Complex Line Bundles over Simplicial Complexes and their Applications

TL;DR

This paper classifies discrete line bundles over simplicial complexes using Discrete Differential Geometry, introduces a discrete analogue of Weil's theorem, and generalizes the Laplace operator to these bundles, with applications in physics and computer graphics.

Contribution

It provides a complete classification of discrete vector bundles over finite simplicial complexes and introduces a discrete Dirichlet energy generalizing the cotangent Laplacian.

Findings

Complete classification of discrete hermitian line bundles over simplicial complexes.

Discrete analogue of Weil's theorem on line bundle classification.

A generalized discrete Dirichlet energy for hermitian line bundles.

Abstract

Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular, we obtain a discrete analogue of a theorem of Andr\'e Weil on the classification of hermitian line bundles. Moreover, we associate to each discrete hermitian line bundle with curvature a unique piecewise-smooth hermitian line bundle of piecewise constant curvature. This is then used to define a discrete Dirichlet energy which generalizes the well-known cotangent Laplace operator to discrete hermitian line bundles over Euclidean simplicial manifolds of arbitrary dimension.