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变形等变有向算子。我们将利用这些算子的良好谱性质来构造新的流形不变量。特别地，我们引入了开Kaehler流形上等变全纯向量丛的正则化上同调。该项目的目标之一是将零化定理和半连续定理推广到非紧环境。这些应用将包括Witten全纯Morse不等式和Mumford商的上同调公式的非紧版本，以及这些结果在紧情况下的新证明。我们还引入了一个关于正则上同调行列式的Quillen度量的类似，并计划研究这个度量。这将重新阐明紧致流形上的Quillen度量的性质。特别是，在与A·阿巴诺夫的一个联合项目中，我们建议对描述超导性的非线性西格玛模型进行数学上的严格描述。与P.E.Paradan一起，我们计划使用变形的Dirac算子来研究李群的离散级数表示。紧流形上的椭圆算子具有非常好的性质。在这个项目中，我们在具有相似性质的非紧流形上引入了一类算子。对这些算子的研究不仅将许多定理从紧致流形推广到非紧致流形，而且为紧致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数学理论到表象理论和复杂几何。

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.