1. Introduction
Higher spin (HS) theory in four dimensions, in its simplest form and when expanded about its (anti-)de Sitter vacuum solution, describes a self-interacting infinite tower of massless particles of spin
. The full field equations, proposed long ago by Vasiliev [
1,
2,
3] (for reviews, see [
4,
5]), are a set of Cartan integrable curvature constraints on master zero-, one- and two-forms living on an extension of spacetime by a non-commutative eight-dimensional twistor space. The latter is fibered over a four-dimensional base, coordinatized by a Grassmann-even
-spinor oscillator
, and the fiber is coordinatized by another oscillator
; the master fields are horizontal forms on the resulting twelve-dimensional total space, valued an infinite-dimensional associative algebra generated by
, that we shall denote by
, and subject to boundary conditions on the base manifold.
A key feature of Vasiliev’s equations is that they admit asymptotically (anti-)de Sitter solution spaces, obtained by taking the HS algebra
to be an extension of the Weyl algebra, with its Moyal star product, by involutory chiral delta functions [
6,
7], referred to as inner Klein operators, relying on a realization of the star product using auxiliary integration variables [
4]. Introducing a related class of forms in
Z-space, that facilitates a special vacuum two-form in twistor space, the resulting linearized master fields can be brought to a special gauge, referred to as the Vasiliev gauge, in which their symbols defined in a certain normal order are real analytic in twistor space, and the master zero- and one-forms admit Taylor expansions in
Y at
in terms of Fronsdal fields on the mass shell and subject to physical boundary conditions.
Although the Vasiliev equations take a compact and elegant form in the extended space, their analysis in spacetime proceeds in a weak field expansion which takes an increasingly complicated form beyond the leading order. Indeed, they have been determined so far only up to quadratic order. In performing the weak field expansion, a number of challenges emerge. Firstly, obtaining these equations requires boundary conditions in twistor space, referring to the topology of
Z space and the classes of functions making up
[
2,
8,
9]. The proper way of pinning down these aspects remains to be determined. Second, the cosmological constant,
, which is necessarily nonvanishing in Vasiliev’s theory (as the transvection operators of the isometry algebra are realized in
as bilinears in
Y), appears in the effective equations to its first power via critical mass terms, but also to arbitrary negative powers via non-local interactions [
4,
10]. Thus, letting
denote a generic Fronsdal field, it follows that
on-shell, and hence interactions with any number of derivatives are of equal relevance (at a fixed order in weak field amplitudes). This raises the question of just how badly nonlocal are the HS field equations, the attendant problem of divergences arising even at the level of the amplitudes [
11,
12], and what kind of field redefinitions are admissible. One guide available is the holographic construction of the bulk vertices [
13,
14,
15]. Clearly, it would be desirable to find the principles that govern the nonlocal interactions, based on the combined boundary conditions in twistor space as well as spacetime, such that an order by order construction of the bulk vertices can proceed from the analysis of Vasiliev equations. The simple and geometrical form of Vasiliev equations, in turn, may pave the way for the construction of an off-shell action that will facilitate the computation of the quantum effects.
In an alternative approach to the construction of HS equations in spacetime, it has been proposed to view Vasiliev’s equations as describing stationary points of a topological field theory with a path integral measure based on a Frobenius-Chern-Simons bulk action in nine dimensions augmented by topological boundary terms, which are permitted by the Batalin-Vilkovisky formalism, of which only the latter contribute to the on-shell action [
16,
17].
This approach combines the virtues of the on-shell approach to amplitudes for massless particles flat spacetime with those of having a background independent action, in the sense that the on-shell action is fixed essentially by gauge symmetries and given on closed form, which together with the background independence of Vasiliev’s equations provides a machinery for perturbative quantum computations around general backgrounds.
In this context, it is clearly desirable to explore in more detail how the choice of boundary conditions in the extended space influences the classical moduli space of Vasiliev’s equations, with the purpose of spelling out the resulting spaces, computing HS invariant functionals on-shell, and examining how the strongly coupled spacetime nonlocalities are converted into physical amplitudes using the aforementioned auxiliary integral representation of star products in twistor space.
The aim of this article is to review three methods that have been used to find exact solutions of the Vasiliev equations, and to describe two schemes for analyzing perturbations around them. In particular we will describe the gauge function method [
18,
19] for finding exact solutions and summarize the first such solution found in [
20], as well as its generalization to de Sitter spacetime studied in [
21] together with the solutions of for a chiral version of the theory with Kleinian
signature. As we shall see, this method uses the fact that the spacetime dependence of the master fields can be absorbed into gauge functions, upon which the problem of finding exact solutions is cast into a relatively manageable deformed oscillator problem in twistor space. The role of different ordering schemes for star product as well as gauge choices to fix local symmetries in twistor space will also be discussed.
