It is important to note that only certain parts of these theories are known at present (in particular, standard action principles are not known) and not many examples have been worked out in detail except some specific toy models (such as the higher-spin extension of pure Chern–Simons,[1][2] Jackiw–Teitelboim,[3] selfdual (chiral)[4][5] and Weyl gravity theories[6][7]).
Systematic study of massless arbitrary spin fields was initiated by Christian Fronsdal.
Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above.
[9][10] This massive theory is important because, according to various conjectures,[11][12][13] spontaneously broken gauges of higher-spins may contain an infinite tower of massive higher-spin particles on the top of the massless modes of lower spins s ≤ 2 like graviton similarly as in string theories.
The linearized version of the higher-spin supergravity gives rise to dual graviton field in first order form.
One of the most well-known is the Weinberg low energy theorem[20] that explains why there are no macroscopic fields corresponding to particles of spin 3 or higher.
The Weinberg theorem can be interpreted in the following way: Lorentz invariance of the S-matrix is equivalent, for massless particles, to decoupling of longitudinal states.
Massless higher spin particles also cannot consistently couple to nontrivial gravitational backgrounds.
[22] An attempt to simply replace partial derivatives with the covariant ones turns out to be inconsistent with gauge invariance.
Other no-go results include a direct analysis of possible interactions[24][25] and show, for example, that the gauge symmetries cannot be deformed in a consistent way so that they form an algebra.
In particular, it was shown by Fradkin and Vasiliev[26] that one can consistently couple massless higher-spin fields to gravity at the first non-trivial order.
The difference between the flat space result and the AdS one is that the gravitational coupling of massless higher-spin fields cannot be written in the manifestly covariant form in flat space[27] as different from the AdS case.
It then can be shown that the asymptotic higher-spin symmetry in anti-de Sitter space implies that the holographic correlation functions are those of the singlet sector a free vector model conformal field theory (see also higher-spin AdS/CFT correspondence below).
Let us stress that all n-point correlation functions are not vanishing so this statement is not exactly the analogue of the triviality of the S-matrix.
[29] In the latter case the holographic S-matrix corresponds to highly nontrivial Chern–Simons matter theories rather than to a free CFT.
As in the flat space case, other no-go results include a direct analysis of possible interactions.
Generic theories with massless higher-spin fields are obstructed by non-localities, see No-go theorems.
The simplest example of a conformal gravity is in four dimensions One can try to generalise this idea to higher-spin fields by postulating the linearised gauge transformations of the form where
is a candidate for the action of a higher-spin theory[34] The idea is that the equations of the exact renormalization group can be reinterpreted as equations of motions with the RG energy scale playing the role of the radial coordinate in anti-de Sitter space.
One then requires the full action to be gauge invariant and solves this constraint at the first nontrivial order in the weak-field expansion (note that
One has to mod out by the trivial solutions that result from nonlinear field redefinitions in the free action.
Unless locality is imposed one can always find a solution to the Noether procedure (for example, by inverting the kinetic operator in
that results from the second term) or, the same time, by performing a suitable nonlocal redefinition one can remove any interaction.
[39] This approach takes advantage of the fact that the kinematics of AdS theories is, to some extent, equivalent to the kinematics of conformal field theories in one dimension lower – one has exactly the same number of independent structures on both sides.
Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space.
However, locality has not been an assumption used in the derivation and, for this reason, some of the results obtained from the equations are inconsistent with higher-spin theories and AdS/CFT duality.
[47] The rationale for the conjectures is that there are some conformal field theories that, in addition to the stress-tensor, have an infinite number of conserved tensors
A generic example of a conformal field theory with higher-spin currents is any free CFT.
Under certain assumptions it was shown by Maldacena and Zhiboedov[28] that 3d conformal field theories with higher spin currents are free, which can be extended[48][49] to any dimension greater than two.
More generally, free and critical vector models belong to the class of Chern–Simons matter theories that have slightly broken higher-spin symmetry.