Duality Hierarchies and Differential Graded Lie Algebras

Roberto Bonezzi,Olaf Hohm

doi:10.1007/s00220-021-03973-8

Roberto Bonezzi, Olaf Hohm

Open Access

https://doi.org/10.1007/s00220-021-03973-8

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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.

Highlights

Gauge theories and supergravity it is often instrumental to append to the physical p−form gauge fields their on-shell dual (D − p − 2)−form fields
The Bianchi identities take a hierarchical structure in which the exterior derivative of a p−form field strength is related to the ( p +1)−form field strength of the higher form field, leading to the notion of tensor hierarchy
The imposition of first-order duality relations requires an entire tower of duality relations, termed duality hierarchy in [10], from which the second-order equations of gauged supergravity follow as integrability conditions

Summary

Introduction

Gauge theories and supergravity it is often instrumental to append to the physical p−form gauge fields their on-shell dual (D − p − 2)−form fields. The observation of [20] was that upon suspension (an overall shift of degree) and upon appending additional vector spaces (including g that carries a Lie algebra structure) the axioms governing and D take the form of a differential graded Lie algebra (dgLa), where becomes a (graded) bracket [·, ·], on which D acts as a derivation, and which satisfies a (graded) Jacobi identity. As the most intriguing result of this paper, we give dynamical equations for the fields of the tensor hierarchy, including scalars with a general scalar potential, in terms of duality relations To this end we have to assume the existence of G–covariant isomorphisms. 3 we introduce the ‘total complex’ encoding the tensor product of this dgLa with the differential forms in order to streamline the gauge theory construction of [19], which is extended by including scalars parametrizing a coset space G/H.

Differential Graded Lie Algebras

Total Complex and dgLa Gauge Theory

Dynamical Equations from Duality Relations

Outlook