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.

arXiv comment60 pages; v2: introduction expanded, typos corrected; v3: minor changes, references added; Final version published in Journal of Physics A

References in corpus (55)

• String Theory and Noncommutative Geometry
• Deformation quantization of Poisson manifolds, I
• Introduction to sh Lie algebras for physicists
• D-branes and Deformation Quantization
• Higher derived brackets and homotopy algebras
• Nonassociative Star Product Deformations for D-brane Worldvolumes in Curved Backgrounds
• Symplectic structures associated to Lie-Poisson groups
• Symmetry, Gravity and Noncommutativity
• Relative formality theorem and quantisation of coisotropic submanifolds
• $L_{\infty }$ Algebras and Field Theory
• WZW-Poisson manifolds
• A Noncommutative M-Theory Five-brane
• Membrane Sigma-Models and Quantization of Non-Geometric Flux Backgrounds
• $L_{\infty }$-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism
• Non-Geometric Fluxes, Quasi-Hopf Twist Deformations and Nonassociative Quantum Mechanics
• BRST theory without Hamiltonian and Lagrangian
• Poisson sigma models and symplectic groupoids
• Topological Open P-Branes
• Position-dependent noncommutativity in quantum mechanics
• Noncommutative D-Brane in Non-Constant NS-NS B Field Background
• Construction of Gauge Theories on Curved Noncommutative Spacetime
• Deformations of coisotropic submanifolds and strong homotopy Lie algebroids
• Star products made (somewhat) easier
• Groupoids, Loop Spaces and Quantization of 2-Plectic Manifolds
• Noncommutative $R^d$ via closed star product
• Higher Poisson Brackets and Differential Forms
• Noncommutative Differential Calculus for D-brane in Non-Constant B Field Background
• Wick type deformation quantization of Fedosov manifolds
• $G_2$-structures and quantization of non-geometric M-theory backgrounds
• Quantum mechanics with coordinate dependent noncommutativity
• Integration of twisted Poisson structures
• Making Non-Associative Algebra Associative
• Triproducts, nonassociative star products and geometry of R-flux string compactifications
• Non-commutative deformation of Chern-Simons theory
• Poisson Algebra of Differential Forms
• Covariant Star Product for Exterior Differential Forms on Symplectic Manifolds
• On the Uniqueness of L${}_{\infty }$ bootstrap: Quasi-isomorphisms are Seiberg-Witten Maps
• Worldvolume Uncertainty Relations for D-Branes
• Symplectic realisation of electric charge in fields of monopole distributions
• $k$-Minkowski-deformation of $U(1)$ gauge theory
• Formal symplectic groupoid
• Smooth 2-Group Extensions and Symmetries of Bundle Gerbes
• Nonassociative Weyl star products
• Geometry and 2-Hilbert Space for Nonassociative Magnetic Translations
• Groupoid Quantization of Loop Spaces
• Universal star products
• $L_{\infty }$-Bootstrap Approach to Non-Commutative Gauge Theories
• Fibre Bundles, Jet Manifolds and Lagrangian Theory. Lectures for Theoreticians
• Recurrence relations for symplectic realization of (quasi)-Poisson structures
• Quantization of Poisson families and of twisted Poisson structures
• Weak associativity and deformation quantization
• Bootstrapping Non-commutative Gauge Theories from L${}_{\infty }$ algebras
• Quantization of Magnetic Poisson Structures
• Homotopy Lie Algebras of Gravity and their Braided Deformations
• Lattice Multiverse Models

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