Unconstrained higher spins of mixed symmetry I. Bose fields

Abstract

This is the first of two papers devoted to the local "metric-like" unconstrained Lagrangians and field equations for higher-spin gauge fields of mixed symmetry in flat space. Here we complete the previous constrained formulation of Labastida for Bose fields. We thus recover his Lagrangians via the Bianchi identities, before extending them to their "minimal" unconstrained form with higher derivatives of the compensator fields and to yet another, non-minimal, form with only two-derivative terms. We also identify classes of these systems that are invariant under Weyl-like symmetries.

Cubic interaction vertices for continuous-spin fields and arbitrary spin massive fields

“…Discussion of other interesting formulations in terms of unconstrained gauge fields can be found, e.g., in refs. [77][78][79]. BRST formulations in terms of traceceless higher-spin gauge fields may be found in refs.…”

Maxwell-like Lagrangians for higher spins

“…Let us finally observe that, in analogy with the spin−1 case, where the divergence of the field strength defines the equations of motion, in our present setting the kinetic tensor 7 Lagrangians for symmetric bosons and fermions of arbitrary spin were first formulated in terms of metric-like curvatures in [29] while an approach similar in spirit was also proposed for mixed-symmetry bosons in [45,46]. Out of the infinitely many options available in principle, the unique Lagrangians leading to the correct propagators were given for symmetric tensors in [28], while their massive deformations were discussed in [31], together with a more detailed analysis of the role of curvatures for fermionic theories.…”

“…The proper generalisation of the index-free notation used in the symmetric case was introduced in [7,8,9,10] and is reviewed in appendix A.2. The basic idea is to employ "family" labels to denote operations adding or subtracting space-time indices belonging to a given group.…”

“…Indeed, for the irreducible case involving traceless fields and transverse-traceless parameters investigated in [5], the Lagrangian (1.7) can be seen to arise from a partial gauge fixing of Fronsdal's Lagrangian itself, whose formulation requires traceless parameters (and doubly traceless fields). In its turn, the Fronsdal-Labastida theory admits minimal unconstrained extensions given in [27,28,7,8], building on previous formulations [29,30,31] where the removal of constraints was linked to the possibility of assigning a dynamical role to the higher-spin curvatures of [32] 4 . For the transverse-invariant Lagrangians that we propose in this work the most natural unconstrained extensions should be identified with the "triplets" associated to the tensionless limit of free open string field theory [34,30,35,36] (see also [37,38,39]), whenever the corresponding actions are available.…”

“…Comparing (1.9) with the constrained Lagrangian of Labastida [6] or with its minimal unconstrained versions given in [7,8] (see also [9,10] for reviews) allows to appreciate the advantages of our present approach 3 : while Lagrangians (1.9) always maintain the same form irrespective of the number N of index-families of the tensor ϕ, in the approach of [6,7] the need to implement the more conventional kind of gauge invariance calls for the introduction of a number of traces of the basic kinetic tensor increasing with N. (See eq. (2.51).)…”