Ray-Singer torsion and the Rumin Laplacian on lens spaces

Abstract

The Rumin complex is the Bernstein-Gelfand-Gelfand complex (BGG complex) of the twisted de Rham complex of a flat vector bundle with respect to contact manifolds. As a typical theorem, the cohomology of the BGG complex coincides with the cohomology of the de Rham complex of a flat vector bundle. Moreover, the Rumin complex arises when we take the sub-Riemannian limit. Let us consider what happens when we replace a concept defined using the de Rham complex with the Rumin complex. In this talk, we adapt this idea to analytic torsion. On flat vector bundles with a unitary holonomy over lens spaces, we express explicitly the analytic torsion functions associated with the Rumin complex in terms of the Hurwitz zeta function. In particular, we determine the analytic torsions, and it is written using the Betti numbers and the Ray-Singer torsion.