#### Partially massless higher-spin theory

Received: December
massless higher-spin theory
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c The Authors. 0
0 31 Caroline St. N, Waterloo , Ontario N2L 2Y5 , Canada
1 CERCA, Department of Physics, Case Western Reserve University
2 Perimeter Institute for Theoretical Physics
We study a generalization of the D-dimensional Vasiliev theory to include a tower of partially massless elds. This theory is obtained by replacing the usual higher-spin algebra of Killing tensors on (A)dS with a generalization that includes \third-order" Killing tensors. Gauging this algebra with the Vasiliev formalism leads to a fully non-linear theory which is expected to be UV complete, includes gravity, and can live on dS as well as AdS. The linearized spectrum includes three massive particles and an in nite tower of partially massless particles, in addition to the usual spectrum of particles present in the Vasiliev theory, in agreement with predictions from a putative dual CFT with the same symmetry algebra. We compute the masses of the particles which are not xed by the massless or partially massless gauge symmetry, nding precise agreement with the CFT predictions. This involves computing several dozen of the lowest-lying terms in the expansion of the trilinear form of the enlarged higher-spin algebra. We also discuss nuances in the theory that occur in speci c dimensions; in particular, the theory dramatically truncates in bulk dimensions D = 3; 5 and has non-diagonalizable mixings which occur in D = 4; 7.
Higher Spin Gravity; Higher Spin Symmetry; AdS-CFT Correspondence
1 Introduction
Review of partially massless elds
Free massive elds
Free partially massless elds
The hs2 algebra
Generalities about the hs2 algebra
Oscillators and star products
Coadjoint orbits
Computation of multi-linear forms
Expectation for the spectrum
Equations of motion
Patterns of unfolding
Truncation to the minimal theory
A partially massless higher-spin theory
Mass computations
Nuances in the spectrum
Reduction to D dimensions
Extracting AdS equations of motion
The scalar masses
The vector mass
The tensor mass
Wavefunctions in the nite theories
Conclusions and future directions
A Bilinear and trilinear forms in hs2
In this paper, we explore an explicit description of a partially massless (PM) higher-spin
(HS) theory, discussed previously in [1{5]. This is a fully interacting theory which can live
on either anti-de Sitter (AdS) or de Sitter (dS), and is expected to be a UV complete and
predictive quantum theory which includes gravity. Like the original Vasiliev theory1 [6{9]
(see [8, 10{15] for reviews), it contains an in nite tower of massless elds of all spins, but in
addition it contains a second in nite tower of particles, all but three of which are partially
massless, carrying degrees of freedom intermediate between those of massless and massive
particles. This tower may be thought of as a partially Higgsed version of the tower in the
The theory on AdS is expected to be the holographic dual to the singlet sector of
the bosonic U(N )
2 free conformal eld theory (CFT) studied in [16] (see also [17{21]),
and on dS is expected to be dual to the Grassmann counterpart CFT, just as the original
Vasiliev theory is expected to be dual to an ordinary free scalar [22{24]. We de ne the
bulk theory as the Vasiliev-type gauging of the CFT's underlying global symmetry algebra,
which we refer to here as hs2. It is a part of a family of theories based on the
theory which contain k towers of partially massless states. We study this theory for several
reasons: in our universe, we've con rmed the existence of seemingly fundamental particles
with spins 0, 12 and 1, and we have good reason to believe that gravity is described by a
particle with spin 2. It is an interesting eld-theoretic question to ask, even in principle,
what spins we are allowed to have in our universe. Famous arguments, such as those
reviewed in [25, 26], would naively seem to indicate that we should not expect particles
with spin greater than 2 to be relevant to an understanding of our universe, but these
nogo theorems are evaded by speci c counterexamples in the form of theories such as string
theory and the Vasiliev theory, both of which contain higher-spin states and are thought to
be complete. Of particular interest is the question of whether partially massless elds fall
into the allowed class. Partially massless elds are of interest due to a possible connection
between partially massless spin-2
eld and cosmology (see e.g., [27] and the review [28]),
which has led to many studies of the properties of the linear theory and possible nonlinear
extensions [21, 27, 29{47]. No examples (other than non-unitary conformal gravity [48{
50]) of UV-complete theories in four dimensions containing an interacting partially massless
eld and a nite number of other elds are known, and so it has remained an open question
whether these particles could even exist. The theory we describe in this paper contains an
in nite tower of partially massless higher-spin particles. Thus, the mere existence of this
theory promotes further studies into partially massless gravity.
Although the past twenty years have seen great progress in our understanding of
quantum gravity in spaces with negative cosmological constant, a grasp of the nature of quantum
gravity in spaces with a positive cosmological constant such as our own remains elusive.
There have been proposals inspired by AdS/CFT for a dS/CFT correspondence, which
would relate quantum gravity on de Sitter to conformal theories at at least one of the past
and future boundaries [51{56]. It was argued in [24] that the future boundary correlators
1Throughout this work, we refer only to the bosonic CP-even Vasiliev theory.
of the non-minimal and minimal Vasiliev higher-spin theories on dS should match the
correlators of the singlet sector of free \U( N )" or Sp(N ) Grassmann scalar eld theories,
respectively. However, a lack of other examples has been an obstacle preventing us from
answering deep questions we would like to understand in dS/CFT, such as how details of
unitarity of the dS theory emerge from the CFT. To that end, it seems a very exciting
prospect to develop new, sensible theories on dS as well as their CFT duals to learn more
about a putative correspondence.
Another interesting puzzle in the same vein is what the connection between the Vasiliev
theory and string theory is. It is well-known that the leading Regge trajectory of string
theory develops an enlarged symmetry algebra in the tensionless limit, (see, e.g. [57]), generally
becoming a higher-spin theory. In particular, the tensionless limit of the superstring on AdS
and the Vasiliev theory appear to be connected, and supersymmetrizing both [58] appears to
the question of how to include in the Vasiliev theory the additional massive states which
are present in the string spectrum is still a challenge. From the point of view of the Vasiliev
theory, there are drastically too few degrees of freedom to describe string theory in full;
string theory contains an in nite set of Vasiliev-like towers of ever increasing masses, and
one would require an in nite number of copies of the elds in the Vasiliev theory in order to
construct a fully Higgsed string spectrum. Without the aid of the hs algebra underlying the
Vasiliev construction, it is not clear how to proceed and add massive states to the Vasiliev
theory to make it more closely resemble that of string theory. The theory we describe here
contains partially massless states, which represent a sort of \middle ground" in the process
of turning a theory with only massless degrees of freedom into one which contains massive
(or partially massless) degrees of freedom as well by adding various Stuckelberg elds.
It is natural to suspect that there should be a smooth Higgsing process by which an
in nite set of massless Vasiliev towers eat each other and become the massive spectrum
of string theory [59{61]. On AdS, there seems to be no obstruction to this, but on dS
the situation is di erent. As we review in section 2, there is a unitarity bound m2
4) for a mass m, spin s particle in D dimensional dS space. Below this
would necessarily be doomed to pass through this non-unitary region before becoming
fully massive. The PM
elds, however, are exceptions to this unitarity bound. They form a
discrete set of points below this bound where extra gauge symmetries come in to render the
non-unitary parts of the elds unphysical (just as massless high-spin particles are unitary on
dS despite lying below the unitarity bound). Thus, one might suspect a discrete Higgs-like
mechanism by which the massless theory steps up along the partially massless points on the
way to full massiveness. These intermediate theories should be Vasiliev-like theories with
towers of partially massless modes (however, the theory we consider here continues to have
a massless tower and we do not know any example of PM theory with no massless elds).