Next, we will describe a refined gauge function method proposed in [
22], where the twistor space equations are solved by employing separation of twistor variables and holomorphicity in the
Z space in a Weyl ordering scheme and enlarging the Weyl algebra in the fiber
Y space by inner Kleinian operators. This approach provides exact solution spaces in a particular gauge, that we refer to as the holomorphic gauge, after which the spacetime dependence is introduced by means of a sequence of large gauge transformations, by first switching on a vacuum gauge function, taking the solutions to what we refer to as the
L-gauge, where the configurations must be real analytic in
Z space, which provides an admissibility condition on the initial data in holomorphic gauge. The solutions can then be mapped further to the Vasiliev gauge, where the linearized, or asymptotic, master fields, are real analytic in the full twistor space and obey a particular gauge condition in
Z space which ensures that they consist of decoupled Fronsdal fields in a canonical basis; the required gauge transformation, from the
L gauge to the Vasiliev gauge, can be constructed in a perturbation scheme, which has so far been implemented mainly at the leading order. We will describe a black hole-like solution in some detail and mention other known solutions obtained by this method so far, including new solutions with six Killing symmetries [
23].
We shall also outline a third method, in which the HS equations are directly tackled without employing gauge functions. In this method, solving the deformed oscillators in twistor space also employs the projector formalism, though the computation of the gauge potentials does not rely on the gauge function method. The black hole-like solution found in this way in [
24] will be summarized.
We shall also review two approaches to the perturbative treatment of Vasiliev’s equations. One of them, which we refer to as the normal ordered scheme, is based on a weak field expansion around (anti-)de Sitter spacetime [
3,
4,
25]. It entails nested parametric integrals, introduced via a homotopy contraction of the de Rham differential in
Z space used to solve the curvature constraints that have at least one form index in
Z space, followed by inserting the resulting perturbatively defined master fields into the remaining curvature constraints with all form indices in spacetime. In an alternative scheme, the equations are instead solved exactly in the aforementioned
L-gauge, and a perturbatively realized large HS gauge transformation is then performed to achieve interpretation in terms of Fronsdal fields in asymptotically (anti-)de Sitter spacetimes in Vasiliev gauge [
7]. The advantages of the latter approach in describing the fluctuations around more general HS backgrounds will be explained.
A word of caution is in order concerning the usage of ‘black hole’ terminology in describing certain types of exact solutions to HS equations. This terminology is, in fact, misleading in some respects since the notion of a line interval associated with a metric field is not HS invariant. Indeed, the apparent singular behaviour at the origin may in principle be a gauge artifact. This point is discussed in more detail in
Section 4.2. Moreover, given the nonlocal nature of the HS interactions, the formulation of causality, which is crucial in describing the horizon of a black hole, is a challenging problem without any proposal for a solution yet in sight; in fact, a more natural physical interpretation of the black hole-like solutions may turn out to be as smooth black-hole microstates [
7,
26]. Another aspect of the known black hole-like solution in HS theory is that they activate fields of all possible spins, and apparently it is not possible to switch of all spins except one even in the asymptotically AdS region.
So, what is meant by a black hole solution in HS theory? Firstly, the symmetry of the solution (which is part of an infinite dimensional extended symmetry forming a subgroup of the HS symmetry group) is in common with the symmetry group arising in the asymptotically AdS BH solution of ordinary AdS gravity. Second, the solution contains a spin-two Weyl tensor field which takes the standard Petrov type D form, with a singularity at the origin; more generally, the spin-s Weyl tensors are of a generalized Petrov type D form, given essentially by the s-fold direct products of a spin-one curvature of the Petrov type D form. The BH terminology is thus used in the context of HS theory with the understanding that it is meant to convey these properties, albeit they do not constitute a rigorous definition of a black hole in HS theory.
The use of HS invariants for exact solutions to capture their physical characteristics has been considered and in some cases they have been computed successfully. These particular invariants alone do not, however, furnish an answer to the question of whether it makes any sense to think about event horizons in HS theory at all, and if so, how to define them; in fact, their existence rather supports the aforementioned microstate proposal, wherein the HS invariants can be interpreted as extensive charges defining HS ensembles.
