Quantum Koszul formula on quantum spacetime (1707.01481v1)

Abstract: Noncommutative or quantum Riemannian geometry has been proposed as an effective theory for aspects of quantum gravity. Here the metric is an invertible bimodule map $\Omega^1\otimes_A\Omega^1\to A$ where $A$ is a possibly noncommutative or quantum' spacetime coordinate algebra and $(\Omega^1,d)$ is a specified bimodule of 1-forms ordifferential calculus' over it. In this paper we explore the proposal of a quantum Koszul formula' with initial data a degree -2 bilinear map $\perp$ on the full exterior algebra $\Omega$ obeying the 4-term relations \[ (-1)^{|\eta|} (\omega\eta)\perp\zeta+(\omega\perp\eta)\zeta=\omega\perp(\eta\zeta)+(-1)^{|\omega|+|\eta|}\omega(\eta\perp\zeta),\quad\forall\omega,\eta,\zeta\in\Omega\] and a compatible degree -1codifferential' map $\delta$. These provide a quantum metric and interior product and a canonical bimodule connection $\nabla$ on all degrees. The theory is also more general than classically in that we do not assume symmetry of the metric nor that $\delta$ is obtained from the metric. We solve and interpret the $(\delta,\perp)$ data on the bicrossproduct model quantum spacetime $[r,t]=\lambda r$ for its two standard choices of $\Omega$. For the $\alpha$-family calculus the construction includes the quantum Levi-Civita connection for a general quantum symmetric metric, while for the more standard $\beta=1$ calculus we find the quantum Levi-Civita connection for a quantum `metric' that in the classical limit is antisymmetric.