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.
Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space.
It is important to note that locality is not properly implemented and the equations give a solution of certain formal deformation procedure, which is difficult to map to field theory language.
The exposition below is organised in such a way as to split the Vasiliev's equations into the building blocks and then join them together.
The same time they can be defined as global symmetries of some conformal field theories (CFT), which underlies the kinematic part of the higher-spin AdS/CFT correspondence, which is a particular case of the AdS/CFT.
will be shown to be related to the Majorana spinors from the space-time point of view and even powers of
Another representation of the same star-product is more useful in practice: The exponential formula can be derived by integrating by parts and dropping the boundary terms.
In both the cases it leaves the Lorentz generators untouched and flips the sign of translations.
-variables is The integral formula here-above is a particular star-product that corresponds to the Weyl ordering among Y's and among Z's, with the opposite signs for the commutator: Moreover, the Y-Z star product is normal ordered with respect to Y-Z and Y+Z as is seen from The higher-spin algebra is an associative subalgebra in the extended algebra.
Any set of differential equations can be put in the first order form by introducing auxiliary fields to denote derivatives.
Unfolded approach[7] is an advanced reformulation of this idea that takes into account gauge symmetries and diffeomorphisms.
is induced by the automorphism of the anti-de Sitter algebra that flips the sign of translations, see below.
The field content of the Vasiliev equations is given by three fields all taking values in the extended algebra of functions in Y and Z: As to avoid any confusion caused by the differential forms in the auxiliary Z-space and to reveal the relation to the deformed oscillators the Vasiliev equations are written below in the component form.
is extended to the full algebra as the latter two forms being equivalent because of the bosonic projection imposed on
The second part makes the system nontrivial and determines the curvature of the auxiliary connection
To prove that the linearized Vasiliev equations do describe free massless higher-spin fields we need to consider the linearised fluctuations over the anti-de Sitter vacuum.
The Lorentz covariant derivative comes from the usual commutator action of the spin-connection part of
In addition one needs to know how to multiply by the Klein operator from the right, which is easy to derive from the integral formula for the star-product: I.e. the result is to exchange the half of the Y and Z variables and to flip the sign.
After some algebra one finds where we again dropped a term with dotted and undotted indices exchanged.
Practically speaking, it is not known in general how to extract the interaction vertices of the higher spin theory out of the equations.
A surprising fact that had been noticed[11][12] before its inconsistency with the AdS/CFT was realized is that the stress-tensor can change sign and, in particular, vanishes for
This would imply that the corresponding correlation function in the Chern-Simons matter theories vanishes,
It was first shown[13] that some of the three-point AdS/CFT functions, as obtained from the Vasiliev equations, turn out to be infinite or inconsistent with AdS/CFT, while some other do agree.
These non-localities are not present in higher spin theories as can be seen from the explicit cubic action.
[15][16][17][18][19] Some of these tests explore the extension of the Klebanov–Polyakov Conjecture to Chern–Simons matter theories where the structure of correlation functions is more intricate and certain parity-odd terms are present.
General analysis of the Vasiliev equations at the second order[20] showed that for any three fixed spins the interaction term is an infinite series in derivatives (similar to
All the problems can be attributed to the assumptions used in the derivation of the Vasiliev equations: restrictions on the number of derivatives in the interaction vertices or, more generally, locality was not imposed, which is important for getting meaningful interaction vertices, see e.g. Noether Procedure.
The problem how to impose locality and extract interaction vertices out of the equations is now under active investigation.
[21] As is briefly mentioned in Other dimensions, extensions, and generalisations there is an option to introduce infinitely many additional coupling constants that enter via phase factor
For example, the perturbative corrections at the second order to the stress-tensors of the matter fields lead to infinite correlation functions.