Flat vector bundles and analytic torsion on orbifolds

TL;DR

This paper explores flat orbifold vector bundles, establishing a correspondence with orbifold fundamental group representations, and extends analytic torsion results to orbifolds, including a Bismut-Zhang anomaly formula and a dynamical zeta function relation.

Contribution

It introduces a bijection between flat orbifold vector bundles and fundamental group representations, and extends analytic torsion results to orbifolds, including anomaly formulas and zeta function relations.

Findings

Bijection between flat orbifold bundles and fundamental group representations

Anomaly formula for the Ray-Singer metric on orbifolds

Equality of analytic torsion and dynamical zeta function value

Abstract

This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of representations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determine line of the cohomology of a compact orbifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of the reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds.