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101
Noncommutative geometry, quantum fields and motives
 Colloquium Publications, Vol.55, American Mathematical Society
, 2008
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DBranes, RRFields and Duality ON NONCOMMUTATIVE MANIFOLDS
, 2006
"... We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of Tduality as well as establishing a very general formula for Dbrane charges. This formula is closely related to a noncommutative GrothendieckRiemannRoch theorem that is ..."
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Cited by 25 (1 self)
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We develop some of the ingredients needed for string theory on noncommutative spacetimes, proposing an axiomatic formulation of Tduality as well as establishing a very general formula for Dbrane charges. This formula is closely related to a noncommutative GrothendieckRiemannRoch theorem that is proved here. Our approach relies on a very general form of Poincaré duality, which is studied here in detail. Among the technical tools employed are calculations with iterated products in bivariant Ktheory and cyclic theory, which are simplified using a novel diagram calculus reminiscent of Feynman diagrams.
Index theory, eta forms, and Deligne cohomology
, 2008
"... The paper sets up a language to deal with Dirac operators on manifolds with corners of arbitrary codimension. In particular we develop a precise theory of boundary reductions. 2. We introduce the notion of a taming of a Dirac operator as an invertible perturbation by a smoothing operator. Given a Di ..."
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Cited by 24 (13 self)
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The paper sets up a language to deal with Dirac operators on manifolds with corners of arbitrary codimension. In particular we develop a precise theory of boundary reductions. 2. We introduce the notion of a taming of a Dirac operator as an invertible perturbation by a smoothing operator. Given a Dirac operator on a manifold with boundary faces we use the tamings of its boundary reductions in order to turn the operator into a Fredholm operator. Its index is an obstruction against extending the taming from the boundary to the interior. In this way we develop an inductive procedure to associate Fredholm operators to Dirac operators on manifolds with corners and develop the associated obstruction theory. 3. A central problem of index theory is to calculate the Chern character of the index of a family of Dirac operators. Local index theory uses the heat semigroup of an associated superconnection in order to produce differential forms representing this Chern character. In this paper we develop a version of local index theory for families
Equivariant Kasparov theory and generalized homomorphisms, Ktheory 21
, 2000
"... Abstract. Let G be a locally compact group. We describe elements of KKG (A, B) by equivariant homomorphisms, following Cuntz’s treatment in the nonequivariant case. This yields another proof for the universal property of KKG: It is the universal split exact stable homotopy functor. To describe a Ka ..."
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Cited by 22 (7 self)
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Abstract. Let G be a locally compact group. We describe elements of KKG (A, B) by equivariant homomorphisms, following Cuntz’s treatment in the nonequivariant case. This yields another proof for the universal property of KKG: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, φ, F) for A, B by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L2G)⊗A ′ and more generally if the group action on A is proper in the sense of Exel and Rieffel. 1.
Unbounded bivariant Ktheory and correspondences in noncommutative geometry
"... Abstract. By adapting the algebraic notion of universal connection to the setting of unbounded KKcycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion ..."
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Abstract. By adapting the algebraic notion of universal connection to the setting of unbounded KKcycles, we show that the Kasparov product of such cycles can be defined directly, by an algebraic formula. In order to achieve this it is necessary to develop a framework of smooth algebras and a notion of differentiable C ∗module. The theory of operator spaces provides the required tools. Finally, the above mentioned KKcycles with connection can be viewed as the morphisms in a category whose objects are spectral triples.
SPECTRAL SECTIONS AND HIGHER ATIYAHPATODISINGER INDEX THEORY ON GALOIS COVERINGS
 GAFA GEOMETRIC AND FUNCTIONAL ANALYSIS
, 1998
"... In this paper we consider Γ → ˜ M → M, a Galois covering with boundary and ˜ D/, a Γinvariant generalized Dirac operator on ˜M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary op ..."
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Cited by 20 (7 self)
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In this paper we consider Γ → ˜ M → M, a Galois covering with boundary and ˜ D/, a Γinvariant generalized Dirac operator on ˜M. We assume that the group Γ is of polynomial growth with respect to a word metric. By employing the notion of noncommutative spectral section associated to the boundary operator ˜ D/ 0 and the bcalculus on Galois coverings with boundary, we develop a higher AtiyahPatodiSinger index theory. Our main theorem extends to such ΓGalois coverings with boundary the higher index theorem of ConnesMoscovici.
Groupoids and an index theorem for conical pseudomanifolds
, 2006
"... We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommutative alge ..."
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Cited by 16 (5 self)
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We define an analytical index map and a topological index map for conical pseudomanifolds. These constructions generalize the analogous constructions used by Atiyah and Singer in the proof of their topological index theorem for a smooth, compact manifold M. A main ingredient is a noncommutative algebra that plays in our setting the role of C0(T ∗ M). We prove a Thom isomorphism between noncommutative algebras which gives a new example of wrong way functoriality in Ktheory. We then give a new proof of the AtiyahSinger index theorem using deformation groupoids and show how it generalizes to conical pseudomanifolds. We thus prove a topological index theorem for conical pseudomanifolds.