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 s o ( d , 2 ) {\displaystyle so(d,2)} 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 s o ( 3 , 2 ) ∼ s p ( 4 , R ) {\textstyle so(3,2)\sim sp(4,\mathbb {R} )} and can be built upon two-dimensional quantum Harmonic oscillator. In the latter case two pairs of creation/annihilation operators a 1 , a 1 † , a 2 , a 2 † {\textstyle a_{1},a_{1}^{\dagger },a_{2},a_{2}^{\dagger }} are needed. These can be packed into the quartet
Y ^ A , A = 1 , . . . , 4 {\textstyle {\hat {Y}}^{A},A=1,...,4} of operators obeying the canonical commutation relations
[ Y ^ A , Y ^ B ] = 2 i C A B , {\displaystyle [{\hat {Y}}^{A},{\hat {Y}}^{B}]=2iC^{AB}\,,}
where C A B = − C B A {\textstyle C^{AB}=-C^{BA}} is the s p ( 4 ) {\textstyle sp(4)} invariant tensor, i.e. it is anti-symmetric. As is well known, the bilinears provide an oscillator realization of s p ( 4 ) {\textstyle sp(4)} :
T A B = − i 4 { Y ^ A , Y ^ B } , [ T A B , T C D ] = T A D C B C + 3 more . {\displaystyle T^{AB}=-{\frac {i}{4}}\{{\hat {Y}}^{A},{\hat {Y}}^{B}\}\,,\qquad [T^{AB},T^{CD}]=T^{AD}C^{BC}+{\text{3 more}}\,.}
The higher-spin algebra is defined as the algebra of all even functions f ( Y ^ ) , f ( Y ^ ) = f ( − Y ^ ) {\textstyle f({\hat {Y}}),f({\hat {Y}})=f(-{\hat {Y}})} in Y ^ A {\textstyle {\hat {Y}}^{A}} . That the functions are even is in accordance with the bosonic content of the higher-spin theory as Y ^ A {\textstyle {\hat {Y}}^{A}} will be shown to be related to the Majorana spinors from the space-time point of view and even powers of Y ^ A {\textstyle {\hat {Y}}^{A}} correspond to tensors. It is an associative algebra and the product is conveniently realised by the Moyal star product:
( f ⋆ g ) ( Y ) = f ( Y ) exp i ( ∂ ← ∂ Y A C A B ∂ → ∂ Y B ) g ( Y ) , {\displaystyle (f\star g)(Y)=f(Y)\exp i\left({{\frac {\overleftarrow {\partial }}{\partial Y^{A}}}C^{AB}{\frac {\overrightarrow {\partial }}{\partial Y^{B}}}}\right)g(Y)\,,}
with the meaning that the algebra of operators f ( Y ^ ) {\textstyle f({\hat {Y}})} can be replaced with the algebra of function f ( Y ) {\textstyle f(Y)} in ordinary commuting variables Y A {\textstyle {Y}^{A}} (hats off) and the product needs to be replaced with the non-commutative star-product. For example, one finds
( Y A ⋆ g ) ( Y ) = ( Y A + i C A B ∂ B ) g ( Y ) , ( f ⋆ Y B ) ( Y ) = ( Y B − i C B A ∂ B ) f ( Y ) , {\displaystyle (Y^{A}\star g)(Y)=(Y^{A}+iC^{AB}\partial _{B})g(Y)\,,\qquad (f\star Y^{B})(Y)=(Y^{B}-iC^{BA}\partial _{B})f(Y)\,,}
and therefore Y A ⋆ Y B − Y B ⋆ Y A = [ Y A , Y B ] ⋆ = 2 i C A B {\textstyle Y^{A}\star Y^{B}-Y^{B}\star Y^{A}=[Y^{A},Y^{B}]_{\star }=2iC^{AB}} as it would be the case for the operators. Another representation of the same star-product is more useful in practice:
