Braided Symmetries in Noncommutative Field Theory

TL;DR

This paper introduces braided $L_$-algebras as a new framework for noncommutative field theories, overcoming previous limitations and providing explicit examples of braided gauge theories.

Contribution

It develops the concept of braided $L_$-algebras and applies them to construct new classes of noncommutative gauge theories with braided symmetries.

Findings

Formulation of braided gauge symmetries using Drinfel'd twist techniques

Construction of a braided scalar field theory and a braided $BF$ theory

Introduction of a braided noncommutative Yang-Mills theory for arbitrary gauge algebras

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.