Exotic tensor gauge theory and duality

TL;DR

This work provides a systematic framework to describe gauge fields in exotic Lorentz representations using bi-forms and, more generally, multi-forms. It constructs gauge potentials, field strengths, Bianchi identities, and gauge-invariant actions for two-column representations $[p,q]$, and reveals how linearised gravity admits multiple dual descriptions via $R=d\tilde{d}h$ and its Hodge-dual partners $S$ and $G$, with matching physical degrees of freedom. The multi-form extension unifies arbitrary GL$(D,\mathbb{R})$ representations and enables $2^N$ dual descriptions for a gauge field of type $[p_1,...,p_N]$, providing a compact algebraic toolkit for higher-spin/string-inspired theories and higher-dimensional gravity dualities. These results illuminate the structure of exotic gauge theories, their dualities, and their gauge-invariant actions, with potential applications to string theory and generalized geometrical frameworks such as $\mathcal{W}$-geometry.

Abstract

Gauge fields in exotic representations of the Lorentz group in D dimensions - i.e. ones which are tensors of mixed symmetry corresponding to Young tableaux with arbitrary numbers of rows and columns - naturally arise through massive string modes and in dualising gravity and other theories in higher dimensions. We generalise the formalism of differential forms to allow the discussion of arbitrary gauge fields. We present the gauge symmetries, field strengths, field equations and actions for the free theory, and construct the various dual theories. In particular, we discuss linearised gravity in arbitrary dimensions, and its two dual forms.