On noncommutative geometry of orbifolds

Abstract

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]

Morita equivalence and spectral triples on noncommutative orbifolds

“…Therefore, we need to introduce additional axiom for the Morita equivalence and require that a Morita equivalence operates as a unitary equivalence on the level of invariant spectral triples. This holds in the case of geometric orbifolds, [7].…”

“…It follows that φ # preserves the lengths of gradients. Now we have supt|apxq´apx 1 q| : a P CpX´Σ G q G , ||rð, as|| ď 1u (7) " supt|bprxsq´bprx 1 sq| : b P CppX´Σ G q{Gq, ||rφ # ð, bs|| ď 1u.…”

“…In both cases we have assumed an effectiveness condition. In the spectral triple language the effectiveness of a group action on a manifold translates to the faithfulness of the representation of the crossed product algebra on the Hilbert space of spinors, [7]. However, it should be noted that in both cases, Example 1 and Example 2, all the spectral triple axioms apart from the faithfulness of the representation remain to hold if the effectiveness is no longer required.…”

“…This is the differential form on Y which is uniquely determined by the equality ˚p νq " α˚pφ # νq. One can apply local sections of α to pull ˚p νq to a globally defined form on Y , [7].…”

“…We have the unitary equivalence of spectral triples [4]. The dimension condition M4 holds because in the coordinates of Y , the induced spinor bundle φ # F Σ and the induced Dirac operator φ # ð, are a actual spinor bundle and a Dirac operator on Y , [7] 4.3, and in this case the spectral dimension will be preserved because the dimension of the base is preserved.…”