Abstract
\pdfglyphtounicode

New classes of bi-axially symmetric solutions

[+5pt]to four-dimensional Vasiliev higher spin gravity

[0pt] Per Sundell111 per.anders.sundell@gmail.com and Yihao Yin222 yinyihao@gmail.com

[0pt] Departamento de Ciencias Físicas, Universidad Andres Bello

Republica 220, Santiago de Chile

[0pt]

Abstract

We present new infinite-dimensional spaces of bi-axially symmetric asymptotically anti-de Sitter solutions to four-dimensional Vasiliev higher spin gravity, obtained by modifications of the Ansatz used in arXiv:1107.1217, which gave rise to a Type-D solution space. The current Ansatz is based on internal semigroup algebras (without identity) generated by exponentials formed out of the bi-axial symmetry generators. After having switched on the vacuum gauge function, the resulting generalized Weyl tensor is given by a sum of generalized Petrov type-D tensors that are Kerr-like or 2-brane-like in the asymptotic AdS region, and the twistor space connection is smooth in twistor space over finite regions of spacetime. We provide evidence for that the linearized twistor space connection can be brought to Vasiliev gauge.

1 Introduction

Vasiliev’s equations [Vasiliev:1990en] (for a recent review, see [Didenko:2014dwa]) provide a fully nonlinear description of higher spin gauge fields in four dimensions coupled to gravity and matter fields. The basic feature of Vasiliev’s theory is that the full field configurations are captured by master fields that live on an extension of spacetime by a noncommutative twistor space. The equations admit an exact solution given by the direct product of anti-de Sitter spacetime and an undeformed twistor space. In a specific gauge, certain linearized perturbations of the noncommutative twistor space structure give rise to Fronsdal fields. This suggests a holographic relationship to three-dimensional conformal field theories [WittenSeminar2001NovJHSchwarz60Birthday, Sezgin:2002rt, Klebanov:2002ja]; see also [Sundborg:1999ue, Leigh:2003gk, Koch:2010cy]. In [Giombi:2009wh, Giombi:2010vg] this relation was examined under the assumption that the Gubser-Klebanov-Polyakov-Witten (GKPW) prescription [Gubser:1998bc, Witten:1998qj] for on-shell computations of Witten diagrams can be applied to classical field configurations obtained from Vasiliev’s equations.

However, the Fronsdal fields embedded into Vasiliev’s master fields have non-local interactions [Sezgin:2002ru, Kristiansson:2003xx, Boulanger:2015ova] 111For a review, see [Bekaert:2010hw]. that belong to a functional class widely separated [Taronna:2016ats] from that of the quasi-local Fronsdal theory [Bekaert:2015tva], which is built by applying the canonical Noether approach to Fronsdal fields in anti-de Sitter spacetime. The GKPW prescription applies to the quasi-local theory by construction, as its action has self-adjoint kinetic terms, and the resulting holographic correlation functions indeed correspond to free three-dimensional conformal field theories.222The functional class encountered in the quasi-local Fronsdal theory in [Bekaert:2015tva] (within the AdS/CFT context) has not yet been identified completely; for a discussion, see [Taronna:2016ats], [Bekaert:2016ezc] and Section 7 of [Sleight:2016hyl]. At the cubic order, the separation between the functional classes of this theory and the Vasiliev theory has been spelled out in [Sleight:2016dba]. Recent work [Vasiliev:2016xui] shows that there exists an explicit field redefinition that maps Vasiliev’s theory to a quasi-local theory on-shell, obtained by carefully fine-tuning the perturbative expansion on the Vasiliev side, though it remains to be seen whether it coincides with that of [Bekaert:2015tva]. Moreover, as later shown in [Taronna:2016xrm] the required field redefinition is large, and hence it is unclear to what extent the method can be used to actually compute any holographic correlation functions. Thus, to our best understanding, the issue of whether holographic amplitudes can be extracted by applying the GKPW prescription to the Fronsdal fields embedded into Vasiliev’s master fields remains an open problem.

