论文标题
分级的广义几何形状
Graded Generalized Geometry
论文作者
论文摘要
广义几何形状在字符串理论的某些方面的数学描述中找到了许多应用。简而言之，它探讨了与给定歧管相关的广义切线束上的各种结构。特别是，可以通过规范的dorfman支架来制定几种整合性条件，这是Courant代数的一个例子。另一方面，可以将光滑的歧管概括为涉及$ \ mathbb {z} $ - 分级变量的功能，不一定是通勤。这导致了分级流形的数学理论。通过探索与给定分级歧管相关的广义切线束上的结构来结合两种理论是很自然的。 在回忆基本级别的几何形状后，引入了分级矢量束上的分级courant代数。我们表明，与分级歧管相关的广义切线束上有一个规范支架。探索了狄拉克结构和广义复合结构的分级类似物。我们介绍了差异分级的库兰特代数，可以将其视为Q-Manifolds的概括。给出了分级谎言的定义和示例。
Generalized geometry finds many applications in the mathematical description of some aspects of string theory. In a nutshell, it explores various structures on a generalized tangent bundle associated to a given manifold. In particular, several integrability conditions can be formulated in terms of a canonical Dorfman bracket, an example of Courant algebroid. On the other hand, smooth manifolds can be generalized to involve functions of $\mathbb{Z}$-graded variables which do not necessarily commute. This leads to a mathematical theory of graded manifolds. It is only natural to combine the two theories by exploring the structures on a generalized tangent bundle associated to a given graded manifold. After recalling elementary graded geometry, graded Courant algebroids on graded vector bundles are introduced. We show that there is a canonical bracket on a generalized tangent bundle associated to a graded manifold. Graded analogues of Dirac structures and generalized complex structures are explored. We introduce differential graded Courant algebroids which can be viewed as a generalization of Q-manifolds. A definition and examples of graded Lie bialgebroids are given.
