Counter-example at low momentum

The theorem does not work for those representations which have low spins in .

For example, consider the beta-deformation. The corresponding vertex was studied in my paper “Notes on beta-deformations of the pure spinor superstring in ” with O.A. Bedoya, L.I. Beviláqua and V.O. Rivelles, arXiv:1005.0049.

It is parametrized by a constant antisymmetric tensor :


But there is a gauge symmetry parametrized by a constant :


(If is of such form, then the deformation (5) can be undone by the field redefinition.)

This means that actually the space of deformations of the form (5) is not but rather .

But to write the Eq. (5) for the vertex we need to pick a representative . Actually, in this case the covariant vertex does not exist. This is because of the low spin; the existence theorem (2) only works for “high enough spin”.