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。卡普勒这项研究将导致Ray-Singer挠率和η-不变量的新性质。特别是，我们提出了一个改进的版本的Bismut-Lott高解析挠包含更多的信息，更容易研究比原来的高挠。我们还提出了一个版本的复杂的Calabi-Yau流形的精细解析挠。这将导致数论中的应用。具体地，涉及戴德金η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一种新的正则化方法。这个过程可以避免通常的zeta函数正则化过程中的许多异常。它也被证明是最充分的非线性西格玛超导模型的描述。在与A. Abanov我们建议使用这种正则化来得到第一个数学上严格计算的Berry相位在这些models.We提出了一个新的几何不变量的紧致流形，它结合了两个经典的不变量-Ray-Singer挠和Atiyah-Patodi-Singer eta-不变量。我们的建设允许同时研究这两个不变量，并导致发现他们的新属性。复流形的一个类似的不变量在复几何和数论中有了新的应用。新不变量的定义是基于对非自伴微分算子行列式的研究。我们提出了一种新的建设，这样的决定因素，在某些情况下，表现得比通常的，这是更充分的描述某些模型的超导性。使用这种结构，我们建议这些模型的第一个严格的方法。

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.