Riemannian Geometry of Bicovariant Group Lattices

TL;DR

This paper develops a Riemannian geometric framework for bicovariant group lattices, enabling the description of discrete geometries with metric, torsion, and curvature, akin to continuous differential geometry.

Contribution

It introduces a novel noncommutative geometric approach to bicovariant group lattices, establishing tensorial objects and compatibility conditions for metric and connection.

Findings

Metrics describe lengths and angles in discrete geometries

Torsion and curvature are geometrically interpreted on lattices

Curvature is related to parallel transport around plaquettes

Abstract

Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (generalized) differential forms, for the subclass of ``bicovariant'' group lattices considered in this work it is possible to understand central geometric objects like metric, torsion and curvature as ``tensors'' with (left) covariance properties. This ensures that tensor components (with respect to a basis of the space of 1-forms) transform in the familiar homogeneous way under a change of basis. There is a natural compatibility condition for a metric and a linear connection. The resulting (pseudo-) Riemannian geometry is explored in this work. It is demonstrated that the components of the metric are indeed able to properly describe properties of discrete geometries like lengths and angles. A simple geometric understanding in particular of torsion and curvature is achieved. The formalism has much in common with lattice gauge theory. For example, the Riemannian curvature is determined by parallel transport of vectors around a plaquette (which corresponds to a biangle, a triangle or a quadrangle).