Braided homotopy Lie algebras, noncommutative field theory and gravity

Abstract

We develop a general framework that enables the systematical construction of a new class of noncommutative field theories, that we call braided field theories. For this, we employ L∞-algebras, their braided counterparts and Drinfel'd twist deformation quantization. We apply this framework to several field theories of mathematical and physical interest, including general relativity, to produce braided noncommutative versions thereof. We start with a detailed account of the cyclic L∞-algebra formulation of general relativity in the Einstein-Cartan-Palatini formalism on spacetimes of arbitrary dimension. 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. We give a general description of how to extend on-shell redundant symmetries in gauge theories to off-shell correspondences 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. We then define a new homotopy algebraic structure, that we call a braided L∞- algebra, and use it to develop the theory of braided noncommutative field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the eld equations. We use Drinfel'd twist deformation quantization to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to construct braided versions of Chern-Simons, scalar field, BF, Yang-Mills and Einstein-Cartan-Palatini theories. We detail the novelties, similarities and differences of our noncommutative theories compared to previous attempts.