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.

60 pages; v2: introduction expanded, typos corrected; v3: minor changes, references added; Final version published in Journal of Physics A

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Cited by in corpus (18)

• Gauge theories on quantum spaces
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• Poisson gauge theory
• Braided Symmetries in Noncommutative Field Theory
• Poisson gauge models and Seiberg-Witten map
• Topological Field Theories induced by twisted R-Poisson structure in any dimension
• Double Quantization
• Lie-Poisson gauge theories and $\kappa$-Minkowski electrodynamics
• On Poisson Electrodynamics With Charged Fields
• On the L${}_{\infty }$ structure of Poisson gauge theory
• Effects of wave propagation in canonical Poisson gauge theory under an external magnetic field
• Poisson electrodynamics on $\kappa$-Minkowski space-time
• Hamiltonian analysis in Lie-Poisson gauge theory
• Light Propagation in $\kappa$-Minkowski Space-Time: Gauge Ambiguities and Invariance
• Non-commutative gauge symmetry from strong homotopy algebras
• Generalized symmetries as homotopy Lie algebras
• Symplectic realization of generalized Snyder-Poisson algebra
• Four-dimensional noncommutative deformations of $U(1)$ gauge theory and $L_{\infty }$ bootstrap