Braided Symmetries in Noncommutative Field Theory

Abstract

We give a pedagogical introduction to L ∞ -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 ∞ -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.

First quantization of braided Majorana fermions

“…Quite likely, this argument can be circumvented for integrable systems (since they can be linearized with the appropriate choice of action-angle coordinates) but conserves its general validity; it then suggests that a full Second Quantization should be pursued. Quite recently a framework for noncommutative braided field theories, based on L 8 -algebras, was made available (see the review [30] and the references therein). It looks promising to exploit it for a Second Quantization.…”

The $L_{\infty}$-structure of noncommutative gravity

“…From a mathematical perspective, their applicability in these contexts comes from their duality with differential graded commutative algebras [28,29]. Our presentation will be mostly informal, in order to highlight the main ideas without too much technical clutter; see [1,20,21,30] for more elaborate reviews.…”

“…∞ -algebras. Let = ⊕ ∈Z be a Z-graded vector space, with degree shifted graded exterior algebra Λ = ∧ • ( [1]) viewed as a free cocommutative coalgebra. An ∞ -structure on is a coderivation : Λ −→ Λ of degree | | = 1 satisfying 2 = 0.…”

“…In this contribution we briefly review recent developments aimed at understanding gauge symmetries of noncommutative field theories in the framework of homotopy algebras, and in particular ∞ -algebras. For a detailed exposition of the subject and its applications to field theory, see [1]. Here we shall summarise the applications to gravity, which were developed in a series of papers [2][3][4].…”

“…The drawback of these treatments is that so far they are only understood at the semi-classical level, and hence do not capture the complete structure underlying noncommutative gauge symmetries. Instead, one can reformulate some of these theories via deformations with braided gauge symmetries [1,4], which involve the same finite number of brackets as their classical counterparts and are applicable at the full quantized level. This formalism is reviewed in Section 3.…”