Spectral triples on proper étale groupoids

Abstract

A properétale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of the tangent bundle and the obstruction to lift the tangent bundle to a spinor bundle. In the case of an orientable and spin Lie groupoid, an invariant spinor bundle and an invariant Dirac operator will be constructed. This data gives rise to a spectral triple. The algebraic orientability axiom in noncommutative geometry is reformulated to make it compatible with the geometric model. MSC: 22A22, 58B34

On noncommutative geometry of orbifolds

“…1.4. Isomorphism classes of vector bundles on a properétale groupoid X ‚ can be described in terms of cohomology theoretic data, [10]. We return to this in 1.5.…”

“…In general, the groupoid cohomology is a cohomology of a double complex where one direction is determined by the complex associated to the simplicial structure and the other direction is determined by an injective resolution. We shall only employ cohomology in degree one and for this reason we will introduce a localization procedure which makes the injective resolution redundant when the degree one cohomology is considered, [10]. Let X ‚ be a Lie groupoid.…”

“…The isomorphism classes of (real or complex) vector bundles on X ‚ correspond to the classes of H 1 pX ‚ , GL k q. This correspondence was described explicitly in [10]: each vector bundle on X ‚ defines a structure cocycle which is represented in H 1 pX ‚ , GL k q. The structure cocycle g has the components g :…”

“…There is a Dirac spectral triple over the algebra of invariant functions C 8 pXq Θ if X ‚ is compact and there is a Dirac spectral triple over the groupoid convolution algebra C 8 c pΞq if X ‚ is effective. A detailed description of these spectral triples is given in [10].…”

Dirac Operators on Quantum Weighted Projective Spaces

“…Whenever the lift exists one can proceed to define an invariant Dirac operator acting on the spinors. This data can be used to model the quotient space as a spectral triple consisting of the invariant function algebra, the Hilbert space of invariant spinors and the invariant Dirac operator, [5] [12].…”