Dirac结构与Dixmier-Douady丛
Dirac structures and Dixmier-Douady bundles
摘要：
向量丛V上的Dirac结构是V的直和及其对偶的极大迷向子丛E。本文研究了如何将希尔伯特空间上的紧算子与Dixmier-Douady丛A（即具有典型纤维的C ~*-代数的Z/2 Z-分次丛）的Dirac结构相联系。相对于Dixmier-Douady丛的Morita态射，该构造具有良好的函子性质。作为应用，我们证明了紧连通李群（由Atiyah-Segal构造）上的“自旋”Dixmier-Douady丛是乘法的，并且在具有群值矩映射的空间上得到了一个规范的“扭曲自旋-c-结构”.
英文摘要：
A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum of V with its dual. We show how to associate to any Dirac structure a Dixmier-Douady bundle A, that is, a Z/2Z-graded bundle of C*-algebras with typical fiber the compact operators on a Hilbert space. The construction has good functorial properties, relative to Morita morphisms of Dixmier-Douady bundles. As applications, we show that the `spin' Dixmier-Douady bundle over a compact, connected Lie group (as constructed by Atiyah-Segal) is multiplicative, and we obtain a canonical `twisted Spin-c-structure' on spaces with group valued moment maps.
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