Calabi复形与Killing层上同调
The Calabi complex and Killing sheaf cohomology
摘要：
最近人们注意到常曲率Lorentz流形上线性化引力的Poisson括号的退化可以用一个微分算子复形的上同调来描述。这个复形最早是由卡拉比提出的，它的上同调被认为是同构于Killing向量的（局部常数）层。我们审查的Calabi复杂的结构在一个新的方式，与显式计算的基础上表示理论的GL（n），也有一些工具来研究其上同调的局部常数层。我们还推测这些工具将如何适应其他背景下的线性引力和其他规范理论。演示文稿包括明确的公式的微分算子在卡拉比复杂的，参数的地方精确性，讨论广义庞加莱对偶，方法计算上同调的局部常数层，并举例计算杀死层上同调的一些黑洞和宇宙学洛伦兹流形。
英文摘要：
It has recently been noticed that the degeneracies of the Poisson bracket of linearized gravity on constant curvature Lorentzian manifold can be described in terms of the cohomologies of a certain complex of differential operators. This complex was first introduced by Calabi and its cohomology is known to be isomorphic to that of the (locally constant) sheaf of Killing vectors. We review the structure of the Calabi complex in a novel way, with explicit calculations based on representation theory of GL(n), and also some tools for studying its cohomology in terms of of locally constant sheaves. We also conjecture how these tools would adapt to linearized gravity on other backgrounds and to other gauge theories. The presentation includes explicit formulas for the differential operators in the Calabi complex, arguments for its local exactness, discussion of generalized Poincaré duality, methods of computing the cohomology of locally constant sheaves, and example calculations of Killing sheaf cohomologies of some black hole and cosmological Lorentzian manifolds.
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