An alternative approach, pursued in [Boulanger:2015kfa], is to seek a weaker relation between the two theories, namely at the level of two distinct effective actions, derived in their own rights following different principles, and then evaluated subject to suitable dual boundary conditions. To this end, one starts from Hamilton’s principle applied to a covariant Hamiltonian action formulated using Weyl order on a noncommutative manifold whose boundary is given by the direct product of spacetime and twistor space [Boulanger:2011dd, Boulanger:2015kfa]; the Weyl order is required for the noncommutative version of the Stokes’ theorem to hold and for the imposition of boundary conditions. The resulting variational principle yields Vasiliev’s equations in Weyl order, that can be mapped back to Vasiliev’s normal order for special classes of initial data in twistor space following the perturbative scheme set up in [Iazeolla:2011cb, Iazeolla:ToAppear]. The resulting form of the higher spin amplitudes [Colombo:2010fu, Colombo:2012jx, Didenko:2012tv] is closely related to first-quantized topological open string amplitudes [Engquist:2005yt], but nonetheless reproduce exactly the same correlation functions as the Witten diagrams computed in the quasi-local theory. We would like to stress the fact that the Hamiltonian form of the action implies that the dependence of the classical Vasiliev master fields on classical sources are of a different type than for fields obeying equations of motion following from an action with self-adjoint kinetic terms. Indeed, instead of applying the GKPW prescription, the higher spin amplitudes are obtained from functionals given by topological boundary terms added to the Hamiltonian action [Sezgin:2011hq, Colombo:2012jx, Boulanger:2015kfa], whose on-shell values are given by higher spin invariants, as we shall comment on further below.

In this paper, we shall construct new perturbatively defined solution spaces to Vasiliev’s equations in Weyl order, by taking into account classes of functions that resemble closely those used in [Iazeolla:2011cb]. We shall then demonstrate explicitly that they can be mapped to Vasiliev’s normal order, at least at the linearized level, thus providing further evidence in favour of the covariant Hamiltonian approach outlined above.

To this end, we recall that at the linearized level, the fluctuations in the master fields that are asymptotic to anti-de Sitter spacetime form various representation spaces of the anti-de Sitter isometry algebra, including lowest-weight spaces as well as spaces associated to linearized solitons [Iazeolla:2008ix] and generalized Petrov type-D solutions [Didenko:2009td, Iazeolla:2008ix, Iazeolla:2011cb]. Nonlinear completions of various Type-D solution spaces were constructed in [Didenko:2009td, Iazeolla:2008ix, Iazeolla:2011cb]; for a review, see [Iazeolla:2012nf]. Of direct relevance for the work in this paper is the subspace that contains the the black-hole-like solutions,333This subspace is related to the massless spectrum by means of a -operation [Iazeolla:2011cb], reminiscent of a U-duality transformation [Bossard:2015foa]. including spherically symmetric solutions. In these solutions, each individual Fronsdal field has a point-like source at the origin, showing up as a divergence in its Weyl tensor. However, upon packing all curvatures into a master zero-form, one obtains the symbol of a quantum-mechanical operator that approaches a delta function distribution at the origin [Iazeolla:2011cb], which defines a smooth state as seen via classical observables given by zero-form charges [Sezgin:2005pv, Colombo:2012jx, Didenko:2012tv]. In this sense, the black-hole-like Type-D solutions to Vasiliev’s theory are source free at the origin.444It remains to be examined whether additional topological two-forms describing Dirac strings need to be activated in the dynamical two-form [Vasiliev:2015mka, Boulanger:2015kfa]. Furthermore, it is possible to dress these solutions with lowest-weight space modes [Iazeolla:ToAppear] at the fully nonlinear level; in doing so, the latter modes induce Type-D modes already at the second order of classical perturbation theory.555This phenomena resembles some of the scattering processes in U-duality covariant field theory [Bossard:2015foa].

