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Published 2015 | Submitted
Journal Article Open

Scalar curvature for noncommutative four-tori


In this paper we study the curved geometry of noncommutative 4-tori T^4_θ. We use a Weyl conformal factor to perturb the standard volume form and obtain the Laplacian that encodes the local geometric information. We use Connes' pseudodifferential calculus to explicitly compute the terms in the small time heat kernel expansion of the perturbed Laplacian which correspond to the volume and scalar curvature of T^4_θ. We establish the analogue of Weyl's law, define a noncommutative residue, prove the analogue of Connes' trace theorem, and find explicit formulas for the local functions that describe the scalar curvature of T^4_θ. We also study the analogue of the Einstein-Hilbert action for these spaces and show that metrics with constant scalar curvature are critical for this action.

Additional Information

© 2015 European Mathematical Society. Received 02 May, 2013. We are indebted to Alain Connes for several enlightening discussions at different stages of this work. Also, F. F. would like to thank IHES for the excellent environment and kind support during his visit in Winter 2012, where part of this work was carried out.

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