Motivated by the quest for giving a physical interpretation of the exact solution in the context of underlying HS symmetries, a geometrical approach to HS equations was proposed in [
27]. We shall summarize this proposal in which the HS geometry is based on an identification of an infinite dimensional structure group in a fibre bundle setting, and the related soldering phenomenon that leads to a HS covariant definition of classes of (non-unique) generalized vielbeins and related metrics, and as such an infinite dimensional generalization of
geometry. In doing so, we will improve the formulation of [
27] by dispensing with the embedding of the relevant infinite dimensional coset space into a larger one that involves the extended HS algebra that includes the twistor space oscillators.
Finally, we are not aware of any exact solutions of HS theories in dimensions
[
28,
29], while in
, assuming that the scalar field is coupled to HS fields, we can refer to [
8,
30,
31] for the known solutions. Purely topological HS theory, which has no dynamical degrees of freedom, and which allows a more rigorous definition of black holes, is known to admit many exact solutions whose description goes beyond the scope of this review.
5. Direct Method and the Didenko-Vasiliev Solution
While, as we have seen so far, the gauge function method is in general of great help in constructing exact solutions, it is sometimes possible to attack the equations directly, by virtue of some other simplifying Ansatz or gauge condition. We shall generically refer here to any method which does not rely on the use of gauge function as direct method. One such solution has been found so far in this way by Didenko and Vasiliev [
24], which has nonvanishing HS fields, and which contains the Schwarzschild black hole solution in the spin 2 sector.
Indeed, motivated by the phenomenon that solutions of Einstein equations that can be put in Kerr-Schild form solve the linearized as well as the nonlinear form of the equations, the Authors of [
24] thought of an Ansatz that would generalize some distinctive features of black hole solutions in gravity. First, it is based on an AdS timelike Killing vector, in the sense that the Weyl tensor will be a function of some element
as (4.13). More precisely, if
satisfies the Killing vector equation, a proper ansatz for a solution of the linearized twisted adjoint equation will be given by
. Second, they chose
in such a way that the Ansatz linearize the Vasiliev equations. For the latter purpose it is important that the function
is a projector—in fact, a Fock-space vacuum projector that coincides with (4.12). Such choice in particular reduces Equations (2.31) and (2.32) to two copies of the 3D (anti)holomorphic deformed oscillator problem that arises in Prokushkin-Vasiliev HS theory in 3D [
8] in terms of the oscillators in (4.21)
9.
The ansatz [
24]
where
M is a constant and
are functions to be determined, indeed reduces the Equations (2.31) and (2.32) to two deformed oscillator problems in terms of the latter functions. A specific gauge choice on the
, while bringing about a further breaking of the manifest Lorentz covariance, effectively linearizes their equations, which are then solved by means of the standard perturbative methods of
Section 6. A further ansatz [
24]
where
g is another function to be determined is then employed to deal with the remaining equations that involve
W, namely (2.28), (2.29) and (2.30). The resulting exact solution is given by [
24]
where
Note also that
does not satisfy the Vasiliev gauge
, and that
W above has not been redefined as in (2.22). Nonetheless, it has been noted in [
24] that a HS transformation of the form
with
and arbitrary
maps
W to
given by
whose spin 2 component gives the frame field and the associated metric
where
. This is the metric of a black hole of mass
M in
in Kerr-Schild form. The terminology of black hole in HS context requires caution as discussed in the introduction. In addition to the
, the solution summarized above has been shown to also have
of the
supersymmetric HS symmetry of the model, and their infinite dimensional extension thereof [
24].
The solution (5.3) differs from (4.17)–(4.19) both in the form of the internal connection
and in that of the gauge field generating functions. As for the internal connection, the difference can be ascribed to the two choices employed in [
6] and [
24] for solving the deformed oscillator problem (referred to as “symmetric” and “most asymmetric”, respectively, in [
6,
22]). One can show that the resulting internal connections can be connected via a gauge transformation (see [
6]), although the small or large nature of this transformation is yet to be investigated. The comparison of
in (4.19) and
W in (5.3) is technically more complicated, as the two differ also by the shift of the Lorentz connection (2.22), and it will be postponed to a future work, but we note that having the same
B identical in both solutions strongly suggests that the physical gauge fields should be equivalent, and in particular equivalent to (5.5).
It is worth mentioning that even by working without the gauge function method, with a specific choice of gauge the Didenko-Vasiliev solution can be simplified in such a way that the
W connection is reduced to the vacuum one
. This simplification was studied in [
45], along with the embedding of the solution in the
and
supersymmetric extensions of the bosonic Vasiliev equations.
6. Perturbative Expansion of Vasiliev Equations
In this Section, we shall summarize the standard perturbative expansion of the Vasiliev equations, benefiting from [
4,
6,
12,
25,
27,
46] (for more recent treatments, see [
47,
48]).
