Spectral Invarinats of Deformed Dirac Operators on Open G-Manifolds

摘要：DMS 0204421。在这个项目中，我们将继续研究开流形上的形变等变狄拉克算子。我们将利用这些算子的谱性质来构造流形的新不变量。特别地，我们引入了开kaehler流形上等变全纯向量束的正则上同调。本课题的目标之一是将消失定理和半连续性定理推广到非紧集。这些应用将包括Witten的全纯morse不等式和Mumfordquotient的上同调公式的非紧化版本，以及紧化情况下这些结果的新证明。我们还在正则上同调的行列式上引入了Quillen度规的类比，并计划对该度规进行研究。这将对紧流形上的Quillen度规的性质有新的启示。特别是，在与a . Abanov的联合项目中，我们提出了对描述超导性的非线性西格玛模型的数学严格描述。与体育-体育联合。最后，我们计划在李群的离散级数表示的研究中使用变形狄拉克算子。紧流形上的椭圆算子具有很好的性质。本文引入了一类性质相似的非紧流形上的算子。这些算子的研究不仅将许多定理从紧流形推广到非紧流形，而且为紧流形理论提供了新的结果和方法。这些结果和方法的应用范围从超导的数学理论到表示理论和复杂几何。

ABSTRACT: DMS 0204421.In this project we will continue to study the deformed equivariant Diracoperators on open manifolds. We will use the nice spectral properties of theseoperators to construct new invariants of manifolds. In particular, we introducethe regularized cohomology of equivariant holomorphic vector bundles over openKaehler manifolds. One of the goals of the project is to extend the vanishingtheorems and the semi-continuity theorem to non-compact setting. Theapplications will include the non-compact versions of Witten's holomorphicMorse inequalities and of the formula for the cohomology of the Mumfordquotient, as well as new proofs of these results in the compact case. We alsointroduce an analogue of the Quillen metric on the determinant of theregularized cohomology and are planning to study this metric. This will shed anew light on the properties of the Quillen metric on compact manifolds. Inparticular, in a joint project with A. Abanov we suggest a mathematicallyrigorous description of the non-linear sigma-model describing thesuperconductivity. Jointly with P.-E. Paradan we a planning to use the deformedDirac operator in the study of discrete series representations of Lie groups.The elliptic operators on compact manifolds have very nice properties. In thisproject we introduce a class of operators on non-compact manifolds with similarproperties. The study of these operators not only leads to a generalization ofmany theorems from compact manifolds to non-compact ones, but also provides newresults and methods in the theory of compact manifolds. The applications ofthese results and methods range from the mathematical theory ofsuperconductivity to representation theory and complex geometry.