Journal of Noncommutative Geometry


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Published online first: 2018-05-24
DOI: 10.4171/JNCG/280

Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants

Raphaël Ponge[1] and Hang Wang[2]

(1) Seoul National University, Republic of Korea
(2) University of Adelaide, Australia and East China Normal University, Shanghai, China

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 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 using various tools from noncommutative geometry, such as twisted spectral triples and cyclic theory. An important step is to establish the conformal invariance of the Connes–Chern character of the conformal Dirac spectral triple of Connes–Moscovici. Ultimately, however, the main results of the paper are stated in a purely differential-geometric fashion.

Keywords: Noncommutative geometry, conformal geometry, index theory, cyclic homology, equivariant cohomology

Ponge Raphaël, Wang Hang: Noncommutative geometry and conformal geometry. I. Local index formula and conformal invariants. J. Noncommut. Geom. Electronically published on May 24, 2018. doi: 10.4171/JNCG/280 (to appear in print)