Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants

@article{Ponge2014NoncommutativeGA,
  title={Noncommutative Geometry and Conformal Geometry. I. Local Index Formula and Conformal Invariants},
  author={Raphael Ponge and Han Wang},
  journal={arXiv: Differential Geometry},
  year={2014}
}
This paper is part of a series of articles on noncommutative geometry and conformal geometry. In this paper, we reformulate the local index formula in conformal geometry in such a way to take into account of the action of conformal diffeomorphisms. We also construct and compute a whole new family of geometric conformal invariants associated with conformal diffeomorphisms. This includes conformal invariants associated with equivariant characteristic classes. The approach of this paper involves… 
View PDF on arXiv
17 Citations
Highly Influential Citations
1
Background Citations
2
Methods Citations
1

17 Citations

Noncommutative geometry and conformal geometry: III. Vafa-Witten inequality and Poincaré duality

NONCOMMUTATIVE GEOMETRY AND CONFORMAL GEOMETRY, II. CONNES-CHERN CHARACTER AND AND THE LOCAL EQUIVARIANT

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…

Noncommutative Geometry and Conformal Geometry. II. Connes-Chern character and the local equivariant index theorem

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…

Index map, $\sigma$-connections, and Connes-Chern character in the setting of twisted spectral triples

Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of…

INDEX MAP, σ-CONNECTIONS, AND CONNES-CHERN CHARACTER IN THE SETTING OF TWISTED SPECTRAL TRIPLES

Twisted spectral triples are a twisting of the notion of spectral triples aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction…

Dynamics of compact quantum metric spaces

We provide a detailed study of actions of the integers on compact quantum metric spaces, which includes general criteria ensuring that the associated crossed product algebra is again a compact…

Nonlocal elliptic problems associated with actions of discrete groups on manifolds with boundary

Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential…

Elliptic boundary value problems associated with isometric group actions

Given a manifold with boundary endowed with an action of a discrete group on it, we consider the algebra of operators generated by elements in the Boutet de Monvel algebra of pseudodifferential…

Regularity of twisted spectral triples and pseudodifferential calculi

We investigate the regularity condition for twisted spectral triples. This condition is equivalent to the existence of an appropriate pseudodifferential calculus compatible with the spectral triple.…

The periodic cyclic homology of crossed products of finite type algebras

References

SHOWING 1-10 OF 91 REFERENCES

Noncommutative geometry and conformal geometry: III. Vafa-Witten inequality and Poincaré duality

The decomposition of Global Conformal Invariants V

This is the fifth in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of “global conformal invariants”; these are defined to be conformally…

Noncommutative Geometry and Conformal Geometry. II. Connes-Chern character and the local equivariant index theorem

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…

On the conformal and CR automorphism groups

This paper concerns the behavior of conformal diffeomorphisms between Riemannian manifolds, and that of CR diffeomorphisms between strictly pseudoconvex CR manifolds, We develop a new approach to the…

Modular Curvature for Noncommutative Two-Tori

In this paper we investigate the curvature of conformal deformations by noncommutative Weyl factors of a flat metric on a noncommutative 2-torus, by analyzing in the framework of spectral triples…

Geometric classification of conformal anomalies in arbitrary dimensions

Hopf Algebras, Cyclic Cohomology and the Transverse Index Theorem

Abstract:In this paper we solve a longstanding internal problem of noncommutative geometry, namely the computation of the index of transversally elliptic operators on foliations. We show that the…

Invariants of conformal Laplacians

The conformal Laplacian D = d*d + (n - 2)s/4(n - 1), acting on functions on a Riemannian manifold Mn with scalar curvature s, is a conformally invariant operator. In this paper we will use D to…

The Logarithmic Singularities of the Green Functions of the Conformal Powers of the Laplacian

Green functions play an important role in conformal geometry. In this paper, we explain how to compute explicitly the logarithmic singularities of the Green functions of the conformal powers of the…

The decomposition of global conformal invariants II: The Fefferman-Graham ambient metric and the nature of the decomposition

This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally…
...