( f ⋆ g ) ( Y ) = 1 ( 2 π ) 4 ∫ d U d V f ( Y + U ) g ( Y + V ) e i U A V B C A B . {\displaystyle (f\star g)(Y)={\frac {1}{(2\pi )^{4}}}\int dUdVf(Y+U)g(Y+V)e^{iU_{A}V_{B}C^{AB}}\,.}
The exponential formula can be derived by integrating by parts and dropping the boundary terms. The prefactor is chosen as to ensure 1 ⋆ 1 = 1 {\displaystyle 1\star 1=1} . In the Lorentz-covariant base we can split A = α , α ˙ ; α = 1 , 2 ; α ˙ = 1 , 2 {\textstyle A=\alpha ,{\dot {\alpha }};\alpha =1,2;{\dot {\alpha }}=1,2} and we also split Y A = y α , y α ˙ {\displaystyle Y^{A}=y^{\alpha },y^{\dot {\alpha }}} . Then the Lorentz generators are L α β = T α β {\textstyle L^{\alpha \beta }=T^{\alpha \beta }} , L ¯ α ˙ β ˙ = T α ˙ β ˙ {\textstyle {\bar {L}}^{{\dot {\alpha }}{\dot {\beta }}}=T^{{\dot {\alpha }}{\dot {\beta }}}} and the translation generators are P α β ˙ = T α β ˙ {\textstyle P^{\alpha {\dot {\beta }}}=T^{{\alpha }{\dot {\beta }}}} . The π {\textstyle \pi } -automorphism can be realized in two equivalent ways: either as π ( y α ) = − y α , π ( y α ˙ ) = y α ˙ {\textstyle \pi (y^{\alpha })=-y^{\alpha },\pi (y^{\dot {\alpha }})=y^{\dot {\alpha }}} or as π ( y α ) = Y α , π ( y α ˙ ) = − y α ˙ {\textstyle \pi (y^{\alpha })=Y^{\alpha },\pi (y^{\dot {\alpha }})=-y^{\dot {\alpha }}} . 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 ◻ 3 ϕ ( x ) = 0 {\displaystyle \square _{3}\phi (x)=0} . 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 Y A {\textstyle {Y}^{A}} , called Z A {\textstyle {Z}^{A}} , which are furthermore entangled with Y. The star-product on the algebra of functions in f ( Y , Z ) {\textstyle f(Y,Z)} in Y , Z {\textstyle {Y},Z} -variables is
F ( Y , Z ) ⋆ G ( Y , Z ) = 1 ( 2 π ) 4 ∫ d U d V F ( Y + U , Z + U ) G ( Y + V , Z − V ) exp [ i U A V B C A B ] . {\displaystyle F(Y,Z)\star G(Y,Z)={\frac {1}{(2\pi )^{4}}}\int dU\,dV\,F(Y+U,Z+U)G(Y+V,Z-V)\exp {[iU_{A}V_{B}C^{AB}]}\,.}
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:
[ Y A , Y B ] = 2 i C A B , [ Z A , Z B ] = − 2 i C A B . {\displaystyle [Y^{A},Y^{B}]=2iC^{AB}\,,\qquad \qquad [Z^{A},Z^{B}]=-2iC^{AB}\,.}
Moreover, the Y-Z star product is normal ordered with respect to Y-Z and Y+Z as is seen from
F ( a , a † ) ⋆ G ( a , a † ) = 1 ( 2 π ) 4 ∫ d U d V F ( a + 2 U , a † ) G ( a , a † + 2 V ) exp [ i U A V B C A B ] , a = Y + Z , a † = Y − Z {\displaystyle {\begin{aligned}F(a,a^{\dagger })\star G(a,a^{\dagger })&={\frac {1}{(2\pi )^{4}}}\int dU\,dV\,F(a+2U,a^{\dagger })G(a,a^{\dagger }+2V)\exp {[iU_{A}V_{B}C^{AB}]}\,,&\quad a=Y+Z\,,a^{\dagger }=Y-Z\end{aligned}}}
The higher-spin algebra is an associative subalgebra in the extended algebra. In accordance with the bosonic projection is given by f ( Y , Z ) = f ( − Y , − Z ) {\displaystyle f(Y,Z)=f(-Y,-Z)} .
