Braided $L_{\infty}$-Algebras, Braided Field Theory and Noncommutative Gravity

TL;DR

This paper introduces braided $L_$-algebras to construct noncommutative field theories with braided gauge symmetries, leading to a braided version of gravity that generalizes classical Einstein-Cartan-Palatini theory.

Contribution

It defines braided $L_$-algebras and develops a systematic framework for braided field theories, including a novel noncommutative gravity model.

Findings

Braided gauge symmetries form a braided Lie algebra with inhomogeneous Noether identities.

Braided noncommutative gravity reduces to classical gravity when deformation vanishes.

The formalism extends classical field theories to include braided noncommutative structures.

Abstract

We define a new homotopy algebraic structure, that we call a braided $L_\infty$-algebra, and use it to systematically construct a new class of noncommutative field theories, that we call braided field theories. Braided field theories have gauge symmetries which realize a braided Lie algebra, whose Noether identities are inhomogeneous extensions of the classical identities, and which do not act on the solutions of the field equations. We use Drinfel'd twist deformation quantization techniques to generate new noncommutative deformations of classical field theories with braided gauge symmetries, which we compare to the more conventional theories with star-gauge symmetries. We apply our formalism to introduce a braided version of general relativity without matter fields in the Einstein-Cartan-Palatini formalism. In the limit of vanishing deformation parameter, the braided theory of noncommutative gravity reduces to classical gravity without any extensions.