L ∞ -algebras of Einstein–Cartan–Palatini gravity

Abstract

We give a detailed account of the cyclic L ∞ -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 ∞ -algebras in the case of parallelizable spacetimes. We show that our L ∞ -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 ∞ -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.

Braided $$\varvec{L_{\infty }}$$-algebras, braided field theory and noncommutative gravity

“…[11,18,20]. We argued in [30] that the L ∞ -algebra formalism should provide the natural receptacle to capture the failure of closure and covariance of field equations under nonassociative gauge transformations. To accommodate these instances, one may in principle deform the classical L ∞ -algebra using cochain twists of the enveloping algebra of vector fields, rather than cocycle twists, which results in a quasi-Hopf algebra.…”

“…In this paper, we discuss a new alternative approach, which arose from our attempts to understand a noncommutative version of the Einstein-Cartan-Palatini theory of gravity in this language; the L ∞ -algebra formalism for the classical theory was developed in detail by [30]. In the standard noncommutative extension of this theory [4], the (extended) local Lorentz symmetry is implemented by the usual star-gauge transformations, and in principle one may construct correspondingly an L ∞ -algebra structure as in the bootstrapping approach to noncommutative gauge theories.…”

“…In this paper, we apply our formalism of braided L ∞ -algebras to produce new braided noncommutative theories of gravity in three and four spacetime dimensions. We consider only the case of parallelizable spacetime manifolds M, for which the twist deformations are straightforward to implement and exhibit the salient features of our constructions more clearly; see [30] for the general treatment in the classical case. In three dimensions, the noncommutative field equations and action functional are naive deformations of the classical ones (with braided commutators); however in four dimensions we uncover non-trivial deformations.…”

“…so that gauge invariance of the action functional δ λ S(A) = 0 is then equivalent to the Noether identities d A F A = 0. For further details and the connections to the BV-BRST formalism see [37], while details of the aspects covered in this paper are found in [30].…”

“…Let us illustrate how to reconstruct Chern-Simons theory in three spacetime dimensions in the L ∞ -algebra framework, which is the prototypical example of a gauge theory with a simple bracket structure, see e.g. [30,37]. Let g be a real Lie algebra with Lie bracket [−, −] g and let M be a three-manifold.…”