First Order Formalism for Mixed Symmetry Tensor Fields

TL;DR

The paper addresses the challenge of formulating gauge-invariant first-order descriptions for massless mixed-symmetry high-spin fields. It develops tetrad-like first-order Lagrangians for the three representative fields Φ_{[μν],α}, T_{[μν α],β}, and R_{[μν],[αβ]}, introducing auxiliary gauge fields to yield simple, geometrically interpretable expressions and enabling straightforward elimination to recover the standard second-order theories. Each sector provides explicit L_I forms with clear gauge structures, and dual formulations in lower dimensions are presented to simplify the descriptions. The framework offers a natural basis for exploring interactions and non-linear extensions of mixed-symmetry high-spin fields, with dimensional constraints d ≥ 3, 4, 5 for the respective cases and potential for covariant, interaction-rich generalizations.

Abstract

In this paper we give explicit first order Lagrangian formulation for mixed symmetry tensor fields Φ_{[μν],α}, T_{[μνα],β} and R_{[μν],[αβ]}. We show that such Lagrangians could be written in a very suggestive form similar to the well known tetrad formalism in gravity. Such description could simplify the investigations of possible interactions for these fields. Some examples of interactions are given.