representations, and equal rank subgroups
• Gravitational fields on a noncommutative space
• The Dirac quantisation condition for fluxes on four-manifolds
• The Connes-Lott program on the sphere
• Connes-Lott model building on the two-sphere

Cited by in corpus (18)

• A New SU(2) Anomaly
• Finite Gauge Theory on Fuzzy CP^2
• The Spectrum of the Dirac Operator on Coset Spaces with Homogeneous Gauge Fields
• Higher Dimensional Quantum Hall Effect as A-Class Topological Insulator
• A universal Dirac operator and noncommutative spin bundles over fuzzy complex projective spaces
• Einstein Metrics on Group Manifolds and Cosets
• Dirac operator on the q-deformed Fuzzy sphere and Its spectrum
• Gauge gravitation theory from the geometric viewpoint
• Scalar and Spinor Field Actions on Fuzzy $S^4$: fuzzy $C{P}^3$ as a $S_F^2$ bundle over $S_F^4$
• Quantum Hall Effect on Odd Spheres
• Dirac operators for coadjoint orbits of compact Lie groups
• Index Theorem for the $q$-Deformed Fuzzy Sphere
• Monopoles, Dirac operator and index theory for fuzzy $SU(3)/(U(1)\times U(1))$
• A projective Dirac operator on CP^2 within fuzzy geometry
• Noncommutative Geometry and Quantum Group Symmetries
• Extended Dynamical Symmetries of Landau Levels in Higher Dimensions
• Some Notes on Dirac operators on the ${\mathrm{S}}^3$ and ${\mathrm{S}}^2$ spheres
• A projective Dirac operator on $\mathbb{C}{P}^n$ and extended SUSY