(L_{\infty}\)-Algebras of Einstein-Cartan-Palatini Gravity

• Published in: arXiv.org 2020-10
• Main Authors: Ćirić, Marija Dimitrijević, Giotopoulos, Grigorios, Radovanović, Voja, Szabo, Richard J
• Format: Article
• Language: English
• Subjects: Algebra; Cosmological constant; Field theory (physics); Gravitation; Isomorphism; Lie groups; Quantum theory; Relativity; Spacetime
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.2003.06173

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 present a local formulation as well as a global covariant framework, and an explicit isomorphism between the two \(L_\infty\)-algebras in the case of parallelizable spacetimes. By duality, 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.