Clearly, the full extent of the moduli space of the theory yet remains to be determined. In this paper, we shall present a new infinite-dimensional class of bi-axially symmetric exact solutions that are asymptotic to anti-de Sitter spacetime and singularity free at the level of zero-form charges. We shall furthermore propose a super-selection mechanism based on requiring that the solutions can be brought to Vasiliev gauge (where the asymptotic linearized fluctuations are in terms of Fronsdal fields).

Our construction method follows closely the one devised in [Iazeolla:2011cb] using gauge functions and separation of twistor space variables, which is in effect equivalent to starting from an Ansatz in Weyl order. The key difference is that we shall expand the master fields over a new set of elements in the associative fiber algebra, thus adding a branch to the existing moduli space. In a generic gauge, the expansion coefficients are functions on the base manifold. However, in the holomorphic gauge of [Iazeolla:2011cb] the Weyl zero-form is a constant while the twistor space one-form is given by a universal set of functions, related to Wigner’s deformed oscillators, originally derived within the context of three-dimensional matter coupled higher spin gravity [Prokushkin:1998bq]. The resulting solution space is then mapped to Vasiliev gauge in which the spacetime one-form consists of nonlinear Fronsdal tensors (after a suitable field redefinition in order to reinstate manifest Lorentz covariance). This map is achieved by means of two consecutive (large) gauge transformations: First, one uses a vacuum gauge function in SO(2,3)/SO(1,3).666Whether a more general vacuum gauge function can introduce additional classical moduli remains an open problem. Provided that the resulting twistor space connection is smooth at the origin of the base of the twistor space, Vasiliev gauge can be reached by means of a second perturbatively defined gauge transformation. As we shall see, the real-analyticity requirement constrains the initial data in the Weyl zero-form already at the linearized level.777An optional criterion is that the fiber algebra is a unitarizable representation of the higher spin algebra and hence the anti-de Sitter isometry algebra; we expect this property to arise at higher orders of classical perturbation theory by requiring positivity of a suitable free energy functional.

More specifically, the new sector of the fiber algebra is isomorphic to the group algebra where is generated by two elements in Sp given by exponentials of a pair of Cartan generators of sp. These correspond to linear symmetries of the two-dimensional harmonic oscillator, and generate the Killing symmetries of the solutions (including higher spin symmetries). As we shall see, the aforementioned super-selection rule amounts to restricting the master fields to a subalgebra of the group algebra not containing the unity.

The paper is organized as follows: In Section 2 we review parts of Vasiliev’s bosonic higher spin gravity model that we shall use in constructing and interpreting the exact solutions. Solution spaces based on (semi)group algebras are constructed in Section 3 using the aforementioned method; the singular nature of the contribution from the identity is pointed out in Section 3.4. In Section 4, we show that the Weyl tensor is given by a sum of Petrov type-D tensors that are Kerr-like or 2-brane-like in the asymptotic AdS region, and we compute higher spin curvature invariants. In Section 5, we show in special cases that the twistor space one-form is real-analytic in twistor space over finite regions of spacetime, and that its linearized part can be brought to Vasiliev gauge. We conclude in Section 6.

2 Bosonic Vasiliev model

In this section, we describe the non-minimal bosonic higher spin gravity model of Vasiliev type [Vasiliev:1990en],888For recent reformulations containing the original Vasiliev system as consistent truncations, see [Vasiliev:2015mka, Boulanger:2015kfa]. for which we shall present exact solutions in the next section. The model is characterized by the fact that it admits a linearization consisting of real Fronsdal fields in four-dimensional anti-de Sitter spacetime of spins with each spin occurring once; for further details, we refer to [Sezgin:2002ru, Iazeolla:2011cb] and the review [Didenko:2014dwa].

We first provide the formal definition in terms of master fields on the direct product of a commuting space and a noncommutative twistor space. We then spell out the component form of the equations, including their reformulation in terms of deformed oscillators. Finally, we remark on choices of bases for the internal algebra, and the Lorentz covariant weak field expansion scheme leading to Fronsdal fields, stressing the role of Vasiliev gauge and smoothness in twistor space.

