On the index theorem for symplectic orbifolds

Fedosov, Boris; Schulze, Bert-Wolfgang; Тарханов, Николай

doi:10.5802/aif.2061

Cited by 5 publications

(5 citation statements)

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“…In this section, we use the local Riemann-Roch formula in Theorem 5.3 to prove an algebraic index theorem on a symplectic orbifold and thus confirm a conjecture by [13]. As an application of this algebraic index theorem, we provide in Section 6.2 an alternative proof of the Kawasaki index theorem for elliptic operators on orbifolds [19].…”

Section: The Algebraic Index For Orbifoldsmentioning

confidence: 84%

“…In the reduced case it proves a conjecture of Fedosov-Schulze-Tarkhanov [13]. Although they work with the different algebra of invariants of Ah(G 0 ) instead of the crossed product, one can show that in the reduced case the two are Morita equivalent, allowing for a precise translation.…”

Section: Introductionmentioning

confidence: 94%

“…, n. It is proved in [Fe00,Thm. 1.1], see also [FeSchTa,Lem. 7.3], that this functional is a γ-twisted trace density, i.e., satisfies equation (3.5).…”

Section: In An Algebramentioning

confidence: 99%

“…Hereby [13,Section 5]). Using the fact that ψ D E (1) = ψ D E +D E (ê Γ ), we thus obtain the following equality:…”

Section: ) G) By Definition Tr * ([E]) Equals Tr(ê)mentioning

confidence: 99%

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An algebraic index theorem for orbifolds

Pflaum

Posthuma

Tang

2007

Advances in Mathematics

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Using the concept of a twisted trace density on a cyclic groupoid, a trace is constructed on a formal deformation quantization of a symplectic orbifold. An algebraic index theorem for orbifolds follows as a consequence of a local Riemann-Roch theorem for such densities. In the case of a reduced orbifold, this proves a conjecture by Fedosov, Schulze, and Tarkhanov. Finally, it is shown how the Kawasaki index theorem for elliptic operators on orbifolds follows from this algebraic index theorem.

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Section: The Algebraic Index For Orbifoldsmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 94%

“…, n. It is proved in [Fe00,Thm. 1.1], see also [FeSchTa,Lem. 7.3], that this functional is a γ-twisted trace density, i.e., satisfies equation (3.5).…”

Section: In An Algebramentioning

confidence: 99%

“…Hereby [13,Section 5]). Using the fact that ψ D E (1) = ψ D E +D E (ê Γ ), we thus obtain the following equality:…”

Section: ) G) By Definition Tr * ([E]) Equals Tr(ê)mentioning

confidence: 99%

See 2 more Smart Citations

An algebraic index theorem for orbifolds

Pflaum

Posthuma

Tang

2007

Advances in Mathematics

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“…For the second claim note first that ι * κ is well-defined indeed, since by the assumption (d) on the equivariant embedding (F, ι), the induced map ι : Conj(Γ) → Conj(Γ ′ ) has to be injective. Next we conclude from [FeSchTa,Cor. 7.5] that…”

Section: Traces On the Deformed Groupoid Algebramentioning

confidence: 85%

Homology of formal deformations of proper étale Lie groupoids

Neumaier

Pflaum

Posthuma

et al. 2006

Journal Fur Die Reine Und Angewandte Mathematik (Crelles Journal)

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In this article, the cyclic homology theory of formal deformation quantizations of the convolution algebra associated to a properétale Lie groupoid is studied. We compute the Hochschild cohomology of the convolution algebra and express it in terms of alternating multi-vector fields on the associated inertia groupoid. We introduce a noncommutative Poisson homology whose computation enables us to determine the Hochschild homology of formal deformations of the convolution algebra. Then it is shown that the cyclic (co)homology of such formal deformations can be described by an appropriate sheaf cohomology theory. This enables us to determine the corresponding cyclic homology groups in terms of orbifold cohomology of the underlying orbifold. Using the thus obtained description of cyclic cohomology of the deformed convolution algebra, we give a complete classification of all traces on this formal deformation, and provide an explicit construction.

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