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 Frønsdal 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. 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 higher-spin AdS/CFT correspondence is reviewed in Higher-spin theory article.
The Vasiliev equations are generating equations and yield differential equations in the space-time upon solving them order by order with respect to certain auxiliary directions. The equations rely on several ingredients: unfolded equations and higher-spin algebras.
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 example of the four-dimensional bosonic Vasiliev's equations is reviewed at length since all other dimensions and super-symmetric generalisations are simple modifications of this basic example. the definition of the higher-spin algebra is given since the higher-spin theory equations turns out to be the equations for two fields taking values in the higher-spin algebra; the specific star-product that the fields that enter Vasiliev's equations take values in is defined; part of the Vasiliev equations is related to an interesting deformation of the Harmonic oscillator, called deformed oscillators, which is reviewed; the unfolded approach is discussed, which is a slightly advanced form of writing the differential equations in the first order form; the Vasiliev equations are given; it is proved that the linearisation of Vasiliev's equations over anti-de Sitter space describes free massless higher-spin fields.
Three variations of Vasiliev's equations are known: four-dimensional, three-dimensional and d-dimensional. They differ by mild details that are discussed below.
Higher-spin algebras
Higher-spin algebras are global symmetries of the higher-spin theory multiplet. 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. Another definition is that higher-spin algebras are quotients of the universal enveloping algebra of the anti-de Sitter algebra by certain two-sided ideals. Some more complicated examples of higher-spin algebras exist, but all of them can be obtained by tensoring the simplest higher-spin algebras with matrix algebras and then imposing further constraints. Higher-spin algebras originate as associative algebras and the Lie algebra can be constructed via the commutator.
In the case of the four-dimensional bosonic higher-spin theory the relevant higher-spin algebra is very simple thanks to and can be built upon two-dimensional quantum Harmonic oscillator. In the latter case two pairs of creation/annihilation operators are needed. These can be packed into the quartet of operators obeying the canonical commutation relations
where is the invariant tensor, i.e. it is anti-symmetric. As is well known, the bilinears provide an oscillator realization of :
The higher-spin algebra is defined as the algebra of all even functions in . That the functions are even is in accordance with the bosonic content of the higher-spin theory as will be shown to be related to the Majorana spinors from the space-time point of view and even powers of correspond to tensors. It is an associative algebra and the product is conveniently realised by the Moyal star product:
with the meaning that the algebra of operators can be replaced with the algebra of function in ordinary commuting variables (hats off) and the product needs to be replaced with the non-commutative star-product. For example, one finds
and therefore as it would be the case for the operators. 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. The prefactor is chosen as to ensure . In the Lorentz-covariant base we can split and we also split . Then the Lorentz generators are , and the translation generators are . The -automorphism can be realized in two equivalent ways: either as or as . In both the cases it leaves the Lorentz generators untouched and flips the sign of translations.
The higher-spin algebra constructed above can be shown to be the symmetry algebra of the three-dimensional Klein–Gordon equation . Considering more general free CFT's, e.g. a number of scalars plus a number of fermions, the Maxwell field and other, one can construct more examples of higher-spin algebras.
Vasiliev star-product
The Vasiliev equations are equations in certain bigger space endowed with auxiliary directions to be solved for. The additional directions are given by the doubles of , called , which are furthermore entangled with Y. The star-product on the algebra of functions in in -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. In accordance with the bosonic projection is given by .
Deformed oscillators
The essential part of the Vasiliev equations relies on an interesting deformation of the Quantum harmonic oscillator, known as deformed oscillators. First of all, let us pack the usual creation and annihilation operators in a doublet . The canonical commutation relations (the -factors are introduced to facilitate comparison with Vasiliev's equations)
can be used to prove that the bilinears in form generators
In particular, rotates as an -vector with playing the role of the -invariant metric. The deformed oscillators are defined by appending the set of generators with an additional generating element and postulating
Again, one can see that , as defined above, form -generators and rotate properly . At we get back to the undeformed oscillators. In fact, and form the generators of the Lie superalgebra , where should be viewed as odd generators. Then, is the part of the defining relations of . One (or two) copies of the deformed oscillator relations form a part of the Vasiliev equations where the generators are replaced with fields and the commutation relations are imposed as field equations.
Unfolded equations
The equations for higher-spin fields originate from the Vasiliev equations in the unfolded form. Any set of differential equations can be put in the first order form by introducing auxiliary fields to denote derivatives. Unfolded approach is an advanced reformulation of this idea that takes into account gauge symmetries and diffeomorphisms. Instead of just the unfolded equations are written in the language of differential forms as
where the variables are differential forms of various degrees, enumerated by an abstract index ; is the exterior derivative . The structure function is assumed to be expandable in exterior product Taylor series as
where has form degree and the sum is over all forms whose form degrees add up to . The simplest example of unfolded equations are the zero curvature equations for a one-form connection of any Lie algebra . Here runs over the base of the Lie algebra, and the structure function encodes the structure constants of the Lie algebra.
Since the consistency of the unfolded equations requires
which is the Frobenius integrability condition. In the case of the zero curvature equation this is just the Jacobi identity. Once the system is integrable it can be shown to have certain gauge symmetries. Every field that is a form of non-zero degree possesses a gauge parameter that is a form of degree and the gauge transformations are
The Vasiliev equations generate the unfolded equations for a specific field content, which consists of a one-form and a zero-form , both taking values in the higher-spin algebra. Therefore, and , . The unfolded equations that describe interactions of higher-spin fields are
where are the interaction vertices that are of higher and higher order in the -field. The product in the higher-spin algebra is denoted by . The explicit form of the vertices can be extracted from the Vasiliev equations. The vertices that are bilinear in the fields are determined by the higher-spin algebra. Automorphism is induced by the automorphism of the anti-de Sitter algebra that flips the sign of translations, see below. If we truncate away higher orders in the -expansion, the equations are just the zero-curvature condition for a connection of the higher-spin algebra and the covariant constancy equation for a zero-form that takes values in the twisted-adjoint representation (twist is by the automorphism ).
Field content
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: gauge connection , whose value at Z=0 gives the connection of the higher-spin algebra . The bosonic projection implies ; zero-form , whose value at Z=0 gives the zero-form of the higher-spin algebra . The bosonic projection implies ; an auxiliary field , where it is sometimes useful to view it as a one-form in the auxiliary Z-space, hence the differentials: This field can be eliminated when solving for the Z-dependence. The bosonic projection for the -field is due to the additional index that is eventually carried by Y,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. The Vasiliev equations can be split into two parts. The first part contains only zero-curvature or covariant constancy equations:
where the higher-spin algebra automorphism is extended to the full algebra as
the latter two forms being equivalent because of the bosonic projection imposed on .
Therefore, the first part of the equations implies that there is no nontrivial curvature in the x-space since is flat. The second part makes the system nontrivial and determines the curvature of the auxiliary connection :
where two Klein operators were introduced
The existence of the Klein operators is of utter importance for the system. They realise the automorphism as an inner one
In other words, the Klein operator behave as , i.e. it anti-commutes to odd functions and commute to even functions in y,z.
These 3+2 equations are the Vasiliev equations for the four-dimensional bosonic higher-spin theory. Several comments are in order. The algebraic part of the system when split into components in accordance with the choice of the -metric
becomes equivalent to two copies of the mutually commuting deformed oscillators:
Therefore, the last two equations are equivalent to the definition relations of two copies of with the and playing the role of the odd generators and with and playing the role of the deformations. Since is the same for the two copies, they are not independent, which does not spoil the consistency.