In the normal order, defined by the star product formula (2.4), the inner Klein operators become real analytic in
Y and
Z space, viz.,
Assuming that the full field configurations are real-analytic on
for generic points in
, one may thus impose initial conditions
where the
x-dependence is understood. In order to compute the
Z-dependence of the fields one may choose
, which is a trivial flat connection, to vanish, and a homotopy contractor for the de Rham differential on
, which entails imposing a gauge condition on
. One may then solve the constraints on
,
and
on
in a perturbative expansion in
C. This procedure gets increasingly complex with increasing order in the expansion, which schematically can be written as
where
,
and
are
n-linear functionals in
C, and
is a linear functional in
. These quantities, which are constructed using the homotopy contractor on
, depend on
Z, and are real-analytic in
provided that
C and
are real analytic in
Y-space and all star products arising along the perturbative expansion are well-defined. As for the remaining equations, that is, that
and
, it follows from the Bianchi identities that they are perturbatively equivalent to
and
, which form a perturbatively defined Cartan integrable system on
for
C and
.
To Lorentz covariantize, one imposes
with
from (2.22), that is
where
w does not contain any component field proportional to
and
in view of (2.23). Upon substituting the above relation into
, it follows from the manifest Lorentz covariance that the dependence of
and
on the Lorentz connection arises only via the Lorentz covariant derivative ∇ and the Riemann two-form
. Thus, the resulting equations in spacetime take the form [
6]
where
Alternatively, in order to stress the perturbation expansion around the maximally symmetric background, including the spin-two fluctuations, it is more convenient to work in terms of the original one-form
. The perturbative expansion up to 3rd order in Weyl curvatures reads
where
are functionals that can be determined from (6.8) and (6.9). One next assumes that the homotopy contraction in
Z-space is performed such that
which we refer to as the Vasiliev gauge, and expands
where
is the maximally symmetric background;
a and
are linearized fields;
and
are
nth order fluctuations. The resulting linearized field equations on
X-space provide an unfolded description of a dynamical scalar field
and a tower of spin-
s Fronsdal fields
where we use the convention that repeated indices are symmetrized. Computing the functional
, the linearized unfolded system is given by [
4]
where
The oscillator expansion of (6.18) furnishes the definition of the spin Weyl tensors, and gives the field equations for spin-s fields which remarkably do not contain higher than second order derivatives, and indeed they are the well known Fronsdal equations for massless spin s-fields in . As for (6.19), its oscillator expansion gives the AdS massless scalar field equation in unfolded form.
Perturbative expansion around
at second order is rather complicated but still manageable. Schematically the equations take the form
Even though these terms have been known in the form of parametric integrals for some time, their detailed structure and consequences for the three point functions were considered much later in [
11,
49], where the field equation for the scalar field was examined, and used for computing the three-point amplitude for spins
. If
, only the first source term in (6.22b) contributes and gives a finite result, in agreement with the boundary CFT prediction. However, if
, only the second source term in (6.22b) contributes and gives a divergent result [
11,
49]. This divergence was confirmed later in [
12,
50] where the divergence in the three point amplitude for spins
, resulting from the last term in (6.22a). Soon after, it was shown that a suitable redefinition of the master zero-form cures this problem [
48,
51], as has been also confirmed with the computation of relevant three-point amplitudes [
52,
53]. A similar redefinition in the one-form sector has also been determined so that the divergence problem arising in the last term in the first equation above has also been removed [
54].
In determining the higher order terms in the perturbative expansions of Vasiliev equations, it remains to be established in general what field redefinitions are allowed in choosing the appropriate basis for the description of the physical fields. In wrestling with this problem, the remarkable simplicity of the holographic duals of this highly nonlinear and seemingly very complicated interactions may provide a handle by means of their holographic reconstruction. Such reconstruction has been achieved for the three and certain four-point interactions [
13,
14,
15]. Putting aside the analysis and interpretation of the nonlocalities [
14,
55,
56], which are present, and nonetheless in accordance with holography by construction, the issue of how to extract helpful hints from them with regard to the nature of the allowed field redefinitions in perturbative analysis of Vasiliev equation remains to be seen.
Of course, ultimately it would be desirable to have a direct formulation of the principles that govern the nonlocal interactions, based on the combined boundary conditions in twistor space as well as spacetime, as we shall comment further below.