The partially massless higher-spin theory we describe in this paper is constructed in a
similar fashion to the Vasiliev theory. It is constructed at the level of classical equations of
motion, although just as in the case of the Vasiliev theory, we believe the dual CFT de nes
the theory quantum-mechanically and in a UV-complete fashion. There's no universally
agreed-upon action for this theory or for the original Vasiliev theory (see [62{69] for e orts
in this direction), but this is believed to be a technical issue rather than a fundamental
issue, and an action is expected to exist. The theory can be de ned on both AdS and dS,
and is essentially nonlocal on the scale of the curvature radius L, though it has a local
expansion in which derivatives are suppressed by the scale L. Nevertheless, this theory
admits a weakly-coupled description and so can be studied perturbatively in AdSD 3; in
particular it can be linearized, which we do in this paper.
Our primary technical tool and handle on the theory is its symmetry algebra. The
original Vasiliev theory in AdSD 4 is the gauge theory of the so-called hs algebra, an in
nitedimensional extension of the di eomorphism algebra which gauges all Killing tensors as well
as Killing vectors on AdS. This algebra is equivalent to the global symmetry algebra of free
scalar eld theory in one fewer dimension, which consists of all conformal Killing tensors
as well as conformal Killing vectors. The algebra we employ in this paper is the symmetry
algebra of the
2 free eld theory, which includes all of the generators of the hs algebra,
and in addition \higher-order Killing tensors", studied in [70]. The representations and the
bilinear form of this algebra were studied by Joung and Mkrtchyan [5], and we make use of
many of their results.2 The structure of this algebra is very rigid, and its gauging completely
xes the structure of the corresponding theory on AdS, giving rise to the PM HS theory.
One crucial distinction between this PM HS theory and the original Vasiliev theory is
that the PM theory on AdS is non-unitary/ghostly. This follows from the non-unitarity of
the dual CFT, as well as the fact that the PM
elds themselves are individually non-unitary
on AdS. Nevertheless, despite being nonunitary, our CFT is completely free, so there cannot
be any issue of instability usually associated with nonunitary/ghostly theories. We may
compute its correlators with no issues, seemingly de ning an interacting nonunitary theory.
The bulk theory should somehow not be unstable, since it is dual to a free theory. Thus we
believe that this theory exists in AdS and is stable despite its nonunitarity, and we believe
that the in nite-dimensional underlying gauge algebra hs2 is so constraining as to prevent
any sort of instability from arising, though we will not attempt here to study interactions
in detail in this theory, deferring such questions instead to future work.
We might suspect that the PM theory on dS is nonunitary as well, but without a
Lagrangian description of the theory, and without the clearcut link between boundary and
bulk unitarity enjoyed by AdS/CFT, we do not have a clear-cut answer as to whether the
PM theory is unitary on dS. The individual particles, including the PM particles, are all
unitary on de Sitter, but unitarity could sill be spoiled if there are relative minus signs
between kinetic term of di erent particles, and without a Lagrangian we cannot directly
check whether this is the case.
2 CFT, we demonstrated in [16] that certain dimensions were special; in
nite CFTs. Furthermore, in d = 3; 6
there was module mixing that took place in the CFT. We will see that this manifests as
2They referred to this algebra as p2; however as this algebra arises from a 2 dual CFT, we refer to this
algebra in this paper simply as the hs2 algebra.
module structures mimic each other comes as no surprise, but does o er evidence that the
PM HS theory is truly the AdS dual of the
2 CFT. Furthermore, the details of the duality
in these cases are new, and are not speci c to the Vasiliev formalism; this constitutes new
evidence that the AdS/CFT duality continues to hold at the non-unitary level.
One interesting and powerful check of the duality between the Vasiliev theory and
free eld theory was the one-loop matching of the partition functions of the boundary
and bulk theories [71, 72]. It has been argued that unitary higher-spin theories where
the symmetry is preserved as we approach the boundary should have quantized inverse
coupling constant [73]. Therefore, when computing the one-loop correction to the inverse
duality and GN1 = N
1 for the minimal/O(N ) duality.
Newton's constant in the Vasiliev theory, one was forced to obtain an integer multiple of
the dual theory's a-type conformal anomaly (even d) or sphere free energy F (odd d), which
was precisely what happened. Despite the fact that the
2 CFT is non-unitary, its N is
nevertheless quantized, and so we continue to expect that the one-loop correction to the
inverse Newton's constant is consistent with its quantization. In the companion paper [74]
we do this computation in several dimensions and
nd a positive result (see also [75]);
we obtain integer multiples of the a-type conformal anomaly or sphere free energy F of a
single real conformally coupled
2 scalar in one dimension fewer. In particular, we obtain
The outline of this paper is as follows: we begin by introducing and reviewing the
properties of partially massless higher-spin free particles in AdS and dS in section 2. We
then turn to reviewing properties and the relevant representation of the algebra hs2 in
section 3, as it is so central to all of the discussions in the paper, and discuss how to compute
trilinear forms in the algebra, which are necessary for later calculations. We gauge this
algebra in section 4, linearize the theory, and discuss how the linearized master elds break
up into unfolding elds for the physical particles. In section 5, we compute the masses of the
four particles whose masses are not xed by gauge invariance. We discuss which boundary
conditions are necessary on the various
elds so as to reproduce CFT expectations. In
agreement with expectations from the dual CFT. Finally, in section 7, we discuss various
future directions for research, as well as implications for dS/CFT. We discuss the one-loop
renormalization of the inverse Newton's constant in the companion paper [74].
We use the mostly plus metric signature, and the curvature conventions
+ S ). The
)T indicates that the enclosed indices are to be symmetrized and made
completely traceless. Throughout this work, we unfortunately must reference three di erent
spacetime dimensions; the dimension of the dual CFT is denoted d, the dimension of the
bulk (A)dS is denoted D, and the dimension of the ambient or embedding space in which
Embedding space coordinates are indexed by A; B; C; : : :, and moved with the at ambient
metric AB. (A)dS spacetime coordinates are indexed by ; ; ; : : :, and moved with the
(A)dS metric g . (A)dS tangent space indices are indexed by a; b; c; : : :, and moved with
the tangent space at metric ab. The boundary CFT indices are i; j; k; : : :, and are moved
with the at boundary metric ij . The background (A)dS space has a vielbein e^ a which
relates AdS spacetime and AdS tangent space indices. L refers to the AdS length scale,
and H refers to Hubble in dS (see section 2).
All Young tableaux are in the manifestly antisymmetric convention, and on tensors we
use commas to delineate the anti-symmetric groups of indices corresponding to columns
(except on the metrics ab, g , AB, ij .). We use the shorthand [r1; r2; : : :] to denote
a Young tableau with r1 boxes in the
rst row, r2 boxes in the second row, etc. All of
the Young tableaux we work with will also be completely traceless, so we do not indicate
tracelessness explicitly. The projector onto a tableau with row lengths r1; r2;
P[r1;r2; ] where the indices to be projected should be clear from the context. The action of
the projector is to rst symmetrize indices in each row, and then anti-symmetrize indices
projector does not include the subtraction of traces. Introductions to Young tableaux can
be found in section 4 of [77] or the book [78].
Review of partially massless
We begin by reviewing some properties of partially massless higher-spin elds in AdS or
dS [79{89], and how they behave as we take them to the boundary, i.e. the properties of the
dual CFT operators. Partially massless elds are elds with more degrees of freedom, and
correspondingly less gauge symmetry, than a massless eld, but fewer degrees of freedom,
and correspondingly more gauge symmetry, than a fully massive eld. For a given spin, the
amount of gauge symmetry
xes the mass on both AdS and dS. Partially massless elds
are necessarily below the unitarity bound in AdS, but are unitary in dS.
s-index eld
1 eld on D dimensional (A)dS with mass m is described by a symmetric
1::: s which satis es the equations of motion
+ H2 (6 + D(s
i.e. it is transverse, traceless, and satis es a Klein-Gordon equation.
curved space Laplacian.
