Determinants of Elliptic Operators in Geometry, Number Theory, and Physics

在这个项目中，我们继续研究在我们与T. Kappeler的联合论文中引入的Ray-Singer解析扭转的复值细化。这项研究将导致Ray-Singer扭转和不变量的新性质。特别地，我们提出了bismutt - lott高解析扭转的改进版本，它包含更多的信息，比原来的高解析扭转更容易研究。我们还提出了复杂Calabi-Yau流形的一种精化解析扭转。这将导致数论的应用。特别是对Dedekind函数的多维推广。我们还考虑了奇维流形上一类伪微分算子的迹和行列式的定义的一种新的正则化过程。这个过程可以避免由通常的ζ函数正则化引起的许多异常。它也被证明是最适合描述超导的非线性sigma模型的。在与a . Abanov的一个联合项目中，我们建议使用这种正则化来获得其中一些模型中Berry阶段的第一个数学上严格的计算。本文提出了一种新的紧流形几何不变量，它结合了两个经典不变量Ray-Singer扭转和Atiyah-Patodi-Singer不变量。我们的构造允许同时研究这两个不变量，并导致它们的新性质的发现。复流形的类似不变量在复几何和数论中有新的应用。新不变量的定义是基于对非自伴随微分算子行列式的研究。我们提出了这种行列式的新结构，在某些情况下，它比通常的行列式表现得更好，并且更适合于描述某些超导模型。使用这种结构，我们建议对这些模型采用第一种严格的方法。

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.