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

几何和物理中非自伴椭圆算子的行列式

• 批准号: 1005888
• 负责人: Maxim Braverman
• 金额: $ 19.58万
• 依托单位: Northeastern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005888&HistoricalAwards=false
• 关键词: Determinants; non; self; adjoint; elliptic

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.

研究了非自伴椭圆算子的正则化行列式定义的几何不变量。特别地，我们继续研究在我们与T。Kappeler利用奇签名算子的分次行列式。这导致了Ray-Singer挠率和η-不变量的新性质。我们建议一个改进的版本的Bismut-Lott高解析挠，它包含更多的信息，更容易研究比原来的高挠。它还提供了一个更高的解析挠率和更高的η-不变的铋和Cheeger之间的联系。我们提出了复Calabi-Yau流形的精化解析挠率的一个版本。这导致了戴德金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计算其中一些模型中的Berry相位。我们将行列式和迹的概念从有限矩阵推广到微分算子，以构造新的数学对象。这导致新的不变量的流形以及新的信息旧的不变量。当应用到某些复杂的流形，它给出了一个经典的戴德金η-函数和数论的新应用的推广。我们应用一个新的行列式的构造来获得一些超导模型的新描述。这种新的描述允许在许多示例中计算所谓的Berry相位。