Here H is the (A)dS curvature scale, i.e. H2 > 0 for dS, in which case H is the Hubble
usual AdS radius). The scalar curvature R and cosmological constant
1=L2 with L the
are related to the
Hubble constant as
R = D(D
1::: s = 0;
2::: s = 0;
3::: s = 0;
In the AdS case, H2 =
L12 , the high spin elds are dual to symmetric tensor
\singletrace" primaries Oi1:::is . For generic m, these satisfy no particular conservation conditions.
Their scaling dimensions are given in terms of the mass by
Here d = D
1 is the dimension of the dual CFT. The positive root corresponds to the
\ordinary quantization" of AdS/CFT, and the negative root corresponds to the \alternate
quantization" of [90].
The unitarity bound [91] for symmetric traceless tensor operators is
s = 0;
s = 0;
For scalars, s = 0, we have
and the unitarity bound is
Solving for m gives
m2L2 =
m2L2 = (
s > 1, only the ordinary quantization is compatible with unitarity. However, in the
(nonunitary) partially massless theory, we will see that we do indeed need to use the alternate
quantization for certain particles with s
s = 0 ;
1, we have m2
0 in the bulk, and there is no analog of the
BreitenlohnerFreedman bound [92, 93]3 allowing for slightly tachyonic but stable scalars. For s
as soon as the mass is negative, we generically expect instabilities owing to the theory
becoming ghostly/non-unitary.
bound below which the particle is generically non-unitary is the Higuchi bound [81, 94, 95],
Below this bound, the kinetic term for one of the Stuckelberg elds is generically of the
wrong sign, indicating that some of the propagating degrees of freedom are ghostly.
However, at special values of the mass between zero and the Higuchi bound, the particle develops
a gauge symmetry which eliminates the ghostly degrees of freedom, and the eld is unitary
at these special points. These points are the partially massless elds, and we turn to them
3The Breitenlohner-Freedman bound for scalars is m2
Partially massless elds occur at the special mass values
ms2;t = H2 (s
1) (D + s + t
4) ; t = 0; 1; 2; : : : ; s
Here, t is called the depth of partial masslessness. At these mass values, the system of
equations (2.3) becomes invariant under a gauge symmetry,
1::: s = r( t+1::: s 1::: t) + : : :
and so t counts the number of indices on the gauge parameter
1::: t . Here : : : stands for
lower-derivative terms proportional to H2. On shell, the gauge parameter is transverse and
traceless and satis es a Klein-Gordon equation
1::: t + : : : = 0;
2::: t = 0;
3::: t = 0:
The : : : terms in (2.13) and (2.12) as well as the mass values (2.11) are completely xed
by demanding invariance of the on-shell equations of motion (2.3) under the on-shell gauge
transformation (2.12).
Just as massive and massless elds carry irreducible representations of the dS group,
partially massless elds also carry irreducible representations, albeit ones which have no
at space counterpart. A generic massive
eld has, in the massless limit, the degrees
of freedom of massless
elds of spin s; s
1; : : : ; 0 (usually called, with some abuse of
terminology, helicity components). The gauge symmetry of a PM
eld removes some of
the lower helicity components; a depth t PM
eld has helicity components
1; : : : ; t + 1:
The highest depth is t = s
containing only helicity components s. We see that on AdS, all but the highest depth PM
elds have negative masses, and are non-unitary. On dS, the masses are positive, and the
This saturates the unitarity/Higuchi bound on dS. Fields with masses below this bound are
ghostly and therefore non-unitary, unless they are at one of the higher depth PM points.
As an illustration of this structure, see gure 1, which shows the Higuchi bound on dS4 as
well as the rst few partially massless particles' masses and spins.
elds are dual to multiply -conserved symmetric tensor single-trace primaries
Oi1:::is , i.e. they satisfy a conservation condition involving multiple derivatives [96],
i1:::is = 0:
For the massless case, t = s
1, this is the usual single-derivative conservation law. More
introduced in [16]), i.e. the number of derivatives you need to dot into the operator to kill it.
All 2s+1 DoF
ticles' masses and spins. Above the Higuchi bound, a massive particle is unitary. Below the bound,
generically the kinetic term for at least one of the helicity components is of the wrong sign, indicating
that some degrees of freedom are ghostly and the particle is non-unitary. However, at the speci c
partially massless points (represented by the location of the numbers in the
gure), the particle
develops a gauge symmetry which eliminates the ghostly degrees of freedom, making the particle
unitary. This comes at the expense of reducing the number of propagating degrees of freedom; how
many degrees of freedom propagate is represented by the number at each partially massless location.
On AdS, the mass-scaling dimension relation (2.5) (with the positive root) tells us that
the dimension of these partially conserved currents should be4
s;t = d + t
1 = d + s
The second equality shows that these operators violate the CFT unitarity bound (2.6)
1, which saturates it.
4As a check, one can see that the general form for the two-point correlation functions,
hOi1:::is (x)Oj1:::js (0)i
We now discuss the symmetry algebra, hs2, which we will ultimately gauge in order to
obtain a partially massless higher spin theory. In the linearized partially massless
higherspin theory, there will be two \master"
elds (a gauge eld and a eld strength), and a
\master" gauge parameter, which are valued in the hs2 algebra. The Vasiliev equations
themselves are also valued in the algebra.
There is a multi-linear form which is de ned on the algebra. This will be used to
extract component equations from the general Vasiliev equations, which in turn will allow
us to calculate the masses of the four particles in the linearized PM HS theory without any
gauge symmetry. Our ultimate goal will be to compute the multilinear forms we will need
to compute the masses.
The reader who is interested purely in the physics of the theory may familiarize
themselves with the generators of the hs2 algebra in subsection 3.1, and then move on to
section 4, skipping the intermediate details of the computation. The content of this
section is mostly a review of, or slight extensions of, previous work [5, 9, 11, 97]. Our main
contribution is the explicit calculation of several of the lowest-lying terms in the expansion
of the trilinear form of this algebra, which are given in appendix A.
First we describe the construction of the algebra abstractly, without reference to any
particular realization. Then, we introduce oscillators with a natural star product which
form a realization of the algebra which is useful for computations. Finally, we implement
the technology of coadjoint orbits which can be used as a bookkeeping device for the
di erent tensor structures which emerge and greatly simpli es calculations.
Generalities about the hs2 algebra
The hs2 algebra is realized as the algebra of global symmetries of a conformal eld theory [1{
5, 16, 70], the
2 CFT described by the action
2 CFT contains as its underlying linearly realized5 symmetry algebra precisely the
algebra hs2. The spectrum of operators and conserved currents form a representation of
We rst discuss this algebra abstractly. hs2 can be abstractly de ned as a quotient of
space Lorentz algebra so(D), by a particular ideal. The abstract generators TAB of so(D)
transform in the
adjoint representation of the so(D) algebra,
5These are not to be confused with the non-linearly realized higher shift symmetries of [98, 99], which
are also present.
6This construction is independent of the signature.
The commutation relations for so(D) are
[TAB; TCD] =
AC TBD + BC TAD
BDTAC + ADTBC ;
where AB is the so(D) invariant metric tensor.
The universal enveloping algebra, and then hs2, will be described as successive
quotients of the algebra of all formal products of the T 's. First, we consider the tensor product
algebra formed from the T 's. We can label the elements of the tensor product algebra by
the irrep under so(D), which we display as a tableau, as well as by the number of powers of
T they came from, which we indicate using a subscript n on the tableau, and which we'll
refer to as the \level". For example, we may decompose the product of two T 's as
TABTCD =
the commutator.
In the top line of (3.4) are terms which are symmetric in the interchange of the two
T s, whereas the bottom line contains terms which are antisymmetric in the interchange
of the two T s. To pass to the UEA, we use the commutation relations (3.3) to eliminate
all anti-symmetric parts in terms of parts with a lower number of T 's, leaving only the
symmetric parts in the top line (see the Poincare-Birkho -Witt theorem).
