On the relation between local and geometric Lagrangians for higher spins

TL;DR

The work addresses how to define unique, curvature-based non-local Lagrangians for free higher-spin fields by starting from local unconstrained Lagrangians and integrating out auxiliary fields. For bosons, spins 3 and 4 on flat space and spin 3 on (A)dS are shown to yield effective non-local actions that coincide with the geometric theories that produce the correct propagators. For fermions, the spin-5/2 case on flat space is analyzed, yielding a non-local kinetic tensor consistent with a curvature-based interpretation and correct current exchanges. Together, the results argue that geometric higher-spin Lagrangians can emerge as effective theories from conventional local formulations, providing a criterion for selecting the physically correct non-local curvature-based descriptions and guiding future work on interactions and triplets.

Abstract

Equations of motion for free higher-spin gauge fields of any symmetry can be formulated in terms of linearised curvatures. On the other hand, gauge invariance alone does not fix the form of the corresponding actions which, in addition, either contain higher derivatives or involve inverse powers of the d'Alembertian operator, thus introducing possible subtleties in degrees of freedom count. We suggest a path to avoid ambiguities, starting from local, unconstrained Lagrangians previously proposed, and integrating out the auxiliary fields from the functional integral, thus generating a unique non-local theory expressed in terms of curvatures.