Quantum Koszul formula on quantum spacetime

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 or `differential 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 -1 `codifferential' 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.