To pass to the hs2 algebra, we quotient by a further ideal. The generators
6)(D + 2) generate an ideal of the UEA (here
4 comes in at
level 4, e.g. from the tensor product
2). We quotient the UEA to the hs2 algebra
6)(D + 2), and
Those generators which remain in the resulting quotient de ne the generators of the
hs2 algebra, and consist of the representations:
The rst line are generators which are in the same representations as the generators of the
massless hs algebra, which we will call T(r) at level r, whereas the second line are generators
new to hs2, which we will call T~(r) at level r. The old generators T(r) correspond to Killing
tensors of AdS and conformal Killing tensors of the CFT, whereas the new generators T~(r)
correspond to so-called order three Killing tensors in AdS and order three conformal Killing
tensors in the CFT, as reviewed in [5, 16], and in the appendix. They are associated with
multiply conserved currents in the CFT, and are gauged by partially massless elds in AdS.
It is noteworthy that hs2 contains hs as a sub-vector space. However, as the values of the
Casimirs do not match, it is not, strictly speaking, a subalgebra.7
7We thank Evgeny Skvortsov for discussions of this point.
In the process of taking the quotient, all of the Casimirs are xed to speci c values:
C2 = TABT AB =
C4 = T ABT BC T CDT DA
A =
~A1B1;:::Ar 2Br 2;Ar 1;Ar T~A1B1;:::;Ar 2Br 2;Ar 1;Ar
All of the generators in the hs2 algebra are traceless two-row Young tableaux, which
we generically call [r; s] with s
r. These generators can be written as elements of the
UEA in the appropriate representations
where P[r;s] is the normalized projector onto the [r; s] tableau (this de nition
normalization of the generators).
These generators carry indices in the fully traceless
tableau of shape [r; s]. We use the anti-symmetric convention, which means that they are
anti-symmetric in any A; B pair, vanishes if we try to anti-symmetrize any A; B pair with
2, which we referred to as T~r.
A general algebra element is a linear combination of the above generators,
coe cient tensors A~(r) have the symmetry of a traceless [r; r
where the coe cient tensors A(r) have the symmetry of a traceless [r; r] tableau and the
The product on the hs2 algebra is the product in the UEA mod the ideal, and we
denote it by ?. It takes the schematic form
The product is bilinear and associative but not commutative. The commutator of the star
product, for any two algebra elements A and B, is
and it gives the hs2 algebra the structure of a Lie algebra which is isomorphic to the Lie
algebra of linearly realized global symmetries of the
There is a natural trace on the algebra which projects onto a singlet, de ned simply as
and a multi-linear form can be de ned using this trace as
Tr(A) = A(0);
M(A; B; : : :)
Tr(A ? B ? : : :):
Note that the bilinear form is diagonal in the degree r, because the product of a rank r
be mixing between algebra elements with the same degree but corresponding to di erent
Young diagrams, which we will have to worry about later.
Oscillators and star products
Although in principle the previous subsection contains all of the ingredients necessary
to de ne the hs2 algebra, it is incredibly cumbersome to use those de nitions directly to
compute anything in the algebra. In this section we review an oscillator construction of the
algebra, as introduced in [5, 9, 11]. The oscillator construction comes with its own natural
star product, which is very convenient for computations, and ultimately reproduces the
results of the computations in the ideal described in the previous section. One reason for the
simpli cation is the introduction of a \quasiprojector" which greatly assists with the step of
modding out by the ideal, and makes it possible to compute the bilinear and trilinear forms
of the algebra to a high enough order to extract what we need from the Vasiliev equations.
We introduce bosonic variables Y A, called oscillator variables, which carry an sl(2)
= +;
in addition to an so(D) index A. (For us, this sl(2) is a completely
auxiliary structure useful for de ning the representation and we do not think of it as being
physical or related to any spacetime.) At the end of the day, all physical quantities will be
singlets under this sl(2). The invariant tensor for sl(2) is
which is anti-symmetric,
Suppose we have two arbitrary polynomials in the Y A variables, F (Y ) and G(Y ).
We may de ne an oscillator star product, , between them. (Note that the oscillator
star product is a priori di erent from the hs2 product ? which we de ned in the previous
subsection; we will discuss how to relate the two further below. We will refer to both as \the
star product" in this paper, leaving the distinction clear from context.) The (oscillator)
star product between them is de ned to be
G = F exp 4
1 AB
is bi-linear and associative. Our goal is to understand how we can use this
easy-to-evaluate product
to evaluate the desired product ?.
8This sl(2) is the Howe dual algebra to the so(D), see e.g. the review [11].
With the star product we de ne the star commutator
[F; G] = F
The star products and commutators among the basic variables are
Y B = Y AY B
In addition, there is an integral version of this same star product [9, 100]:
1 Z
G =
d2DSd2DT F (Y + S)G(Y + T )e2 AB
It should be noted that there are consequently two products available to the Y ; an
ordinary product and a star product. The Y 's commute as ordinary products, despite not
commuting as star products. When we write polynomials in Y , we mean that they are
polynomials in the ordinary product sense.
We de ne antisymmetric so(D) and symmetric sl(2) generators as
T(AYB) = Y AY B
= Y AY B AB:
We may use the above star product to evaluate the star commutators of (3.18), and these
reproduce the commutation relations of decoupled so(D) and sl(2) algebras,
= 0 :
[k ; k ] =
To each element of the algebra A, we may associate a polynomial A(Y ) in the Y 's by
replacing the generators with a product of Y 's
T A1B1;:::;ArBr ! T(AY1)B1 T(AY2)B2 : : : T(AYr)Br ;
T~A1B1;:::;Ar 2Br 2;Ar 1;Ar ! T(AY1)B1 T(AY2)B2 : : : T(AYr)Br Br 1Br :
We would like to be able to use the
product on A(Y ) in place of the ? product on A,
but there is an obstruction in that, in general, we still have nontrivial Casimir elements in
the polynomial A(Y ), which must be xed to particular numbers. We may force all of the
Casimir-type elements to be set to the values required by the hs2 algebra by introducing a
quasiprojector,9
hs2 , which will be useful for setting the Casimirs to their proper values,
and extracting from a general polynomial F an element of hs2 when working within a trace:
9This is referred to as a quasiprojector rather than a projector because the explicit form does not satisfy
hs2 ; rather, its square doesn't converge [100]. This is not a problem at the level of working to
any xed order in the algebra, as we do.
formally obtained by simply setting Y ! 0. Therefore
Tr(Fhs2 ) =
Y =0
Once we have the quasiprojector, we can compute multi-linear forms using the
M(A; B; : : :)
Tr(A ? B ? : : :) =
Y =0
ideals, including replacing the Casimir T A(YB)
T(AYB) with the appropriate number,
We now need to know what
hs2 is. It should implement the modding out by the
Y =0
6)(D + 2) ;
and likewise with all higher powers.
A useful form for the quasi-projector was found in [5],
hs2 = N
with N a normalization factor,
N =
Coadjoint orbits
In order to conveniently deal with the tensor structures which emerge, it is useful to
introduce, following [5], the technology of coadjoint orbits. The coadjoint orbit method allows
us to replace the coe cient tensors A(Ar1)B1;:::;ArBr , A~(Ar1)B1;:::Ar 2Br 2;Ar 1;Ar of a general
algebra element (3.8) with products of a single antisymmetric tensor AAB, called a coadjoint
orbit, which we write in a script font,
A(Ar1)B1;:::;ArBr ! AA1B1 AA2B2 : : : AArBr
! AA1B1 : : : AAr 1Br 1 AArBr Br 1Br :
These coadjoint orbits will serve as placeholders or bookkeeping devices. Expressions for
our multi-linear forms will be written in terms of products of matrix traces of products of
these coadjoint orbits for various hs2 valued elds. These are in one-to-one correspondence
with the di erent tensor structures or ways of contracting the indices. Once we have
obtained the multi-linear form with the coadjoint orbits, we may reconstruct the tensor
structure in question by passing back to spacetime elds.