7. A Proposal for an Alternative Perturbation Scheme
In what follows, we shall show that for physically relevant initial data given by particle and black hole-like states, the solutions obtained using the factorization method can be mapped to the Vasiliev gauge used in the normal ordered perturbation scheme at the linearized level. Whether the Vasiliev gauge is compatible with an asymptotic description in terms of Fronsdal fields to all orders in perturbation theory, or if it has to be modified, possibly together with a redefinition of the zero-form initial data, is an open problem. In finding the proper boundary conditions in both spacetime and twistor space it may turn out to be necessary to require finiteness of a set of classical observables involving integration over these spaces.
One can define formally the aforementioned map to all orders of classical perturbation theory by applying a gauge function
to the holomorphic gauge solution space, where
is a field dependent gauge function to be fixed as to impose the Vasiliev gauge in normal ordering. To begin with, let us consider fluctuations around
for which
consists of particle states, focusing, for concreteness, to the case of scalar particle states worked out in [
7]. Upon switching on the gauge function
L, the field
develops non-analyticities in the form of poles in
Y-space in the particle sector [
7]. Applying
H removes these poles and expresses the results in terms of Fronsdal fields (at the same time restoring the manifest Lorentz covariance, as we shall see), at least in the leading order. This can be see as follows. We want to obtain
such that
The leading order gauge transformation reads
Contracting by
and using the fact that by definition
, one finds
In particular, activating only the scalar ground state (and its negative-energy counterpart) via the parameters
in the projector ansatz for
B (see (4.25)), this was computed in [
7] with the result
where we have set
, and
The matrix
can be found in Section 5.2.1 of [
7].
This result for is regular in but has a pole in .
We observe that is now real-analytic everywhere on .
Furthermore, it was shown in [
7] that the above expressions for
lead to the relation
in agreement with the result obtained in the standard perturbative analysis of Vasiliev equations at leading order. The emergence of
in (7.7) shows that manifest Lorentz covariance is restored, in comparison with the expression for
. The prefactor in (7.7) is a consequence of the fact that we are considering the lowest mode alone in the solution for Fronsdal equation the scalar field, as opposed to summing all the full set of modes.
Expressing the exact solutions obtained in holomorphic gauge in terms of Fronsdal fields amounts to setting up an alternative perturbation scheme in which one constructs the higher orders of the gauge function
subject to
dual boundary conditions, that is, to conditions restricting the both the twistor-space dependence of the master fields and their spacetime asymptotic behaviour. Indeed, one requires that after having switched on
H, the master fields have symbols in normal order that are real-analytic at
, and symbols in Weyl order that belong to a (associative) star-product subalgebra with well-defined classical observables defined by traces over twistor space and integrations over cycles in spacetime. The proposal is that this problem admits a non-trivial solutions, and that it fixes
up to residual small HS gauge transformations and the initial data for the master zero-form
B to all orders in classical perturbation theory. It would be interesting to see whether these type of field redefinitions are related to those recently proposed by Vasiliev in order to obtain a quasi-local perturbation theory in terms of Fronsdal fields [
51,
54]. An important related issue is whether the gauge function
G is large in the sense that it affects the values of the HS zero-form charges, which are special types of classical observables given by traces over twistor space defining zero-forms in spacetime that are de Rham closed [
20,
27,
57].
A nontrivial test of the factorization approach is first to show that the solution is finite after performing the higher order
gauge transformations, and second, to show that the resulting
n-point correlators are in agreement with the result expected from holography. The corrections beyond the leading order remain to be determined, while the computation of correlators has been performed in which the second order solution in standard perturbative scheme has been used. It has been shown that a naive computation of
in the standard perturbative scheme leads to divergences [
11,
12], and later it was shown that these divergences can be removed by a suitable redefinition of
[
48,
51]. Whether there exists a principle based on any notion of quasi-locality in spacetime that governs the nature of such redefinitions to all orders in perturbation is not known, to our best knowledge.
An advantage of the factorization method is that here we start from a full solution to Vasiliev’s theory, defined as a classical field theory on the product of spacetime and twistor space (not referring a priori to the conventional perturbative approach). This provides a convenient framework for the description of the solutions with particles fluctuating around nontrivial backgrounds. The key principle here is that linear superposition principle holds for the zero-form initial data
. For example, if we want to describe the solution to Vasiliev equations for particles propagating around BH solution of
Section 4, one simply takes
. The exact solution for the combined system is obtained in this way, but in small fluctuations of a particle propagating in a fixed and exact black hole background, one may treat the parameter
and
as small and large, respectively. A very interesting open problem is thus to combine this scheme with the aforementioned proposal for dual boundary conditions in order to work out new types of generating functions for HS amplitudes.