Analytic Torsion of Z_2-graded Elliptic Complexes

TL;DR

This paper generalizes analytic torsion to Z_2-graded elliptic complexes, introducing a new framework applicable to various complex types, and studies its properties and metric invariance on different manifold dimensions.

Contribution

It defines a new form of analytic torsion for Z_2-graded complexes, extending previous variants and exploring its properties and metric dependence.

Findings

Analytic torsion is metric-independent on odd-dimensional manifolds.

A relative analytic torsion is metric-independent on even-dimensional manifolds.

The framework applies to flat superconnection and twisted complexes.

Abstract

We define analytic torsion of Z_2-graded elliptic complexes as an element in the graded determinant line of the cohomology of the complex, generalizing most of the variants of Ray-Singer analytic torsion in the literature. It applies to a myriad of new examples, including flat superconnection complexes, twisted analytic and twisted holomorphic torsions, etc. The definition uses pseudo-differential operators and residue traces. We also study properties of analytic torsion for Z_2-graded elliptic complexes, including the behavior under variation of the metric. For compact odd dimensional manifolds, the analytic torsion is independent of the metric, whereas for even dimensional manifolds, a relative version of the analytic torsion is independent of the metric. Finally, the relation to topological field theories is studied.