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 a † , a {\textstyle a^{\dagger },a} in a doublet q α , α = 1 , 2 {\textstyle q_{\alpha }\,,\alpha =1,2} . The canonical commutation relations (the 2 i {\textstyle 2i} -factors are introduced to facilitate comparison with Vasiliev's equations)
[ q α , q β ] = − 2 i ϵ α β , ϵ α β = [ 0 1 − 1 0 ] , {\displaystyle \left[q_{\alpha },q_{\beta }\right]=-2i\epsilon _{\alpha \beta }\,,\qquad \epsilon _{\alpha \beta }={\begin{bmatrix}0&1\\-1&0\end{bmatrix}}\,,}
can be used to prove that the bilinears in q α {\displaystyle q_{\alpha }} form s p ( 2 ) ∼ s l ( 2 ) {\displaystyle sp(2)\sim sl(2)} generators
T α β = i 4 { q α , q β } , [ T α β , q γ ] = q α ϵ β γ + q β ϵ α γ , [ T α β , T γ δ ] = T α δ ϵ β γ + T β δ ϵ α γ + T α γ ϵ β δ + T β γ ϵ α δ . {\displaystyle {\begin{aligned}T_{\alpha \beta }&={\frac {i}{4}}\{q_{\alpha },q_{\beta }\}\,,\\\left[T_{\alpha \beta },q_{\gamma }\right]&=q_{\alpha }\epsilon _{\beta \gamma }+q_{\beta }\epsilon _{\alpha \gamma }\,,\\\left[T_{\alpha \beta },T_{\gamma \delta }\right]&=T_{\alpha \delta }\epsilon _{\beta \gamma }+T_{\beta \delta }\epsilon _{\alpha \gamma }+T_{\alpha \gamma }\epsilon _{\beta \delta }+T_{\beta \gamma }\epsilon _{\alpha \delta }\,.\end{aligned}}}
In particular, T α β {\displaystyle T_{\alpha \beta }} rotates q α {\displaystyle q_{\alpha }} as an s p ( 2 ) {\displaystyle sp(2)} -vector with ϵ α β {\displaystyle \epsilon _{\alpha \beta }} playing the role of the s p ( 2 ) {\displaystyle sp(2)} -invariant metric. The deformed oscillators are defined by appending the set of generators with an additional generating element Q {\displaystyle Q} and postulating
{ q α , Q } = 0 , [ q α , q β ] = − 2 i ϵ α β ( 1 + Q ) . {\displaystyle \{q_{\alpha },Q\}=0\,,\qquad \left[q_{\alpha },q_{\beta }\right]=-2i\epsilon _{\alpha \beta }(1+Q)\,.}
Again, one can see that T α β {\displaystyle T_{\alpha \beta }} , as defined above, form s p ( 2 ) {\displaystyle sp(2)} -generators and rotate properly q α {\displaystyle q_{\alpha }} . At Q = 0 {\displaystyle Q=0} we get back to the undeformed oscillators. In fact, q α {\displaystyle q_{\alpha }} and T α β {\displaystyle T_{\alpha \beta }} form the generators of the Lie superalgebra o s p ( 1 | 2 ) {\displaystyle osp(1|2)} , where q α {\displaystyle q_{\alpha }} should be viewed as odd generators. Then, { q α , q β } = − 4 i T α β {\displaystyle \{q_{\alpha },q_{\beta }\}=-4iT_{\alpha \beta }} is the part of the defining relations of o s p ( 1 | 2 ) {\displaystyle osp(1|2)} . 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 ∂ μ ϕ i ( x ) = f μ i ( ϕ ) {\textstyle \partial _{\mu }\phi ^{i}(x)=f_{\mu }^{i}(\phi )} the unfolded equations are written in the language of differential forms as
d W A = F A ( W ) , {\displaystyle dW^{A}=F^{A}(W)\,,}
where the variables are differential forms W A = W μ 1 . . . μ q A ( x ) d x μ 1 ∧ . . . ∧ d x μ q {\textstyle W^{A}=W_{\mu _{1}...\mu _{q}}^{A}(x)\,dx^{\mu _{1}}\wedge ...\wedge dx^{\mu _{q}}} of various degrees, enumerated by an abstract index A {\textstyle A} ; d {\textstyle d} is the exterior derivative d = d x μ ∂ μ {\textstyle d=dx^{\mu }\partial _{\mu }} . The structure function F A ( W ) {\textstyle F^{A}(W)} is assumed to be expandable in exterior product Taylor series as
F A ( W ) = ∑ q 1 + . . . + q n = q + 1 F B 1 . . . B n A W B 1 ∧ . . . ∧ W B n , {\displaystyle F^{A}(W)=\sum _{q_{1}+...+q_{n}=q+1}F_{B_{1}...B_{n}}^{A}W^{B_{1}}\wedge ...\wedge W^{B_{n}}\,,}
where W A {\textstyle W^{A}} has form degree q {\textstyle q} and the sum is over all forms whose form degrees add up to q + 1 {\textstyle q+1} . The simplest example of unfolded equations are the zero curvature equations d ω = 1 2 [ ω , ω ] {\textstyle d\omega ={\tfrac {1}{2}}[\omega ,\omega ]} for a one-form connection ω {\textstyle \omega } of any Lie algebra g {\textstyle {\mathfrak {g}}} . Here A {\textstyle A} runs over the base of the Lie algebra, and the structure function F A ( ω ) = f B C A