The tensor hierarchy of 8-dimensional field theories

TL;DR

This paper develops the tensor hierarchy framework for 8-dimensional bosonic field theories, detailing the construction of gauge-invariant field strengths, gaugings, and equations of motion, including massive deformations.

Contribution

It provides the first comprehensive construction of the tensor hierarchy for generic 8D bosonic theories with gaugings and massive deformations using the embedding tensor formalism.

Findings

Explicit form of all field strengths up to 6-forms.

Complete tensor hierarchy including gaugings and massive deformations.

Identification of equations of motion as combinations of Bianchi identities.

Abstract

We construct the tensor hierarchy of generic, bosonic, 8-dimensional field theories. We first study the form of the most general 8-dimensional bosonic theory with Abelian gauge symmetries only and no massive deformations. This study determines the tensors that occur in the Chern-Simons terms of the (electric and magnetic) field strengths and the action for the electric fields, which we determine. Having constructed the most general Abelian theory we study the most general gaugings of its global symmetries and the possible massive deformations using the embedding tensor formalism, constructing the complete tensor hierarchy using the Bianchi identities. We find the explicit form of all the field strengths of the gauged theory up to the 6-forms. Finally, we find the equations of motion comparing the Noether identities with the identities satisfied by the Bianchi identities themselves. We find that some equations of motion are not simply the Bianchi identities of the dual fields, but combinations of them.