Generalized symmetries as homotopy Lie algebras

TL;DR

This paper demonstrates that homotopy Lie algebras naturally describe generalized gauge symmetries in field theories, including non-commutative gauge symmetries and double field theory, through specific algebraic structures.

Contribution

It shows that homotopy Lie algebras provide a unified framework for describing generalized gauge symmetries in different field theories, including braided and curved L$_$-algebras.

Findings

Homotopy Lie algebras encode generalized gauge symmetries.

Braided L$_\infty$-algebra describes twisted gauge symmetry.

Curved L$_\infty$-algebra models double field theory symmetries.

Abstract

Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the description of generalized gauge symmetries using two specific examples. The first example deals with the non-commutative gauge symmetry obtained using Drinfel'd twist of the symmetry Hopf algebra. The homotopy Lie algebra compatible with the twisted gauge symmetry turns out to be the recently proposed braided L$_\infty$-algebra. In the second example we focus on the generalized gauge symmetry of the double field theory. The symmetry includes both diffeomorphisms and gauge transformation and can consistently be defined using a curved L$_\infty$-algebra.