Analytic Torsion for Symmetric Contact Manifolds

Abstract

We review the definitions of the Rumin complex and analytic contact torsion for contact manifolds. We then compute the contact torsion for symmetric contact manifolds. To this end, we use a representation-theoretic description of the relevant operators and spaces of differential forms. Our computation of the contact torsion for symmetric contact manifolds relies on a recent result by Rumin that reduces the problem to harmonic forms. Following a construction by Boothby and Wang, the contact manifolds under consideration are total spaces of $S^1$-principal bundles over Kähler manifolds. This allows us to interpret the harmonic forms as Dolbeault cohomology with coefficients in a certain holomorphic line bundle of the base Kähler manifold, thereby simplifying the computation by allowing us to use results from complex geometry. Furthermore, we generalise Rumin's result to the equivariant case and, using the previous considerations, determine the equivariant contact torsion for the case of isolated fixed points.