Refined analytic torsion: comparison theorems and examples

Abstract

Braverman and Kappeler introduced a refinement of the Ray-Singer analytic torsion associated to a flat vector bundle over a closed odd-dimensional manifold. We study this notion and improve the Braverman-Kappeler theorem comparing the refined analytic torsion with the Farber-Turaev refinement of the combinatorial torsion. Using this result we establish, modulo sign, the Burghelea-Haller conjecture, comparing their complex analytic torsion with the Farber-Turaev torsion in the case when the flat connection can be deformed in the space of flat connections to a Hermitian connection. We then compute the refined analytic torsion of lens spaces and answer some of the questions posed in [BK1, Remark 14.9].