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Discrete vector bundles are important in Physics and recently foundremarkable applications in Computer Graphics. This article approaches discretebundles from the viewpoint of Discrete Differential Geometry, including acomplete classification of discrete vector bundles over finite simplicialcomplexes. In particular, we obtain a discrete analogue of a theorem of Andr\'eWeil on the classification of hermitian line bundles. Moreover, we associate toeach discrete hermitian line bundle with curvature a unique piecewise-smoothhermitian line bundle of piecewise constant curvature. This is then used todefine a discrete Dirichlet energy which generalizes the well-known cotangentLaplace operator to discrete hermitian line bundles over Euclidean simplicialmanifolds of arbitrary dimension.

Complex Line Bundles over Simplicial Complexes and their Applications