2.1 Master field equations

Vasiliev’s original formulation of higher spin gravity is given in terms of two master fields and of degrees and , respectively, and two closed and twisted-central elements and of degree , all of which are elements of a differential graded associative algebra of forms on a non-commutative manifold , valued in an internal associative algebra . Letting denote the associative product of , which is assumed to be compatible with , the fully nonlinear master field equations read

 F+B⋆Φ⋆I−¯¯¯¯B⋆Φ⋆¯¯¯I = 0 , (2.1) DΦ = 0 , (2.2)

where

 F:=dA+A⋆A ,DΦ:=dΦ+A⋆Φ−Φ⋆π(A) , (2.3)

and denotes an automorphism of the differential graded associative algebra. The two-forms are characterised by the subsidiary constraints

 dI = 0 ,I⋆f = π(f)⋆I , (2.4)

for any , idem . Finally, the star functions

 B:=∞∑n=0bn(Φ⋆π(Φ))⋆n ,¯¯¯¯B:=∞∑n=0¯bn(Φ⋆π(Φ))⋆n , (2.5)

where . It follows that and hence is covariantly constant, viz.

 dB+A⋆B−B⋆A=0 , (2.6)

idem . As and are covariantly constant as well, it follows that the constraint on is compatible with its Bianchi identity. The integrability of the constraint on , on the other hand, requires to vanish, which is indeed a consequence of the constraint on . The resulting Cartan integrability, i.e. consistency with , holds for any dimension of and any star functions and , which are hence not fixed uniquely by the requirement of higher spin symmetry alone.

In the context of higher spin gravity, it is usually assumed that

 M=X4×Z4 , (2.7)

where is a four-dimensional real commuting manifold, with coordinates , and is a four-dimensional real non-commutative symplectic manifold, with canonical coordinates . The compatibility between the star product and the differential amounts to the Leibniz’ rule

 d(f⋆g)=df⋆g+(−1)deg(f)f⋆dg . (2.8)

The differential star product algebra is assumed to be trivial in strictly positive degrees, in the sense that are taken to be graded anti-commuting elements obeying

 dΞM⋆f=dΞM∧f ,f⋆dΞM=f∧dΞM , (2.9)

which are consistent with associativity. The algebra is also assumed to be equipped with an anti-linear anti-automorphism , for which we use the convention

 (f1⋆f2)†=(−1)deg(f1)deg(f2)f†2⋆f†1 ,(df)†=d(f†) . (2.10)

In case of the basic bosonic models, without internal Yang-Mills symmetries, the internal algebra consists of classes of functions on yet one more four-dimensional real non-commutative symplectic manifold, that we shall denote by , with canonical coordinates . We shall refer to as the full twistor space, and and , respectively, as the internal and external twistor spaces.999Taking the master fields to be smooth functions of yields an anti-de Sitter analog of the Penrose-Newman transformation; to our best understanding, the precise relation between and the original (commuting) twistor space of Penrose remains to be spelled out in detail. The Sp(4;) quartets are split into SL(2;) doublets, viz.101010The doublet indices are raised and lowered using and idem .

 Yα––=(yα,¯y˙α) ,Zα––=(zα,¯z˙α) , (2.11)

obeying

 ¯y˙α=(yα)† , ¯z˙α=−(zα)† , (2.12)

The automorphism and its hermitian conjugate are defined by

 π(xμ;yα,¯y˙α;zα,¯z˙α) = (xμ;−yα,¯y˙α;−zα,¯z˙α) , (2.13) ¯π(xμ;yα,¯y˙α;zα,¯z˙α) = (xμ;yα,−¯y˙α;zα,−¯z˙α) , (2.14)

and idem . Imposing

 Φ† = π(Φ) ,A† = −A ,I† = ¯¯¯I , (2.15)

and

 B†=¯¯¯¯B , (2.16)

that is, , and

 π¯π(Φ) = Φ ,π¯π(A) = A ,π¯π(I) = I ,π¯π(¯¯¯I) = ¯¯¯I (2.17)

yields a model with a perturbative expansion around four-dimensional anti-de Sitter spacetime in terms of Fronsdal fields of all integer spins.

