Discrete Connection Laplacians

Abstract

To every Hermitian vector bundle with connection over a compact Riemannian manifold $M$ one can associate a corresponding connection Laplacian acting on the sections of the bundle. We define analogous combinatorial metric dependent Laplacians associated to triangulations of $M$ and prove that their spectra converge, as the mesh of the triangulations approaches zero, to the spectrum of the connection Laplacian.