Determinants of non-self-adjoint elliptic operators in geometry and physics

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We study geometric invariants defined using regularized determinant of non-self-adjoint elliptic operators. In particular, we continue to study the complex valued refinement of the Ray-Singer torsion introduced in our joint paper with T. Kappeler using the graded determinant of the odd signature operator. This leads to new properties of both the Ray-Singer torsion and the eta-invariant. 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. It also provides a link between the higher analytic torsion and the higher eta-invatriant of Bismut and Cheeger. We suggest a version of the refined analytic torsion for complex Calabi-Yau manifolds. This leads 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 compute the Berry phase in some of these models. We use different extension of the notion of determinant and trace from finite matrices to differential operators in oder to construct new mathematical objects. This leads to new invariants of manifolds as well as to the new information about the old invariant. When applied to certain complex manifolds it gives a generalization of a classical Dedekind eta-function and new applications to number theory. We apply a new construction of a determinant to obtain a new description of some models of superconductivity. This new description allows to compute so called Berry phase in many examples.