Analytic torsions on contact manifolds

Abstract

We propose a definition for analytic torsion of the contact complex on contact manifolds. We show it coincides with Ray-Singer torsion on any 3-dimensional CR Seifert manifold equipped with a unitary representation. In this particular case we compute it and relate it to dynamical properties of the Reeb flow. In fact the whole spectral torsion function we consider may be interpreted on CR Seifert manifolds as a purely dynamical function through Selberg-type trace formulae.Comment: 40 page

Abelian analytic torsion and symplectic volume

“…A proof of Theorem 9 follows from [9,Theorem 5.4], where the analytic torsion is computed on a closed Sasakian three-manifold twisted by a unitary representation ρ : π 1 (X) → U(r). Combining this with a substitution of some known special values of the Riemann-Hurwitz zeta function completes the proof.…”

“…Note that [9] defines and studies a new type of analytic torsion on contact manifolds called the contact analytic torsion, denoted by T C X , and they also introduce a corresponding contact Ray-Singer metric, denoted || · || C . These quantities are defined in terms of the contact complex (E, D H ), originally introduced by M. Rumin [24], on a contact manifold (X, κ).…”

“…This operator is maximally hypoelliptic and invertible in the Heisenberg symbolic calculus [9]; a key property that allows one to make sense of the zeta function for the contact Laplacian ζ(∆ C q )(s). [9] introduce the contact torsion function,…”

“…Note that our definition of K(s) is the negative of the one that occurs in [9]. The contact analytic torsion is then defined to be, (25) T C X := exp…”

“…Our main result, Theorem 9, shows that two mathematically a priori different definitions of the abelian Chern-Simons partitition function derived from [7] are rigorously equivalent. Our main strategy is to use the the work of M. Rumin and N. Seshadri [9] which naturally connects the analytic torsion with contact structures on three-manifolds.…”