Note on antisymmetric spin-tensors

TL;DR

This paper clarifies the existence of gauge invariant Lagrangians for antisymmetric spin-tensors, showing they exist in dimensions greater than four but vanish in four dimensions, and provides a simple form for further generalizations.

Contribution

It provides a direct analysis confirming the existence of gauge invariant Lagrangians for antisymmetric spin-tensors in dimensions greater than four, resolving previous contradictions.

Findings

Gauge invariant Lagrangian exists for d > 4

Lagrangian becomes zero in d = 4

Simple form facilitates deformation to AdS and massive cases

Abstract

It was known for a long time that in d = 4 dimensions it is impossible to construct the Lagrangian for antisymmetric second rank spin-tensor that will be invariant under the gauge transformations with unconstrained spin-vector parameter. But recently a paper arXiv:0902.1471 appeared where gauge invariant Lagrangians for antisymmetric spin-tensors of arbitrary rank n in d > 2n were constructed using powerful BRST approach. To clarify apparent contradiction, in this note we carry a direct independent analysis of the most general first order Lagrangian for the massless antisymmetric spin-tensor of second rank. Our analysis shows that gauge invariant Lagrangian does exist but in d > 4 dimensions only, while in d = 4 this Lagrangian becomes identically zero. As a byproduct, we obtain a very simple and convenient form of this massless Lagrangian that makes deformation to AdS space and/or massive case a simple task as we explicitly show here. Moreover, this simple form admits natural and straightforward generalization on the case of massive antisymmetric spin-tensors of rank n for d > 2n.