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

Abstract

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.

84 pages; v2: minor corrections and changes; v3: exposition improved and references added; Final version to be published in Journal of Mathematical Physics