Differential Geometry of Group Lattices

TL;DR

This paper develops a differential geometric framework on discrete group lattices using noncommutative geometry, exploring structures like vector fields, gauge theories, and connections on Cayley graphs.

Contribution

It introduces a novel differential geometric approach to group lattices, extending calculus to higher orders and formulating gauge theories on discrete structures.

Findings

Differential calculus on group lattices corresponds to Cayley graph structures.

A lattice gauge theory (Yang-Mills) action is constructed for arbitrary group lattices.

Discrete vector fields satisfy a Lie-Cartan identity, enabling geometric analysis.

Abstract

In a series of publications we developed "differential geometry" on discrete sets based on concepts of noncommutative geometry. In particular, it turned out that first order differential calculi (over the algebra of functions) on a discrete set are in bijective correspondence with digraph structures where the vertices are given by the elements of the set. A particular class of digraphs are Cayley graphs, also known as group lattices. They are determined by a discrete group G and a finite subset S. There is a distinguished subclass of "bicovariant" Cayley graphs with the property that ad(S)S is contained in S. We explore the properties of differential calculi which arise from Cayley graphs via the above correspondence. The first order calculi extend to higher orders and then allow to introduce further differential geometric structures. Furthermore, we explore the properties of "discrete" vector fields which describe deterministic flows on group lattices. A Lie derivative with respect to a discrete vector field and an inner product with forms is defined. The Lie-Cartan identity then holds on all forms for a certain subclass of discrete vector fields. We develop elements of gauge theory and construct an analogue of the lattice gauge theory (Yang-Mills) action on an arbitrary group lattice. Also linear connections are considered and a simple geometric interpretation of the torsion is established. By taking a quotient with respect to some subgroup of the discrete group, generalized differential calculi associated with so-called Schreier diagrams are obtained.