Tensor Hierarchies of 5- and 6-Dimensional Field Theories

TL;DR

This paper develops a systematic construction of tensor hierarchies for bosonic field theories in 5 and 6 dimensions, introducing new deformation tensors and form potentials, with methods extendable to arbitrary dimensions.

Contribution

It provides a general framework for constructing tensor hierarchies in higher-dimensional field theories, including the role of deformation tensors and associated form potentials.

Findings

Constructed tensor hierarchies for 5- and 6-dimensional theories.

Identified relations between deformation tensors and form potentials.

Presented a method generalizable to any dimension.

Abstract

We construct the tensor hierarchies of generic, bosonic, 5- and 6-dimensional field theories. The construction of the tensor hierarchy starts with the introduction of two tensors: the embedding tensor which tells us which vector is used for gauging and another tensor Z which tells us which vector is eaten by a 2-form. In dimensions five or more these two (deformation) tensors are in principle unrelated. Besides the embedding tensor and Z there can be further deformation tensors describing other couplings unrelated to (but compatible with) gauge symmetry. For each deformation tensor there appears a (d-1)-form potential and for each constraint satisfied by the deformation tensors there appears a d-form potential in the tensor hierarchy. For each symmetry of the undeformed theory there is an associated (d-2)-form appearing in the tensor hierarchy. Our methods easily generalize to arbitrary dimensions and we present a general construction for the d-, (d-1)- and (d-2)-form potentials for a tensor hierarchy in d dimensions.