Tensor Hierarchies of 5- and 6-Dimensional Field Theories

We construct the tensor hierarchies of generic, bosonic, 5and 6-dimensional field theories. The construction of the tensor hierarchy starts with the introduction of two tensors: the embedding tensor θ 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 d ≥ 5 these two (deformation) tensors are in principle unrelated. Besides θ 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.