The coadjoint orbits AAB satisfy what we will call here the coadjoint orbit conditions:
A BA DA F A H
G) BD F H = 0;
[AB C]D = 0:
These two together serve to enforce that products of r copies of AAB in (3.29) have the
symmetry properties of either a trace-ful [r; r] or trace-less [r; r
2] tableau. (We often
view A as a matrix in what follows, and use h: : :i to denote a matrix trace.) To see this,
with a second coadjoint orbit B to show
ABA =
Note that this identity also implies that A
3 = 0. Therefore, if we consider the quantity
AA1B1 : : : AArBr , then it is in the [r; r] representation, but it is not traceless (which is why
the trace has to be explicitly subtracted in (3.29)), and taking a single trace of, say, any two
B indices puts the resulting tensor in the [r; r
2] representation, which is automatically
In the computations we will do, we will have several di erent elds present in each
multi-linear form, so we'll introduce several di erent, independent coadjoint orbits, one for
eld, each satisfying their own coadjoint orbit conditions (and each with the script
version of the letter associated to the particular eld).
As mentioned, we must subtract the single traces manually from the [r; r] elds. There
are no traces to subtract at level 0 or 1 in the algebra; we must rst subtract traces at level
2, and (as we will see) we'll need trace-free replacements up to level 4. The explicit form of
the traces can be worked out by adding all possible trace terms with arbitrary coe cients,
and demanding that the resulting tensor is in the [r; r] representation and is totally traceless
A(A31)B1;A2B2;A3B3 ! AA1B1 AA2B2 AA3B3 +
A(A41)B1;A2B2;A3B3;A4B4 ! AA1B1 AA2B2 AA3B3 AA4B4
There are no such subtleties with the [r; r
2] tensors, which are already traceless given
the coadjoint orbit conditions, so we may simply replace
A~(A31)B1;A2;A3 ! AA1B1 AA2C AC
A~(A41)B1;A2B2;A3;A4 ! AA1B1 AA2B2 AA3C AC
With these replacements we may pass to coadjoint orbits, perform our computations of
the multilinear form, and then pass back by the inverse operation:
AA1B1 AA2B2 AA3B3 ! A(A31)B1;A2B2;A3B3
In practice we will not need the initial replacement (3.34). We will instead compute the
multilinear form of particular elements in the algebra directly in terms of the coadjoint
orbits, and then reconstruct the elds with the inverse operation (3.35).
Computation of multi-linear forms
Now we move onto the computation of the multilinear form. As stated above, it is
convenient to use the quasiprojector (3.27) for computations. The strategy for evaluating the
nth multi-linear form
M(W1; : : : ; Wn) is detailed at length in [5], which we review here
for completeness' sake. For each of the n algebra elements in the argument of multi-linear
Gaussian eY+ Wi Y
for each i. Finally, we may evaluate the trace by using the integral
version of the star product to star together the n Gaussians as well as the Gaussian from
the alternate quasiprojector (3.27). In total, we have:
M(W1; : : : ; Wn)
Y =0
After this has been evaluated, it may be series-expanded in each W to extract the relevant
part of the multilinear form, which can be written in terms of products of traces of products
Now we describe the process of evaluating the multilinear form to obtain a series
expansion for the answer in the desired form, products of traces. First, we need to star
in the Gaussian form of the quasiprojector. Then we evaluate the star products with the
integral version of the star product. This returns a determinant to be evaluated on the
in powers of W and carry out the resulting x-integrals term-by-term. In all, we have:
M =
= N
Y =0
Y =0
Qjn=1 (2
x) D2 6 2F1 3; 1; 32 ; 1 1x
px) Qjn=1 2 Wj
From here it is conceptually straightforward but computationally quite intensive to
Taylor expand the log, perform the trace, series expand the exponential, then nally expand
piece of information we need are the values of the integrals. (Although a p
the 1 1::: , all the while exploiting the coadjoint conditions satis ed by W. The only other
x appears in
the determinant in (3.37), only integer powers of x come out at the end of the day.) We
are then able to do the integrals over x nding
coe cients.
xm =
( 2) m(2m + 1)!!(D + 4m)
We may collect the forms by trace structures and powers of each W and read o the
In section 5, we will see that we need only the bilinear form, which we'll call B, and
trilinear form, which we'll call T , for the mass computations we're interested in doing,
T (W1; W2; W3)
Y =0
Y =0
Suppose that W1;2;3 are hs2
elds valued in only particular levels of the algebra; call the
levels n1, n2, and n3. We denote the corresponding bilinear and trilinear forms B(n1;n2) and
T(n1;n2;n3), respectively. We have computed the bilinear form up to fth order in both W1
and W2, as well as the trilinear form up to fth order in W1, rst order in W2, and fth order
in W3 which include all the cases we will need to compute the linearized mass spectrum of
the hs2 theory. The results are rather lengthy, and so we list them in appendix A.
A partially massless higher-spin theory
Just as the original Vasiliev theory can be thought of, in a sense, as a Yang-Mills-like gauge
theory with gauge algebra hs on AdS, so too can the partially massless Vasiliev theory be
thought of as a Yang-Mills-like gauge theory based on the hs2 algebra on AdS. We now turn
to providing a description of the degrees of freedom of the partially massless higher-spin
theory and the way in which they are embedded into hs2 valued elds.
The full non-linear theory can be constructed using the generalized formalism of [4], and
should also be reconstructible from the dual CFT (3.1), along the lines of e.g. [69, 101, 102].
We are interested here in studying the linear theory and subtleties of the spectrum, and
matching to the dual CFT. Rather than linearize the full theory, it will be easier for us to
directly construct the linear theory from the hs2 algebra.
AdS/CFT tells us that the spectrum of physical elds in the bulk should match the
spectrum of single trace primary operators in the CFT. The spectrum of single trace primaries
for the U(N )
2 CFT has been worked out in [2, 16]. There is a tower of conserved higher
= d + s
2. These should correspond to massless bulk elds with full massless gauge symmetry. On top of this, there is a tower of \triply-conserved" currents with spins s = 3; 4; 5; : : : and dimensions = d + s 4.
These should correspond to t = s
3 partially massless bulk elds. In addition, there are
= d
2, an s = 1 of dimension
= d
3, and an s = 2 of dimension
= d
In the case of the theory on AdS, there is a straightforward map between unitarity of
the boundary CFT and unitarity of the bulk theory. In particular, the sign of the kinetic
term of a eld in the bulk theory is the same as the sign of the coe cient of the two point
function of the eld's dual operator. We may therefore deduce the signs of the kinetic terms
of the elds from the calculations of the two-point functions in [16], and we can see precisely
elds are non-unitary due to a wrong sign kinetic term (in addition to the already
non-unitary nature of the PM
elds). Unfortunately, as there's no universally agreed-upon
action for the Vasiliev theory, we cannot directly check the signs of the bulk kinetic terms to
verify this correspondence. Note that only the relative sign between elds is relevant, as the
overall sign can be changed by multiplying the entire bulk action (and CFT action) by
For the theory on dS, however, there is not a straightforward connection between
unitarity in the bulk and unitarity of the boundary CFT. We know that the partially
elds are themselves unitary on dS, but because we lack an action or a clean
link to boundary unitarity, we cannot say whether the relative kinetic signs between the
elds and other elds of the theory on dS is positive, and thus we cannot make any
de nitive claim about unitarity of the bulk dS theory.
