A Scalar Curvature Formula For the Noncommutative 3-Torus

Abstract

We compute the scalar curvature of a curved noncommutative 3-torus. To perturb the flat metric, the standard volume form on the noncommutative 3-torus is conformally perturbed and the corresponding perturbed Laplacian is analyzed. Using Connes' pseudodifferential calculus for the noncommutative 3-torus, we explicitly compute the first three terms of the small time heat kernel expansion for the perturbed Laplacian. The third term of the expansion gives a local formula for the scalar curvature. Finally, we show that in the classical limit when the deformation parameters vanish, our formula coincides with the formula for the commutative case.

References in corpus (6)

• Modular curvature and Morita equivalence
• On the Gauss-Chern-Bonnet theorem for the noncommutative 4-sphere
• On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for $C^{\ast }$-Dynamical Systems
• Modular curvature for toric noncommutative manifolds
• On Certain Spectral Invariants of Dirac Operators on Noncommutative Tori
• The Moyal Sphere

Cited by in corpus (5)

• Pseudodifferential calculus on noncommutative tori, I. Oscillating integrals
• Spectral geometry of functional metrics on noncommutative tori
• Gauss-Bonnet for matrix conformally rescaled Dirac
• The Ricci Curvature for Noncommutative Three Tori
• Pseudodifferential calculus on noncommutative tori, II. Main properties