We give a detailed account of the cyclic $L_\infty$-algebra formulation of general relativity with cosmological constant in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension and signature, which encompasses all symmetries, field equations and Noether identities of gravity without matter fields. We develop a local formulation as well as a global covariant framework, and derive an explicit isomorphism between the two $L_\infty$-algebras in the case of parallelizable spacetimes. We show that our $L_\infty$-algebras describe the complete BV-BRST formulation of Einstein-Cartan-Palatini gravity. We give a general description of how to extend on-shell redundant symmetries in topological gauge theories to off-shell correspondences between symmetries in terms of quasi-isomorphisms of $L_\infty$-algebras. We use this to extend the on-shell equivalence between gravity and Chern-Simons theory in three dimensions to an explicit $L_\infty$-quasi-isomorphism between differential graded Lie algebras which applies off-shell and for degenerate dynamical metrics. In contrast, we show that there is no morphism between the $L_\infty$-algebra underlying gravity and the differential graded Lie algebra governing $BF$ theory in four dimensions.

中文翻译：

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L∞-爱因斯坦-嘉当-帕拉蒂尼引力的代数

我们详细说明了广义相对论的循环 $L_\infty$-代数公式，在爱因斯坦-嘉当-帕拉蒂尼形式主义中关于任意维度和签名的时空，包括所有对称性、场方程和诺特引力恒等式没有物质场。我们开发了一个局部公式以及一个全局协变框架，并在可并行时空的情况下推导出两个 $L_\infty$-代数之间的显式同构。我们展示了我们的 $L_\infty$-代数描述了爱因斯坦-嘉当-帕拉蒂尼引力的完整 BV-BRST 公式。我们给出了如何将拓扑规范理论中的壳上冗余对称性扩展到 $L_\infty$-代数的准同构方面的对称性之间的壳外对应关系的一般描述。我们使用它在三个维度上将引力和陈-西蒙斯理论之间的壳上等价扩展到适用于壳外和退化动力学度量的微分分级李代数之间的显式 $L_\infty$-准同构。相比之下，我们表明 $L_\infty$-代数基础引力和控制四维 $BF$ 理论的微分分级李代数之间没有态射。