In tables 1 through 6, we display the expected spectrum on AdSD derived from the
U(N ) version of the CFT dual, for all dimensions D
3 (the masses of the de Sitter version
of the theory may be obtained by simply replacing L2 !
H12 ). In the lower dimensions,
priori whether the AdS theory is the dual of the \ nite" or \log" CFTs discussed in [16],
however, we will see below that indeed the PM HS theories, as based on the hs2 algebra,
are the duals of the nite theories and not the log theories. Furthermore, as we'll see, the
number of modes expected from a full propagating degree of freedom. This corresponds to
the fact that the primaries in the nite CFT have a
nite number of descendants.
As we will discuss below, there is a consistent truncation of this theory where we keep
only the even-spin particles, just as in the Vasiliev theory, and this is the dual of the O(N )
CFT. The resulting spectra may be read o from the tables below by simply dropping all
odd-spin particles.
and that the massive eld strength is related to the massive vector by the usual eld
strength relation
(D + 1)(D + 2)L
The nal equation then becomes a eld equation for the massive vector. Using the de
nitions (5.55), plugging (5.57), (5.56) into (5.51), we nd
which is the eld equation for a massive vector eld with mass
c~(2) = 0;
m2 =
matching the expected result (5.2).
will come back to in sections 6.2, 6.1 respectively.
The tensor mass
There are three even symmetric traceless tensor elds in C: c(a2;b)
; c~(a4;b). One combination
of these will be the massive graviton, the other two combinations will be second level
elds for the two scalars. To identify the combination corresponding to the
massive graviton, we look at the three even vector equations of the form d (
) =
= 0 ;
= 0 ;
2D(D + 1)(D + 2)
2D2(D + 1) (D
1)(D+3) 1
16D2(D+1)(D+2)(D+4)
1)(D + 2)(D + 5)
2)(D+1)(D+3)(D+4) r c~(4;) = 0 :
vector elds c(1) and c~(a3), so there is one combination for which the vector elds do not
a
) =
2)(D + 1)(D + 2)
1)(D + 1) (D + 2)L c~(3); + r[ c~(2];)
1)2(D + 5)
2)(D + 1)2(D + 4)2
= 0 ;
= 0 ; (5.66)
1)(D + 2)(D + 5)L c~(3); + r[ c~(4];)
= 0 :
appear and we obtain a constraint equation on a combination of the tensor elds. This
combination is the massive graviton, which we call h ; ,
r h ; = 0;
D(D + 1)(D + 4)
Equation (5.63) is the proper transversality constraint (2.2) for a massive spin-2.
There is one mixed symmetry
eld which is even, c~(a3b);c, so this should be the rst
eld for the massive tensor. The equations from which we should solve for
this eld are the even mixed symmetry symmetric traceless tensor equations of the form
= 0 :
There is one linear combination of these three equations for which the combination h ;
appears under the derivatives. That combination reads
c~(3); = r[ h ];
dimension in which the CFT predicted that mixings would occur in the tensor, so we now
assume D 6= 4 and return to the case D = 4 later.
equation of the form d
The tensor equation of motion comes from the single even symmetric traceless tensor
D(D + 1)2(D + 2)
D(D + 1)(D
The combination of tensor elds that appears here algebraically is precisely the
combination (5.64) which is the massive graviton, and so (5.69) becomes
tion (5.63), we nd a wave equation for h ; ,
Solving (5.68) for c~(3); and plugging into (5.70), and using the transversality
equa
= 0:
h ; = 0 :
m2 =
Comparing with (2.1), this is the equation of motion for a massive spin 2 on AdS with
precisely matching the expected value (5.3).
Finally, we can tell which combinations of the even tensor elds c(a2;b)
spond to the unfolding elds of the scalars by looking at the two even symmetric traceless
tensor equations of the form d ( ) =
1)(D + 1)(D + 2)2
= 0 ;
= 0 :
Taking linear combinations and using the equations (5.41), (5.42) for the rst level auxiliary
elds, we arrive at
2D(D + 1)(D3)+(D2)+L25) c(2;)
5)(D
1)(D + 1)L2 c~(2;) = r
(D + 1)(D + 2)(D
(D + 2)(D + 4)L2 c~(4;) = r
The linear combinations on the left hand side are the second level unfolding elds for the
we will come back to in sections 6.2, 6.4, 6.1 respectively.
In [16], we saw that peculiarities occur in the spectrum of the dual
2 CFT in speci c low
the propagator to be analytic, resulting in a theory with a
nite number of single trace
primaries. There were also the \log CFTs", but as we'll now show, the AdS theories are the
arise, which we will see imitated by the AdS theory. In this section, we demonstrate the
AdS duals of these nuances in D = 3; 4; 5; 7.
In d = 4, the spectrum of the
2 CFT completely degenerates when we choose the basic
have any dynamical elds, with the exception of a single scalar which should allow for only
a single mode.
We can see that this indeed happens in AdS5: all of the equations we have derived,
with the exception of (5.39), come with a prefactor D
5 and hence degenerate in D = 5.
The origin of this is the truncation of the algebra, as observed in11 [5]. This truncation
may be seen directly by looking at the trilinear form given in the appendix, and noting
that every term in it is proportional to D
6. The only equation which is non-vanishing in the entire theory is (5.39), which leaves only
r c(0) = 0:
Therefore c(0) is a eld that allows only one mode, a constant, consistent with our
expectation. It is the only eld in the AdS5 theory; every massless, partially massless, and
other massive states do not even have equations of motion. This is consistent with the
statement that there are no non-trivial bulk dynamics other than a constant solution for
c0, dual to a constant two-point function hj0(0)(x)j0(0)(0)i at the boundary.
In d = 2, the spectrum of the
2 CFT degenerates when we choose the propagator to
be analytic, leaving two scalar primary states of dimensions 0; 2 and a spin one primary
state with dimension
1. These states all have a nite number of descendants, thus we expect the bulk theory to have two scalar elds (one of which allows only a constant mode, as above) and a massive vector eld, all allowing only a nite number of modes.
11Note that the resulting
nite dimensional algebras underlying the nite theories discussed here are
different from the
nite dimensional algebras discussed in [110, 111], which occur in the massless hs algebra in
certain dimensions where parametrized families of algebras are possible and certain values of the parameters
give nite truncations.
mass eigenstates using (5.47), the equations become
Eliminating c(1) between (6.2) and (6.3), we nd that c0(0) satis es,
3L c(1) = 0 ;
L c(1) = 0 ;
L c~(2) = 0 :
r c0(0) = 0:
m2 =
c~0(2) = 0;
c(1) =
L2 c~(2) = 0;
m2 =
This is the same equation as (6.1), so we identify c0(0) as the bulk dual of the
= 0 scalar.
Using (6.2) or (6.3) to solve for c(1), plugging into (6.4), and using (6.5), we nd a
Klein-Gordon equation for c~0(2),
allowing us to identify c~0(2) as dual to the
2 scalar operator. Looking at the ordinary
Klein-Gordon equation (6.6), it's not apparent that the eld is in a nite-dimensional
representation. In order to see that it indeed is, as expected from the CFT dual, we must attempt
to quantize this particle and nd the allowed modes. We will do this below in section 6.5.
us that c(1) is the eld strength of c~(2),
Plugging this into (5.51), we nd
which is the eld equation for a massive vector eld with mass
1 vector operator. This will also live in a nite-dimensional representation,
though we will not explicitly construct it here.
