A Riemann-Roch Theorem For One-Dimensional Complex Groupoids

Denis Perrot (1, 2)

1 CPT - Centre de Physique Théorique - UMR 6207
2 CPT - Centre de Physique Théorique - UMR 7332

We consider a smooth groupoid of the form \Sigma\rtimes\Gamma where \Sigma is a Riemann surface and \Gamma a discrete pseudogroup acting on \Sigma by local conformal diffeomorphisms. After defining a K-cycle on the crossed product C_0(\Sigma)\rtimes\Gamma generalising the classical Dolbeault complex, we compute its Chern character in cyclic cohomology, using the index theorem of Connes and Moscovici. This involves in particular a generalisation of the Euler class constructed from the modular automorphism group of the von Neumann algebra L^{\infty}(\Sigma)\rtimes\Gamma.

Domaines: Physique mathématique [math-ph]; Géométrie différentielle [math.DG]; Physique des Hautes Energies - Théorie [hep-th]

Denis Perrot. A Riemann-Roch Theorem For One-Dimensional Complex Groupoids. 2001. ⟨hal-00130378⟩