- Free Higher Spin Fields
- No-go Theorems Flat Space Anti-de Sitter Space
- Various Approaches to Higher Spin Theories Conformal Higher Spin Theories Collective Dipole Holographic RG Flow Noether Procedure Reconstruction Three dimensions and Chern-Simons Vasiliev Equations
- Higher Spin AdS/CFT Correspondence Klebanov-Polyakov Conjecture Gaberdiel-Gopakumar Conjecture
- References
Higher Spin Theory or Higher Spin Gravity is a common name for field theories that contain massless fields of spin greater than two. Usually, the spectrum of such theories contains
the graviton as a massless spin-two field, which explains the second name. Massless fields are gauge fields and the theories should be (almost) completely fixed by these higher spin symmetries. Higher spin theories are supposed to be consistent quantum theories and, for this reason, to give examples of quantum gravity. Most of the interest in the topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher spin theories to weakly coupled conformal field theories. 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[1], selfdual (chiral)[2][3] and Weyl gravity theories[6][7]).
Free Higher Spin Fields
Systematic study of massless arbitrary spin fields was initiated by Christian Fronsdal. A free spin-s field can be represented by a tensor gauge field.[4]
This (linearised) gauge symmetry generalises that of massless spin-one (photon) 
and that of massless spin-two (graviton) 
. Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above. For example, the equations are
where in the first bracket one needs 
terms more to make the expression symmetric and in the second bracket one needs 
permutations. The equations are gauge invariant provided the field is double-traceless 
and the gauge parameter is traceless 
.
Essentially, the higher spin problem can be stated as a problem to find a nontrivial interacting theory with at least one massless higher spin field (higher in this context usually means greater than two).
No-go Theorems
Possible interactions of massless higher spin particles with themselves and with low spin particles are (over)constrained by the basic principles of quantum field theory like Lorentz invariance. Many results in the form of no-go theorems have been obtained up to date[5]
Flat Space
Most of the no-go theorems constrain interactions in the flat space.
One of the most well-known is the Weinberg low energy theorem[6] 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. The latter is equivalent to gauge invariance under the linearised gauge symmetries above. These symmetries lead, for 
, to 'too many' conservation laws that trivialise scattering so that 
.
Another well-known result is the Coleman-Mandula theorem.[7] that, under certain assumptions, states that any symmetry group of S-matrix is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group. This means that there cannot be any symmetry generators transforming as tensors of the Lorentz group - S-matrix cannot have symmetries that would be associated with higher spin charges.
Massless higher spin particles also cannot consistently couple to nontrivial gravitational backgrounds.[8] 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[9][10] and show, for example, that the gauge symmetries cannot be deformed in a consistent way so that they form an algebra.
Anti-de Sitter Space
In anti-de Sitter space many of the flat space no-go results are invalid. In particular, it was shown by Fradkin and Vasiliev[11] that one can consistently couple massless higher spin fields to gravity at the first non-trivial order.
Nevertheless, an analog of the Coleman-Mandula theorem was obtained by Madacena and Zhiboedov.[12] AdS/CFT correspondence replaces the flat space S-matrix with the holographic correlation functions. 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 (which would be that the conformal field
theory is a generalized free field one).
Various Approaches to Higher Spin Theories
The existence of many higher spin theories is well-justified on the basis of AdS/Correspondence, but it should be noted that none of these hypothetical theories is known in full detail.
Most of the common approaches to the higher spin problem are described below.
Conformal Higher Spin Theories
Usual massless higher spin symmetries generalise the action of the linearised diffeomorphisms from the metric tensor to higher spin fields. In the
context of gravity one may also be interested in Conformal gravity that enlarges diffeomorphisms with Weyl transformations 
where 
is an arbitrary function. 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 higher spin generalisation of the Weyl symmetry. As different from massless higher spin fields, conformal higher spin
fields are much more tractable: they can propagate on nontrivial gravitational background and admit interactions in flat space. In particular, the action of conformal higher spin
theories is known to some extent[13][14] - it can be obtained as an effective action for a free conformal field theory coupled to the conformal higher spin background.
Collective Dipole
The idea is conceptually similar to the reconstruction approach just described, but performs a complete reconstruction in some sense. One begins with the free 
model partition function and performs a change of variables by
passing from the scalar fields 
, 
to a new bi-local variable 
. In the limit
of large 
this change of variables is well-defined, but has a nontrivial Jacobian. The same partition function can then be rewritten as a path integral over bi-local 
. It can also be shown that in the free approximation the bi-local variables describe free massless fields of all spins 
in anti-de Sitter space.
