Some notes on analytic torsion of the Rumin complex on contact manifolds

TL;DR

This paper introduces a new definition of analytic torsion for the Rumin complex on contact manifolds, demonstrating its invariance and analyzing its variation through local and global terms.

Contribution

It defines analytic torsion for the Rumin complex using zeta functions of a modified Laplacian, establishing contact invariance and variation formulas.

Findings

The regular value of the zeta function combination is a contact invariant.

Analytic torsion is given by the derivative at zero of a specific zeta function combination.

Variation of torsion involves local integrals and global null-space contributions.

Abstract

We propose a definition for analytic torsion of the Rumin complex on contact manifolds. This is given by the derivative at zero of a well-chosen combination of zeta functions of a fourth-order modified Rumin Laplacian. The regular value at zero (before differentiation) of this well-chosen combination of zeta functions is shown to be a contact invariant. The variation of our analytic torsion is given as the integral of local terms, together with a global term coming from the null-space of the Laplacian.