Tensors. In
the exception
of (5.61), (5.66), (5.73), which become, respectively,
3L c(1) = 0 ;
r c~(2];) = 0 ;
2L c~(2;) = 0 :
c~(2;) =
All the elds except c~(2;) decouple. c~(2;) is determined algebraically in terms of c(1) by (6.13),
which is given in terms of the scalar c~(2) by (6.4), giving
Thus c~(2;) is the second level unfolding eld for c~(2). Plugging (6.14) into (6.12), we nd
that (6.12) is identically satis ed, and plugging (6.14) into (6.11), we
nd that (6.11)
reduces to the gradient of the c~(2) equation of motion (6.6). Thus, as expected from the
into one extended module, due to the presence of a particular state,
primary and descendant. Consequently, the other linear combination of operators at that
scaling dimension and spin, ~j0(1), was forced into being an operator which was neither a
primary nor a descendant, but was nevertheless in the extended Verma module of j0(0).
We illustrated this module in
gure 4 of [16]. We would like to see how the dual of this
phenomenon arises in the partially massless higher spin theory.
We saw the rst sign of this in the transformation (5.47) to the mass eigenstates, whose
equations of motion (5.45), we have the un-demixable equations
ically normalize the kinetic terms)
L =
The kinetic terms in (6.16) are required to have the opposite relative sign; it is not
possible to write an action with correct sign kinetic terms that reproduces the equations (6.15).
The internal eld space is thus Lorentzian, and the transformations which preserve the
kinetic structure are boosts in
eld space. We can attempt to do a such a boost in
space to diagonalize the mass terms in (6.16), but it cannot be done because the required
precisely because the mass terms in the 7d AdS action are non-diagonalizable, and we are forced
to include Witten diagrams such as the one shown here.
boost would be in nite. The mass term in inherently mixed and cannot be unmixed. Note
that this phenomenon cannot happen in the case of normal kinetic terms, where any mass
terms can always be diagonalized with a Euclidean rotation in eld space.
This is a eld theoretic realization of the spin-0/spin-0 \extended module" uncovered
in [16] for the
there are mass mixing terms means that when we construct Witten diagrams to evaluate
boundary correlators of this theory, we will have diagrams of the form shown in
where we have bulk mixing of the c(0), c~(2) degrees of freedom through non-diagonalizable
mass insertions.
Although we have not yet done so, it would be very interesting to attempt to
quan
The other nuance discussed in [16] for the
2 CFT concerned the d = 3 mixing of the
j(0) and j2(0) modules. Thus in AdS4 we expect there to be irreducible mixing between the
0
genceless eld h
will no longer have an auxiliary
eld and consequently will not satisfy
does not by itself carry the graviton degrees of freedom, and we will have to use a di erent
strategy to identify the graviton.
We start by using (5.73) and (5.74) to eliminate two of the tensor elds in terms of
the third tensor and the two scalars c(0), c~(2). We will choose to eliminate c~(2;) and c~(4;) ,
c~(2;) =
c~(4;) =
where we have replaced the vector elds c(1) and c~(3) with their values in terms of the
scalars from (5.41) and (5.42),
c(1) =
Lr c(0); c~(3) =
terms of rst derivatives of c(2;) ,
Having eliminated c~(2;) and c~(4;) , our tensor degrees of freedom must now be carried by c(2;) .
We now look at the unfolding equations (5.65), (5.66), (5.67), and we nd that once
we eliminate c~(2;) and c~(2;) using (6.17), all three equations (5.65), (5.66), (5.67) reduce to
the same equation, which is independent of the scalars and allows us to solve for ~c(3); in
by the scalars
and rede ning the tensor eld as
)T c~(2) = 0 :
The divergence of (6.22) vanishes upon use of (6.21) and (6.20), and so provides no new
We have now collected all the independent equations of motion, which are the
tensor equation (6.22), the divergence equation (6.21), and the scalar equations of
motion (6.20). We can see that one of the scalars decouples by changing to the mass eigenstate
c~ ; =
Looking now at the divergence equations (5.60), (5.61), (5.62), we nd that upon use
of (6.18), as well as use of the scalar equations of motion (5.45),
c(0) =
c~(2) =
they all reduce to the single equation
r c(2;) =
which xes the divergence of c(2;) in terms of the scalars.
nally to the equation of motion (5.69), using (6.19), and using (6.21) to
eliminate the divergences of c(2;) , we nd an equation of motion for c(2;) which is sourced
c(0) = c0(0)
4 c~0(2); c~(2) = c~0(2) + 16c0(0) = 0
c(2;) = c0(2;)
the equations (6.22), (6.22) and the scalar equations of motion (6.20) become
c0(0) = 0;
)T c~0(2) = 0 ;
c~0(2) = 0 :
We see that the scalar c0(0) decouples, but the scalar c0(2) remains intrinsically mixed with
the tensor. Note that a transformation c0(2;) ! c0(2;) +
)T c~0(2), for any constant ,
leaves the equations invariant, so there is no further de-mixing that can be performed.
Note also that by comparing with (2.1), we can see that the tensor part of (6.25) is that
of a graviton with m2 =
2, as expected.
tions (6.25), (6.26), (6.27),
L = LPM
where now c02; is a trace-ful symmetric tensor, and
LPM = p g
2 r c0(2;) r c0(2) ; + r c0(2;) r c0(2) ;
is the standard Fierz-Pauli [112] Lagrangian for a partially massless graviton on AdS4
(see [28, 113, 114] for reviews).
The equations (6.25), (6.26), (6.27) can be derived from (6.28) as follows. Taking the
following combination of the tensor equations of motion, all the higher derivatives and
tensor dependence cancels, and we recover the scalar equation of motion,
(Note that r r
Lc0P(2M;) vanishes identically, due to the Noether identity following
from the PM gauge symmetry of LPM.) Taking the following combination of the tensor and
scalar equations of motion, we recover a constraint telling us that the tensor is traceless,
The divergence of the tensor equation becomes,
which upon use of (6.31) to set c0(2)
c0L(2;) , after eliminating divergences using (6.32), eliminated traces using (6.31) and using
the scalar equation (6.30), reproduces the tensor equation of motion (6.25).
The Lagrangian (6.28) cannot be unmixed into separate Fierz-Pauli and Klein-Gordon
Lagrangians for a scalar and a tensor. It is a eld theoretic realization of the spin-0/spin-2
\extended module" uncovered in [16] for the
2 in d = 3. Here we see the AdS4 dual of
this phenomenon.
Wavefunctions in the
nite theories
In sections 6.1 and 6.2 above, we saw that the theory dramatically truncates in dimensions
in the nite dual CFT's discussed in [16]. Not only do these CFT's have a nite number of
primaries, each primary has a nite number of descendants. In AdS, this should correspond
to the elds having a
nite number of modes.
This corresponds to the fact that the dual CFT has a single
= 0 scalar operator which
has no descendants.
primaries in the dual nite CFT. One of these was c00 which satis ed an equation of motion
operator with no descendants. However the other scalar satis ed a full dynamical
Klein= 0 scalar
to a scalar operator with
that we should quantize with the alternate boundary conditions.
The conformal algebra dictates that a scalar operator with
only a nite number of modes. The fact that this scalar lives in a nite-dimensional module
has been known for some time (to our knowledge it was rst uncovered in [115]). We review
the construction of the wavefunctions here for completeness' sake.
The idea is to construct the ground state wavefunction by solving the Klein-Gordon
equation on AdS3, and then act with isometries which act as raising operators, adding
momentum to the state. We will see that this representation has a \speed limit" of sorts;
adding too much momentum annihilates the state, spanning a nine-dimensional Verma
module, exactly as in the dual nite CFT.
= d2
q d42 + L2m2 with d = 2, telling us
ds2 =
We will use the notation
;` for a wavefunction dual to a state of scaling dimension
and angular momentum `, suppressing dependence on the spacetime coordinates. The
ground state wavefunction will be
2;0. The ground state wavefunction will solve the
Klein-Gordon equation:
2;0 = cos2
2;0 = m2
The general solutions to this are the wavefunctions
= c+ei +t cos+
= 4; 2
but since we are choosing the alternate boundary conditions, we choose the smaller root,
and so our ground state wavefunction is
2;0 = e 2it cos 2 :
on AdS3. The scalar lives in a nine-dimensional module, with
2;0 being the ground state. We may
move up by acting with P
(red arrows) or down by acting with K
(blue arrows), but attempting
to act with a third P+ or a third P
annihilates the state.
