• Noether procedure
• local symmetries
• higher spins
• gauge theories
• cubic interactions

This work deals with Higher-Spin Gauge Theories, an old branch of Relativistic Field Theory that attracted new interest in the last few years due to its role in instances of holographic dualities as well as in force of the possible role of higher-spin symmetries in clarifying the geometrical basis of String Theory.
Higher-spin interactions were long considered to be inconsistent, due to a number of no-go theorems mainly focused on S-matrix arguments and on minimal coupling to gravity in flat backgrounds. Starting from the end of the eighties, however, central results due to M. A. Vasiliev and collaborators showed how to bypass those negative conclusions and how to construct fully interacting non-linear equations of motion. Still, Vasiliev’s equations are intrinsically non-Lagrangian, thus motivating the effort of building an action principle for interacting higher-spin fields by some alternative route. In addition, recent findings points to some possible ambiguities in Vasiliev’s formulation: when some auxiliary fields needed in that context are removed infinite higher derivative corrections to the vertices apparently emerge, and the proper way to interpret them is currently under debate. A better understanding of higher-spin interactions formulated directly in terms of physical fields is certainly desirable.
These observations serve to motivate the central theme of this work: we construct cubic Lagrangian vertices for higher-spin gauge fields described at free level by a new class of Lagrangians, termed Maxwell-like, generalising the Fronsdal formulation to the case of reducible multiplets of massless particles. Our main tool is the so-called Noether procedure. The general idea is that, given a free gauge theory, one can construct the corresponding interacting theory in a perturbative way, adding vertices of increasing order (cubic, quartic and so on), while at the same time allowing for field-dependent corrections to the gauge transformation. The requirement that gauge invariance is kept at all orders fixes in principle all these quantities thus providing a sort of “bootstrap approach” to the construction of interacting gauge theories.
Our work covers three main aspects:

(1) We provide an overview on the main features of higher spin theories at free and interacting level. We review the group-theoretical definition of elementary particles in Relativistic Field Theory and indicate how to recover the Lagrangians for free massless fields in the formulation due to Fronsdal. We provide a somewhat streamlined critical discussion of some of the main no-go theorems against the possibility of interacting theories involving higher spins, such as the Weinberg theorem, trying to emphasize hypothesis and physical implications, while also highlighting the main mechanisms explored so far to the goal of evading the no-go results.

(2) We discuss our methodology by means of a detailed illustration of the Noether procedure. After a presentation of the general structure of the method, we show how to implement it for the low-spin cases of Maxwell fields (spin 1) and gravitons (spin 2). In the former case we show how to recover Yang-Mills theory by the mere requirement that a bunch of Maxwell fields interact in a way that be consistent with their free, Abelian gauge symmetry. In the latter case, starting from free massless spin 2 particles, we find the Einstein-Hilbert cubic vertex expanded around Minkowski space-time, together with the associated deformation of the gauge transformation, observing that it gives rise to the Lie derivative. The geometrical structure of the corresponding theories, such as the need for an underlying Lie algebra for spin−one massless fields and diffeomorphism invariance for interacting gravitons emerge as an outcome of the self-consistency procedure and need not be known in advance.

(3) The third section collects our original results. We apply the Noether procedure in the context of the Maxwell-like formulation for higher-spin gauge fields. Their relevant features are of two types: on the one hand, the kinetic tensor building the free Lagrangian is simpler with respect to the Fronsdal one, and one may hope that this simplicity is kept at the interacting level. In addition, the corresponding equations of motion propagate a reducible spectrum of particles, which matches the description of massless fields of arbitrary spin emerging from the tensionless limit of free open strings. Thus, for instance, Maxwell-like equations of motion for a rank−two symmetric tensor propagate a massless spin−two particle together with a massless scalar, as opposed to the Fronsdal case (corresponding for spin two to the linearised Einstein-Hilbert theory) that describes starting from the same tensor a single massless particle with spin two. The Maxwell-like cubic vertices resulting from our approach display a sizeable simplification with respect to the Fronsdal ones: while the latter involve a total of ten types of terms the former only make use of four classes of cubic contributions.
We find that the structure of the free gauge symmetry of Maxwell-like Lagrangians calls for a generalization of the Noether procedure that, to our knowledge, was not considered so far, at least in the higher-spin context. More precisely, we are led to consider the Noether method for gauge parameters subject to some constraints at the free level. These constraints may in principle enter the procedure in an algorithmic way, forcing an additional equation encoding their possible corrections order by order. For the Maxwell-like case we find indeed that such an additional equation is needed and we compute the explicit deformation of the constraint associated to every vertex involving arbitrary spins. In fact, with hindsight, one can realize that the deformation of the constraints can actually encode crucial information about the underlying geometry. We discuss in these terms the explicit example of a self-interacting rank-two Maxwell-like field (providing a bottom-up approach to Unimodular Gravity) and comment on the possibility that also Fronsdal trace constraints get similarly deformed, an option that so far was almost completely neglected.
Finally, we present a new class of couplings, containing particular combinations of the divergences of the fields, that are naturally suggested in our generalized Noether setting, and that was considered to be forbidden in previous treatments.