Formulation of metric compatibility of a connection in noncommutative geometry

Abstract: The goal of the talk is to formulate the notion of Levi-Civita connections in noncommutative geometry. More precisely, we will work in the set up of differential calculus over a ( possibly ) noncommutative algebra. Given a pseudo-Riemannian metric g on the calculus, a connection on the space of one-forms will be called a Levi-Civita connection for g if the connection is torsionless and compatible with g. The torsion of a connection in noncommutative geometry is well-known. So our main focus would be to define metric compatibility condition of a connection. We need the calculus to satisfy some conditions to make sense of our metric compatibility condition and also the symmetry of the pseudo-Riemannian metric g. It turns out that these conditions are also sufficient to ensure the existence of a unique Levi-Civita connection for any bilinear pseudo-Riemannian metric. Examples of such calculus include the fuzzy 3-sphere, the quantum Heisenberg manifold and a class of Rieffel deformations of classical manifolds under free and isometric toral actions. The talk is based on a joint work with D. Goswami and G. Landi.

general relativity and quantum cosmologyHEP - theorymathematical physicsquantum algebra

Audience: researchers in the topic

Mathematical Physics Seminar

Series comments: Description: Noncommutative geometry, field theory, gravity.