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.

arXiv comment79 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"

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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
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• Non-commutative gauge symmetry from strong homotopy algebras
• Generalized symmetries as homotopy Lie algebras