A Local Equivariant Index Theorem for Sub-Signature Operators

This paper is the second part of a series of papers on noncommutative geometry and conformal geometry. In this paper, we compute explicitly the Connes–Chern character of an equivariant Dirac spectral triple. The formula that we obtain is used in the first paper of the series. The computation has two main steps. The first step is the justification that the CM cocycle represents the Connes–Chern character. The second step is the computation of the CM cocycle as a byproduct of a new proof of the local equivariant index theorem of Donnelly–Patodi, Gilkey and Kawasaki. The proof combines the rescaling method of Getzler with an equivariant version of the Greiner–Hadamard approach to the heat kernel asymptotics. Finally, as a further application of this approach, we compute the short-time limit of the JLO cocycle of an equivariant Dirac spectral triple.

In this paper, we prove a local odd dimensional equivariant family index theorem which generalizes Freed’s odd dimensional index formula. Then we extend this theorem to the noncommutative geometry framework. As a corollary, we get the odd family Lichnerowicz vanishing theorem and the odd family Atiyah-Hirzebruch vanishing theorem.

We report on the construction of secondary invariants in connection with the Atiyah-Singer index theorem for families, and the Theorem of Riemann-Roch-Grothendieck. The local families index theorem plays an important role in the construction. In complex geometry, the corresponding objects are the analytic torsion forms and the analytic torsion currents. These objects exhibit natural functorial properties with respect to composition of maps. Gillet and Soulé have used these objects to prove a Riemann-Roch theorem in Arakelov geometry. Also we state a Riemann-Roch theorem for flat vector bundles, and report on the construction of corresponding higher analytic torsion forms.

We intend to produce a theory which generalises topological spaces; we call it non-commutative topology. In non-commutative differential geometry the basic homology theory is the periodic cyclic homology, based on the bi-complex (b, B). In non-commutative topology this structure will be replaced by the bi-complex \((\tilde {b}, d)\); the boundary \(\tilde {b}\) is called topological Hochschild boundary. These ideas combine Connes’ work (Noncommutative geometry, Academic Press, 1994) with ideas of the articles by Teleman and Teleman (St Cerc Math Tom 18(5):753–762, Bucarest, 1966; Rev Roumaine Math Pures Applic 12:725–731, 1967; Central European journal of mathematics. C*-algebras and elliptic theory II. Trends in mathematics, 251–265, Birkhauser, 2008; Local3 index theorem. arXiv: 1109.6095v1, math.KT, 28 Sep. 2011; textitLocal Hochschild homology of Hilbert-Schmidt operators on simplicial spaces. arXiv hal-00707040, Version 1, 11 June 2012; Local algebraic K-theory, 26 Aug 2013 - arXiv.org > math > arXiv:1307.7014, 2013; The local index theorem, HAL-00825083, arXiv 1305.5329, 22 May 2013; \(K_ {i}^{loc}(\ mathbb {C}) \), i = 0, 1 10 set 2013 - arXiv:1309.2421v1 [math.KT] 10 Sep 2013).

The noncommutative geometry of graph [formula omitted]-algebras I: The index theorem

Local atiyah-singer index theorem for families of dirac operators

Superconnection and family Bergman kernels

In this paper, we give a local Riemann-Roch theorem for certain almost Hermitian manifolds by a canonical connection and the theory of almost Hermitian structure of null eccentricity. In particular, we obtain a local Riemann-Roch theorem for all closed almost complex manifolds of dimension 4. Also, by using the heat equation method, we give an analytic approach to a local index theorem for generalized Dirac operators.

We show that one can define (p,∞)-summable spectral triples using degenerate metrics on smooth manifolds. Furthermore, these triples satisfy Connes–Moscovici's discrete and finite dimension spectrum hypothesis, allowing one to use the Local Index Theorem [1] to compute the pairing with K-theory. We demonstrate this with a concrete example.

Brownian Chen series and Atiyah–Singer theorem

On considere des conditions suffisantes pour qu'un theoreme d'indice local soit vrai pour un operateur de Dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de Levi-Civita L

This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group [Formula: see text] and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a finite sum of locally defined intersection indices. The local indices are integers arising from the geometry of the stratification. The result is used to study a well-known problem in chemical physics, namely, the problem of enumerating the electronic excitations (excitons) of a molecule equipped with scattering data.

This is an account of a lecture given at the 7th ISAAC Congress, where I described a joint work with E. Leichtnam, X. Tang and A. Weinstein giving a proof of the Atiyah-Weinstein index formula. This concerns the index of an operator closely related to Toeplitz operators, for which analogues of the Atiyah-Singer index formula do not make sense. Instead we used an equivariant asymptotic index formula, which does; it is an outgrowth of Atiyah and Singers theory of equivariant index for transversally elliptic pseudodifferential operators.

In this article, we begin by recalling the inversion formula for the convolution with the box spline. The equivariant cohomology and the equivariant $K$-theory with respect to a compact torus $G$ of various spaces associated to a linear action of $G$ in a vector space $M$ can both be described using some vector spaces of distributions, on the dual of the group $G$ or on the dual of its Lie algebra $\mathfrak{g}$. The morphism from $K$-theory to cohomology is analyzed, and multiplication by the Todd class is shown to correspond to the operator (deconvolution) inverting the semi-discrete convolution with a box spline. Finally, the multiplicities of the index of a $G$-transversally elliptic operator on $M$ are determined using the infinitesimal index of the symbol.

We give a new short proof of the index formula of Atiyah and Singer based on combining Getzler’s rescaling with Greiner’s approach of the heat kernel asymptotics. As an application we can easily compute the CM cyclic cocycle of even and odd Dirac spectral triples, and then recover the Atiyah-Singer index formula (even case) and the Atiyah-Patodi-Singer spectral flow formula (odd case).