Friday, July 12, 2013

1307.2778 (Shahn Majid)

Reconstruction and quantization of Riemannian structures    [PDF]

Shahn Majid
We show how the Riemannian structure on a manifold can be conveniently reconstructed from its codifferential $\delta$, including a formula for the Levi-Civita connection. In terms of 1-forms, $\nabla_\omega\eta={1\over 2}([\omega,\eta]+\CL_\omega\eta+\lceil_\eta\extd\omega)$ where $[\omega,\eta]=\delta(\omega\eta)-(\delta\omega)\eta+\omega(\delta\eta)$ is the Lie bracket of vector fields in terms of forms, $\CL$ is the Lie derivative along the vector field corresponding to a 1-form and $\lceil$ is interior product via the inverse metric. Our approach arises naturally out of a central extension problem in noncommutative geometry and leads to a `quantum' differential graded algebra $\Omega_\lambda(M)$ associated to any classical Riemannian manifold $M$. We also provide a semidirect product of any differential graded algebra by the quantum differential algebra $\Omega(t,\extd t)$ in one variable, to introduce a noncommutative `time'. Composing these two constructions recovers a quantisation $\Omega_\lambda(M\times\R)$ in \cite{Ma:bh} and extends it explicitly to all degrees.
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