Gauge Theories, Duality Relations and the Tensor Hierarchy

TL;DR

This paper constructs and analyzes tensor hierarchies in gauge theories, showing how to derive on-shell equations and actions for gauged supergravity in 3 and 4 dimensions, including duality relations and scalar fields.

Contribution

It provides a comprehensive construction of off-shell tensor hierarchies and their on-shell duality relations, connecting gauge transformations with equations of motion in gauged supergravity.

Findings

Complete 3- and 4-dimensional tensor hierarchies derived.

Duality relations encode equations of motion including scalars.

Gauge-invariant actions constructed for all hierarchy fields.

Abstract

We compute the complete 3- and 4-dimensional tensor hierarchies, i.e. sets of p-form fields, with $1\le p\le D$, which realize an off-shell algebra of bosonic gauge transformations. We show how these tensor hierarchies can be put on-shell by introducing a set of duality relations, thereby introducing additional scalars and a metric tensor. These so-called duality hierarchies encode the equations of motion of the bosonic part of the most general gauged supergravity theories in those dimensions, including the (projected) scalar equations of motion. We construct gauge-invariant actions that include all the fields in the tensor hierarchies. We elucidate the relation between the gauge transformations of the p-form fields in the action and those of the same fields in the tensor hierarchy.