The equations given so far provide a formal definition of the basic bosonic model.

2.2 Star product, twisted central element and traces

In what follows, we shall use Vasiliev’s original realization of the -product given by

 f1(y,¯y,z,¯z)⋆f2(y,¯y,z,¯z) =

We shall encounter -product compositions leading to Gaussian integrals involving indefinite bilinear forms. To define these we use the fact that the auxiliary integration is a formal representation of the original Moyal-like contraction formula, which means that the integration must be performed by means of analytical continuations of the eigenvalues of the bilinear forms.

Symbol calculus.

The star product rule implies that

 [f1(y,¯y),f2(z,¯z)]⋆=0 , (2.19)

that is, the variables and are mutually commuting. Moreover, from

 yα⋆yβ=yαyβ+iεαβ% , \ yα⋆zβ=yαzβ−iεαβ , \ zα⋆yβ=zαyβ+iεαβ , \ zα⋆zβ=zαzβ−iεαβ , (2.20)

it follows that

 a±α:=12(yα±zα) , (2.21)

obey

 [a−α,a+β]⋆=[a+α,a−β]⋆=iεαβ , [a+α,a+β]⋆=[a−α,a−β]⋆=0 . (2.22)

Letting and denote the Wigner maps that send a classical function to the operator with symbol in the Weyl and normal order, respectively, where an operator is said to be in normal order if all stand to the left of all . As a result, one has

 ONormal(f1(y,z)⋆f2(y,z))=ONormal(f1(y,z))ONormal(f2(y,z)) . (2.23)

One also has

 OWeyl(f(y))=ONormal(f(y)) ,OWeyl(f(z))=ONormal(f(z)) , (2.24)

resulting in that

 OWeyl(f1(y)⋆f2(y)) = OWeyl(f1(y))OWeyl(f2(y)) , (2.25) OWeyl(f1(z)⋆f2(z)) = OWeyl(f1(z))OWeyl(f2(z)) , (2.26)

and also

 ONormal(f1(y)⋆f2(z)) = OWeyl(f1(y)f2(z)) = OWeyl(f1(y))OWeyl(f2(z)) . (2.27)

Twisted central element.

The condition (2.4) can be solved by

 I=jz⋆κy ,jz=i4dzα∧dzβεαβκz , \qquadκy=2πδ2(y) , \qquadκz=2πδ2(z) , (2.28)

where is an inner Klein operator obeying

 κy⋆f(y)⋆κy=f(−y) , \qquadκy⋆κy = 1 , (2.29)

idem . Thus, one may write

 I=i4dzα∧dzβεαβκ ,κ:=κy⋆κz=exp(iyαzα) , (2.30)

where thus

 κ⋆f(y,z)=κf(z,y) ,f(y,z)⋆κ=κf(−z,−y) , (2.31)
 κ⋆f(y,z)⋆κ=π(f(y,z)) ,κ⋆κ=1 . (2.32)

By hermitian conjugation one obtains

 ¯¯¯I=−I†=i4d¯z˙α∧d¯z˙βε˙α˙β¯κ . (2.33)

The two-forms and can be extended to globally defined forms on a non-commutative space having the topology of a direct product of two complexified two-spheres [Iazeolla:2011cb, Boulanger:2015kfa], with nontrivial flux

 ∫Z4jz⋆¯j¯z=−14 . (2.34)

In this topology, it is furthermore assumed that belongs to a section that is bounded at infinity, while the twistor-space one-form is a connection whose curvature two-form falls off at infinity.

