Scalar Curvature, Gauss-Bonnet Theorem and Einstein-Hilbert Action for Noncommutative Tori
(Western University & IHÉS)
Amphitéâtre Léon Motchane (IHES)
Amphitéâtre Léon Motchane
Le Bois Marie
35, route de Chartres
Geometric spaces are described by spectral triples (A, H, D) in non-commutative geometry. In this context, A is an involutive noncommutative algebra represented by bounded operators on a Hilbert space H, and D is an unbounded selfadjoint operator acting in H which plays the role of the Dirac operator, namely that it contains the metric information while interacting with the algebra in a bounded manner. The local geometric invariants such as the scalar curvature of (A, H, D) are extracted from the high frequency behavior of the spectrum of D and the action of A via special values and residues of the meromorphic extension of zeta functions ζa to the complex plane, which are defined for a in A by
ζa (s) = Trace (a ⎜D⎜-s), ℜ(s) >> 0.
Following the seminal work of A. Connes and P. Tretkoff on the Gauss-Bonnet theorem for the canonical translation invariant conformal structure on the noncommutative two torus Tθ2, there have been significant developments in understanding the local differential geometry of these C*-algebras equipped with curved metrics. In this talk, I will review my joint works with M. Khalkhali, in which we extend this result to general translation invariant conformal structures on Tθ2 and compute the scalar curvature. Our final formula for the curvature matches precisely with the independent result of A. Connes and H. Moscovici. I will also present our recent work on noncommutative four tori, in which we compute the scalar curvature and show that the metrics with constant curvature are extrema of the analog of the Einstein-Hilbert action.