Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

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.

A gravitational action with stringy Q and R fluxes via deformed differential graded Poisson algebras

We study a deformation of a 2-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a 2-form B-field and a bivector Π, that we consider as gauge fields of the geometric and non-geometric fluxes H, f, Q and R arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a U(1) gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: with an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle E ≅ TM ⊕ T∗M, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and Q and R fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and H-flux.