L ∞ -algebras of Einstein–Cartan–Palatini gravity

Abstract

We give a detailed account of the cyclic L∞-algebra formulation of general relativity with a 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∞-algebras in the case of parallelizable spacetimes. By duality, we show that our L∞-algebras describe the complete Batalin-Vilkovisky-Becchi-Rouet-Stora-Tyutin 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∞-algebras. We use this to extend the on-shell equivalence between gravity and Chern–Simons theory in three dimensions to an explicit L∞-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∞-algebra underlying gravity and the differential graded Lie algebra governing BF theory in four dimensions.