Braided Symmetries in Noncommutative Field Theory

Abstract

We give a pedagogical introduction to $L_{\infty }$-algebras and their uses in organising the symmetries and dynamics of classical field theories, as well as of the conventional noncommutative gauge theories that arise as low-energy effective field theories in string theory. We review recent developments which formulate field theories with braided gauge symmetries as a new means of overcoming several obstacles in the standard noncommutative theories, such as the restrictions on gauge algebras and matter fields. These theories can be constructed by using techniques from Drinfel'd twist deformation theory, which we review in some detail, and their symmetries and dynamics are controlled by a new homotopy algebraic structure called a 'braided $L_{\infty }$-algebra'. We expand and elaborate on several novel theoretical issues surrounding these constructions, and present three new explicit examples: the standard noncommutative scalar field theory (regarded as a braided field theory), a braided version of $BF$ theory in arbitrary dimensions (regarded as a higher gauge theory), and a new braided version of noncommutative Yang-Mills theory for arbitrary gauge algebras.

79 pages; v2: references added; v3: reference added; Final version to appear in the Special Issue of Journal of Physics A on "Noncommutative Geometry in Physics"

References in corpus (20)

• A Gravity Theory on Noncommutative Spaces
• Symmetry, Gravity and Noncommutativity
• T-duality and closed string non-commutative (doubled) geometry
• Twisted Gauge Theories
• Noncommutative D=4 gravity coupled to fermions
• Deformation theory via differential graded Lie algebras
• Double Copy from Homotopy Algebras
• Nonassociative geometry in quasi-Hopf representation categories I: Bimodules and their internal homomorphisms
• Braided quantum field theories and their symmetries
• Symplectic embeddings, homotopy algebras and almost Poisson gauge symmetry
• Braided $L_{\infty }$-Algebras, Braided Field Theory and Noncommutative Gravity
• BV Yang-Mills as a Homotopy Chern-Simons via SFT
• Poisson gauge theory
• $L_{\infty }$-Bootstrap Approach to Non-Commutative Gauge Theories
• Hopf Algebra Symmetry and String Theory
• Spontaneous symmetry breaking: a view from derived geometry
• Batalin-Vilkovisky quantization of fuzzy field theories
• Higher Lie theory
• Twist Quantization of String and B Field Background
• Homotopy Lie Algebras of Gravity and their Braided Deformations

Cited by in corpus (6)

• First quantization of braided Majorana fermions
• Multiparticle states in braided lightlike $\kappa$-Minkowski noncommutative QFT
• T-Minkowski noncommutative spacetimes II: classical field theory
• T-Minkowski noncommutative spacetimes I: Poincaré groups, differential calculi and braiding
• Non-commutative gauge symmetry from strong homotopy algebras
• Generalized symmetries as homotopy Lie algebras