Twisted K-theory on a manifold X , with twisting in the 3rd integral cohomology, is discussed in the case when X is a product of a circle T and a manifold M . The twist is assumed to be decomposable as a cup product of the basic integral one form on T and an integral class in H 2 .M;Z/. This case was studied some time ago by V. Mathai, R. Melrose, and I.M. Singer. Our aim is to give an explicit construction for the twisted K-theory classes using a quantum field theory model, in the same spirit as the supersymmetric WessZumino-Witten model is used for constructing (equivariant) twisted K-theory classes on compact Lie groups.

An orbifold is a Morita equivalence class of a properétale Lie groupoid. A unitary equivalence class of spectral triples over the algebra of smooth invariant functions are associated with any compact spin orbifold. In the case of an effective spin orbifold we construct a collection of spectral triples over the smooth convolution algebras of the representatives of the Morita equivalence class. MSC 58B34, 22A22, 57R18arXiv:1405.7139v4 [math.DG]

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

The quantum weighted projective algebras CrWP k,l,q s are coinvariant subalgebras of the quantum group algebra CrSU2,qs. For each pair of indices k, l, two 2-summable spectral triples will be constructed. The first one is an odd spectral triple based on coinvariant spinors on CrSU2,qs. The second one is an even spectral triple.MSC: 17B37, 58B34.