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
