In this project we continue to study the complex valued refinement of the Ray-Singer analytic torsion introduced in our joint paper with T. Kappeler. This study will lead to new properties of both the Ray-Singer torsion and the eta-invariant. In particular, we suggest a refined version of the Bismut-Lott higher analytic torsion which contains more information and is easier to study than the original higher torsion. We also suggest a version of the refined analytic torsion for complex Calabi-Yau manifolds. This will lead to applications in number theory. In particular, to a multi-dimensional generalization of the Dedekind eta-function. We also consider a new regularization procedure for definition of the trace and the determinant of certain class of pseudo-differential operators on odd-dimensional manifolds. This procedure allows to avoid many anomalies coursed by usual zeta-function regularization. It also turns out to be the most adequate for description of non-linear sigma-models of superconductivity. In a joint project with A. Abanov we suggest to use this regularization to get a first mathematically rigorous computation of the Berry phase in some of these models.
We propose a new geometric invariant of compact manifolds which combines two classical invariants - the Ray-Singer torsion and the Atiyah-Patodi-Singer eta-invariant. Our construction allows to study both invariants simultaneously and leads to discovery of new properties of them. A similar invariant for complex manifolds leads to new applications in complex geometry and number theory. The definition of the new invariant is based on the study of determinants of non-self-adjoint differential operators. We suggest a new construction of such determinants, which, in some cases, behaves better than the usual one, and which is more adequate for description of certain models of superconductivity. Using this construction we suggest a first rigorous approach to these models.