Therefore, the action in term of the bi-local is a candidate for the action of a higher spin theory[15]
Holographic RG Flow
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. This idea can be applied to the conjectural duals of higher spin theories, for example, to the free model.[16][17]
Noether Procedure
Noether procedure is a canonical perturbative method to introduce interactions. One begins with a sum of free (quadratic) actions 
and linearised gauge symmetries 
, which are given by Fronsdal Lagrangian and by the gauge transformations above. The idea is to add all possible corrections that are cubic in the fields 
and, at the same time, allow for field-dependent deformations 
of the gauge transformations. 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 
because the free action is gauge invariant). Therefore, the first condition is 
. One has to mod out by the trivial solutions that result from nonlinear field redefinitions in the free action. The deformation procedure may not stop at this order and one may have to add quartic terms 
and further corrections 
to the gauge transformations that are quadratic in the fields and so on. The systematic approach is via BV-BRST techniques.[18] Unfortunately, the Noether procedure approach has not given yet any complete example of a higher spin theory, the difficulties being not only in the technicalities but also in the conceptual understanding of locality in higher-spin theories. 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. At present, it seems that higher spin theories cannot be fully understood as field theories due to quite non-local interactions they have.[19]
Reconstruction
The Higher Spin AdS/CFT Correspondence can be used in the reverse order - one can attempt to build the interaction vertices of the higher spin theory in such a way that they reproduce
the correlation functions of a given conjectural CFT dual[20]. 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. In particular, the cubic part of the action of the Type-A higher spin theory was found[21] by inverting the three-point functions of the higher spin currents in the free scalar CFT. Some
quartic vertices have been reconstructed too.[22]
Three dimensions and Chern-Simons
In three dimensions neither gravity nor massless higher spin fields have any propagating degrees of freedom. It is known[23] that the Einstein-Hilbert action with negative cosmological constant can be rewritten in the Chern-Simons form for 
where there are two independent 
-connections, 
and 
. Due to isomorphisms 
and 
the algebra can be understood as the Lorentz algebra in three dimensions. These two connections are related to vielbein 
and spin-connection 
(Note that in three dimensions, the spin-connection, being anti-symmetric in 
is equivalent to an 
vector via 
, where 
is the totally anti-symmetric Levi-Civita symbol). Higher-spin extensions are straightforward to construct:[24] instead of 
connection one can take a connection of 
, where 
is any Lie algebra containing the 'gravitational' subalgebra. Such theories have been extensively studied[25][26] due their relation to AdS/CFT and W-algebras as asymptotic symmetries.
Vasiliev Equations
Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Fronsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang-Mills gaugings. 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. Locality issues remain to be clarified.
Higher Spin AdS/CFT Correspondence
Higher spin theories are of interest as models of AdS/CFT correspondence.
Klebanov-Polyakov Conjecture
In 2002 Klebanov and Polyakov put forward a conjecture[27] that the free and critical vector models, as conformal field theories in three dimensions, should be dual to a theory in four-dimensional anti-de Sitter space with infinite number
of massless higher-spin gauge fields. This conjecture was further extended and generalised to Gross-Neveu and super-symmetric models.[28][29] The most interesting extension is to a class of Chern-Simons matter theories.[30]
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 
, where spin runs over all positive integers 
(in the model the spin is even). The stress-tensor corresponds to the 
case.
By the standard AdS/CFT lore, the fields that are dual to conserved currents have to be gauge fields. For example, the stress-tensor is dual to the spin-two graviton field. A generic example of a conformal field theory with higher spin currents is any free CFT. For instance, the free model is defined by
where 
. It can be shown that there exist an infinite number of quasi-primary operators
that are conserved. Under certain assumptions it was shown by Maldacena and Zhiboedov[12] that conformal field theories with higher spin currents are free ones. Therefore, higher spin theories are generic duals of free conformal field theories. A theory that is dual to the free scalar CFT is called Type-A in the literature and the theory that is dual to the free fermion CFT is called Type-B.
Another example is the critical vector model, which is a theory with action
taken at the fixed point. This theory is interacting and does not have conserved higher spin currents. However, in the large N limit it can be shown to have 'almost' conserved
higher spin currents and the conservation is broken by 
effects.
Gaberdiel-Gopakumar Conjecture
The conjecture put forward by Gaberdiel and Gopakumar[31] is an extension of the Klebanov-Polyakov conjecture to 
. It states that the 
minimal models in the large limit should be dual to theories with massless higher spin fields and two scalar fields. Massless higher spin fields do not propagate in three dimensions, but can be described, as is discussed above, by the Chern-Simons action. However, it is not known to extend this action as to include the matter fields required by the duality.
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