We note that the form of given in Eq. (2.30) is useful in deriving the perturbative expansion in terms of Fronsdal fields in Vasiliev gauge, while the factorized form in Eq. (2.28) is useful in finding exact solutions.

Trace operations.

The detailed form of the symbol of an operator depends on the basis with respect to which it is defined. Its trace, on the other hand, is basis independent, and in addition gauge invariant. The star product algebra admits two natural trace operations. The basic operation is given by the integral over phase space using the symplectic measure, viz.

 Trf:=∫Z4×Y4jy⋆¯j¯y⋆κy⋆¯κ¯y⋆f ,f∈Ω(Z4)⊗A , (2.35)

where is given by replacing by in defined in Eq. (2.28). An alternative trace operation, of relevance to higher spin gauge theory, can be defined if admits the decomposition

 A=⨁n,¯n=0,1An,¯n⋆(κy)n⋆(¯κ¯y)¯n , (2.36)

where consist of operators whose symbols in Weyl order are regular at the origin of . One may then define the trace operation

 Tr′f:=∫Y4jy⋆¯j¯y⋆f1,¯1=−14f1,¯1|y=0=¯y , (2.37)

using the decomposition (2.36), with the convention that

 κy⋆¯κ¯y⋆f=±f⇒Tr′f=∓18f|y=0=¯y . (2.38)

One may view as a regularized version of in the sense that if admits a decomposition of the form (2.36) then

 Trf = ∑n,¯n=0,1Trfn,¯n⋆(κy)n⋆(¯κ¯y)¯n (2.39) = Tr′f+Tr(f0,¯0+f1,¯0⋆κy+f0,¯1⋆¯κ¯y) , (2.40)

that is,

 Tr′f=Trf−Tr(f0,¯0+f1,¯0⋆κy+f0,¯1⋆¯κ¯y) . (2.41)

Indeed, in several applications it turns out that is ill-defined while is well-defined, as for example in the case that is a polynomial on .

2.3 Equations in components and deformed oscillators

We decompose the master one-form into locally defined components as follows:

 A=Uμdxμ+Vαdza+V˙αd¯z˙α , (2.42)

The reality condition (2.15) and the bosonic projection (2.17) imply

 U†μ = −Uμ ,V†α = ¯V˙α , (2.43) π¯π(Uμ) = Uμ ,π¯π(Vα) = −Vα . (2.44)

Decomposing master equations into components using inner derivatives , and , where idem , one has

 ∂[μUν]+U[μ⋆Uν] = 0 , (2.45) ∂μΦ+Uμ⋆Φ−Φ⋆π(Uμ) = 0 , (2.46)

the mixed components

 ∂μVα−∂αUμ+[Uμ,Vα]⋆=0 , \qquad∂μ¯V˙α−∂˙αUμ+[Uμ,¯V˙α]⋆=0 , (2.47)

which are related by hermitian conjugation, and

 ∂[αVβ]+V[α⋆Vβ]+i4εαβB⋆Φ⋆κ = 0 ,∂[˙α¯V˙β]+¯V[˙α⋆¯V˙β]+i4ε˙α˙β¯¯¯¯B⋆Φ⋆¯κ = 0 , (2.48) = 0 ,∂˙αΦ+¯V˙α⋆Φ−Φ⋆π(¯V˙α) = 0 , (2.49) ∂α¯V˙α−∂˙αVα+[Vα,¯V˙α]⋆ = 0 , (2.50)

where the two equations in Eq. (2.48) are related by hermitian conjugation idem Eq. (2.49).

The twistor space equations (2.48)–(2.50) can be rewritten by introducing Vasiliev’s deformed oscillators[Vasiliev:1990en]

 Sα=zα−2iVα , \ ¯S˙α=¯z˙α−2i¯V˙α , (2.51)

for which the reality condition and the bosonic projection take the form:

 (Sα)† = −¯S˙α , (2.52) π¯π(Sα) = −Sα . (2.53)

In terms of the new fields, the aforementioned equations read

 [Sα,Sβ]⋆ = −2i