Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry

TL;DR

This paper develops a new framework for semi-classical gauge transformations in noncommutative gauge theories using symplectic embeddings, simplifying constructions and extending to curved backgrounds with complex gauge algebras.

Contribution

It introduces explicit symplectic embedding techniques to construct gauge algebras in noncommutative theories, generalizing previous methods and applying to curved backgrounds.

Findings

Constructs explicit $P_$-algebras from symplectic embeddings.

Shows gauge symmetries form $L_$-algebras in curved backgrounds.

Provides simpler, all-orders constructions for noncommutative gauge theories.

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.