Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

Abstract

We formulate general definitions of semi-classical gauge transformations for noncommutative gauge theories in general backgrounds of string theory, and give novel explicit constructions using techniques based on symplectic embeddings of almost Poisson structures. In the absence of fluxes the gauge symmetries close a Poisson gauge algebra and their action is governed by a $P_\infty$-algebra which we construct explicitly from the symplectic embedding. In curved backgrounds they close a field dependent gauge algebra governed by an $L_\infty$-algebra which is not a $P_\infty$-algebra. Our technique produces new all orders constructions which are significantly simpler compared to previous approaches, and we illustrate its applicability in several examples of interest in noncommutative field theory and gravity. We further show that our symplectic embeddings naturally define a $P_\infty$-structure on the exterior algebra of differential forms on a generic almost Poisson manifold, which generalizes earlier constructions of differential graded Poisson algebras, and suggests a new approach to defining noncommutative gauge theories beyond the gauge sector and the semi-classical limit based on $A_\infty$-algebras.

Four-dimensional noncommutative deformations of U(1) gauge theory and L∞ bootstrap.

“…The last requirement guarantees that the noncommutative transformations (2.1) reproduce the standard Abelian gauge transformations (1.9) in the undeformed theory. A general result has been established in [6] in the context of symplectic embeddings, that is valid for any Θ which is linear in x. It suggests a solution of eq.…”

“…", contain higher derivatives. From now on we neglect these terms, namely we consider the semi-classical limit [6,7]. Therefore our noncommutative gauge algebra becomes the Poisson gauge algebra:…”

“…Interestingly, the field-dependent deformation of gauge transformations (2.1) has been derived in [6] as the result of a symplectic embedding of the Poisson manifold (M, Θ) into the symplectic manifold (T * M, ω), where T * M denotes the cotangent bundle and ω an appropriate symplectic form such that π * ω −1 = Θ, π : T * M → M being the projection map.…”

“…The relation between the two approaches, which has been established in [6,7], may be summarised as follows.…”

“…One main motivation for confronting once again with noncommutative gauge theory is a series of recent publications proposing the framework of L ∞ algebras and a bootstrap approach as appropriate for formulating gauge noncommutativity in a consistent way [4,5]. Moreover, an interesting connection has been established between the L ∞ bootstrap and symplectic embeddings of non-commutative algebras [6,7]. It is worth mentioning that the role of L ∞ algebras in gauge and field theory is already investigated in [8][9][10], see also [11][12][13][14][15][16] for recent progresses in studies of the L ∞ structures in the field theoretic context.…”