Discrete Vector Bundles with Connection

TL;DR

This paper introduces a combinatorial framework for vector bundles with connection on simplicial complexes, extending discrete exterior calculus to bundle-valued forms with algebraic identities akin to the smooth case.

Contribution

It develops a discrete exterior covariant derivative and demonstrates its properties, including curvature, gauge transformations, and a cochain complex for twisted de Rham cohomology.

Findings

Discrete covariant derivative satisfies algebraic identities like Bianchi identity.

Flat discrete connections compute twisted de Rham cohomology.

Provides a comparison with recent discrete bundle frameworks.

Abstract

We develop a combinatorial theory of vector bundles with connection on locally ordered simplicial complexes. This is a first step towards a discrete exterior calculus for bundle-valued forms. The basic building block is the discrete exterior covariant derivative, a forward-difference operator defined on bundle-valued cochains. Many standard objects in differential geometry (e.g., curvature, connection 1-forms, gauge transformations) can be understood via the discrete covariant derivative operator, with their defining formulas identical to the smooth setting. These discrete objects satisfy all of the expected algebraic identities, such as naturality with respect to simplicial maps, and a Bianchi identity for discrete curvature. We also show that flat discrete connections determine a cochain complex that computes twisted de Rham cohomology in a local coefficient system determined by the discrete vector bundle, with twisted Poincare duality (of densities) being one application. Finally, a coarsening operation applied to bundle-valued cochains provides a direct and concrete comparison with the recent framework for discrete bundles of Christiansen and Hu.