Morita Equivalence and Spectral Triples on Noncommutative Orbifolds

On Noncommutative Geometry of Orbifolds

An orbifold is a Morita equivalence class of a proper {\' e}tale 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.

Residue formula for Morita-Futaki-Bott invariant on orbifolds

In this work we prove a residue formula for Morita-Futaki-Bott invariant with respect any holomorphic vector fields with isolated (possibly degenerated) singularities in terms of Grothendieck's residues.

良轨道上的扭指数理论I：非对易Bloch理论

Twisted index theory on good orbifolds, I: noncommutative Bloch theory

本文与第二部分一起推广了数学DG/9803051的结果。在第一部分中，我们研究了紧良轨道的轨道覆盖空间上的椭圆算子的扭指数理论，其中良轨道在轨道基本群的投射作用下是不变的。我们应用这些结果，以获得定性结果的真实和复杂的双曲空间在2和4维，有关的推广的Bethe-Sommerfeld猜想和十马提尼问题，频谱上的自伴椭圆算子是不变的投射作用下的离散余紧群。

Quasifolds, Diffeology and Noncommutative Geometry

After embedding the objects quasifolds into the category {Diffeology}, we associate a C*-agebra with every atlas of any quasifold, and show how different atlases give Morita equivalent algebras. This builds a new bridge between diffeology and noncommutative geometry (beginning with the today classical example of the irrational torus) which associates a Morita class of C*-algebras with a diffeomorphic class of quasifolds.

Modular curvature and Morita equivalence

The curvature of the noncommutative torus $T^2_θ$ ($θ$ irrational) endowed with a noncommutative conformal metric has been the focus of attention of several recent works. Continuing the approach taken in the paper [A. Connes and H. Moscovici, http://arxiv.org/abs/1110.3500] we extend the study of the curvature to twisted Dirac spectral triples constructed out of Heisenberg bimodules that implement the Morita equivalence of the $C^*$-algebra $A_θ= C(T^2_θ)$ with other toric algebras $A_{θ'}$. In the enlarged context the conformal metric on $T^2_θ$ is exchanged with an arbitrary Hermitian metric…

Noncommutative Poisson structures on Orbifolds

In this paper, we compute the Gerstenhaber bracket on the Hoch-schild cohomology of $C^\infty(M)\rtimes G$ for a finite group $G$ acting on a compact manifold $M$. Using this computation, we obtain geometric descriptions for all noncommutative Poisson structures on $C^\infty(M)\rtimes G$ when $M$ is a symplectic manifold. We also discuss examples of deformation quantizations of these noncommutative Poisson structures.