Duality Hierarchies and Differential Graded Lie Algebras

TL;DR

This paper explores the algebraic structures underlying tensor hierarchies in gauge theories related to supergravity, revealing their reinterpretation as differential graded Lie algebras to streamline hierarchy construction and dynamics formulation.

Contribution

It demonstrates that tensor hierarchies can be understood as differential graded Lie algebras, providing a unified framework for their construction and dynamics in gauge theories.

Findings

Tensor hierarchies are axiomatized by infinity-enhanced Leibniz algebras.

These structures can be reinterpreted as differential graded Lie algebras.

The reformulation simplifies the construction and dynamics of tensor hierarchies.

Abstract

The gauge theories underlying gauged supergravity and exceptional field theory are based on tensor hierarchies: generalizations of Yang-Mills theory utilizing algebraic structures that generalize Lie algebras and, as a consequence, require higher-form gauge fields. Recently, we proposed that the algebraic structure allowing for consistent tensor hierarchies is axiomatized by `infinity-enhanced Leibniz algebras' defined on graded vector spaces generalizing Leibniz algebras. It was subsequently shown that, upon appending additional vector spaces, this structure can be reinterpreted as a differential graded Lie algebra. We use this observation to streamline the construction of general tensor hierarchies, and we formulate dynamics in terms of a hierarchy of first-order duality relations, including scalar fields with a potential.