From here we may move up in the Verma module by acting with isometries which act as
raising operators, P , or lowering operators K
(so named because their actions at the
boundary match that of the raising and lowering operators of the conformal algebra)
= ieit i
= ie it i
sin @t + i cos @
A straightforward computation shows that the wavefunction vanishes if we act with either
more than twice, and furthermore (as expected) the ground state is annihilated
in a nite-dimensional Verma module. We illustrate the structure of the module in gure 9.
There are nine states in total, which matches the expectations from the conformal algebra.
4, quantized with alternate boundary conditions.
There should be a
nite number of
modes in correspondence with the nite number of descendants of the dual
We have presented a construction of a partially massless higher-spin theory which extends
the Vasiliev higher-spin theory to include additional partially massless states. The theory
is based on a Vasiliev-type gauging of the hs2 algebra, the global symmetry algebra of a
2 CFT. We have worked out the rst few dozen terms in the trilinear form of the hs2
algebra, as needed to unfold the C master eld equation of motion and work out the masses
of the four fully massive particles which do not have any linearized gauge symmetry. We've
identi ed the eld content and gauge symmetries of the master elds, demonstrating that
they are in agreement with what is expected from a free scalar
In certain dimensions, the
there are two di erent theories; one is the honest
2 theory with log correlators, and the
other is a CFT with a well-de ned operator algebra, but only a nite number of single-trace
pendent Verma modules are in fact conjoined into a single extended Verma module. This
that the hs2 algebra truncates dramatically into a
nite-dimensional algebra. This
maninite, rather than the log, theories which are the duals of the PM HS theory. Furthermore, we
those particles we would expect from the dual CFT. The module mixing in AdS manifested
itself as the non-diagonalizability of the equations of motion and corresponding free actions.
Finally, in the companion paper [74], we provide evidence that this theory is sensible
at the one-loop level, carrying out the one-loop matching of the coupling of the theory, GN1
to N of the CFT, with identical ndings to what was found for the Vasiliev theory [72].
We believe that all of these checks together constitute signi cant evidence for the
completeness and sensibility of the PM HS theory. Furthermore, as the equations of motion
The dual CFT can be constructed with anti-commuting scalars as well, and our arguments
for an AdS/CFT duality lead us to conjecture that, following [24], the dS/Grassmann
versions of these theories are dual as well, and constitute a new example of the dS/CFT
correspondence. There are many unanswered questions which we hope this new example
will help make progress in; one of the most important such questions to address is what
about the CFT informs our understanding of the unitarity of the dS theory. We plan to
explore this issue in upcoming work.
We also hope that the existence of these extended examples of AdS/CFT and dS/CFT
open the door to many exciting future directions, both within and outside of higher-spin
holography. Given a sensible theory of interacting partially massless higher-spin particles,
it is worth asking if they may play a role in our own universe. Perhaps in the early universe
where massive higher spins may be Hubble scale and possibly partially massless, they might
be detectable in future cosmology experiments [116, 117]. Such a study would go
handin-hand with a study of what interactions might be allowed by partially massless
higherspin particles. In principle, the nonlinear Vasiliev theory gauged with the hs2 algebra
should produce interaction vertices, though it is not clear a priori whether this will produce
all allowable such interactions. In practice, it may be simpler to reconstruct them from
correlators of the dual CFT, following some procedure as in [69, 101, 102].
In particular, we demonstrated that in AdS3/CFT2, the linearized PM HS theory
includes just a single propagating scalar (in addition to a scalar with only a zero mode) in
the minimal theory, plus an additional single vector in the non-minimal theory, in agreement
with CFT predictions. Nevertheless, at
nite N , the CFT is still exactly solvable, but
should now be dual to an interacting non-unitary eld theory (without gravity) on AdS3.
These two theories would be stable by virtue of the nite number of single-particle states,
and the presence of an unbroken hs2 symmetry. Explicitly constructing this theory, and
obtaining its action, would be very interesting, and may provide one of the simplest exactly
solvable examples of AdS/CFT.
In addition, we believe that this only scratches the surface of non-unitary higher-spin
holography. There should be partially massless higher-spin theories dual to the
y@=k
we discussed in [16], with more and more \Regge trajectories." We could also consider the
fermionic counterparts, de ned by a
action, and also the supersymmetric
combination of bosonic and fermionic terms. Perhaps other interesting eld-theoretic mechanisms
exist there as well, and perhaps these new additional examples could also be turned into
useful examples of dS/CFT, some or all of which will hopefully one day play a role in
unlocking the mysteries of quantum gravity in spaces with positive cosmological constants
and the higher spin Higgs mechanism.
Acknowledgments
We thank Nima Arkani-Hamed, Xavier Bekaert, Frederik Denef, Tudor Dimofte, Davide
Gaiotto, Simone Giombi, Bob Holdom, Euihun Joung, Igor Klebanov, Rachel Rosen,
Evgeny Skvortsov, and Matt Walters for helpful discussions and comments. Research at
Perimeter Institute is supported by the Government of Canada through Industry Canada
and by the Province of Ontario through the Ministry of Economic Development and
Inno
Bilinear and trilinear forms in hs2
Here we list the rst few results for the hs2 bilinear forms B and trilinear forms T , de ned
in (3.39), using the techniques described in section 3.3. (Note that the bilinear form is
known to all orders; it was computed in [5]). Our results below for the bilinear forms
match theirs. Our results for the trilinear forms are new, and include all the trilinear forms
necessary for the mass computations of section 5
We compute these by directly expanding and evaluating (3.37). The resulting answers
may be expressed in terms of powers of the cocycles Wj corresponding to each argument
appear. We use notation M(i1;:::;in), where ij indicates how many powers there are of Wj
in that term. We work up to
fth level in two of the cocycles, but restrict the third to
be at level one (because the background master eld W^ which appears in the equations
of motion only has support at level one). We use angle brackets to denote matrix traces.
Finally, we use a few terms in the text that do not appear in the table below; those are all
related to the ones that appear below by the cyclicity of the trace Tr and relabelling. The
results of the computation are listed in table 9. Note that there are potentially multiple
terms appearing in a given part of the expansion; this is in one-to-one correspondence with
the di erent tensor structures which emerge.
hW1W3)hW2W3i
hW1W2W3W3i
hW1W1W2W2i
hW1W3ihW1W2W3i
hW1W1W2W3W3i
hW1W3i hW2W3i
hW1W1W3W3ihW2W3i
hW1W3ihW1W2W3W3i
1 (D 6)(D+2)
1 (D 6)(D+2)
32 (D 1)D(D+1)
3 (D 6)(D+6)
8 (D 1)D(D+1)
3 (D 6)(D+6)
16 (D 1)D(D+1)
32 (D 1)D(D+1)
3 (D 6)(D+6)
16 (D 1)D(D+1)
64 (D 1)D(D+1)
(D 1)D(D+1)(D+3)
(D 1)D(D+1)(D+3)
32 (D 1)D(D+1)(D+3)
16 (D 1)D(D+1)(D+3)
32 (D 1)D(D+1)(D+3)
(D 1)D(D+1)(D+3)
64 (D 1)D(D+1)(D+3)
128 (D 1)D(D+1)(D+3)
64 (D 1)D(D+1)(D+3)
1024 (D 1)(D+1)(D+3)(D+5)
hW1W2ihW1W1W2W2i
hW1W3i hW1W2W3i
hW1W2W3ihW1W1W3W3i
hW1W3ihW1W1W2W3W3i
hW1W1W3W3W1W2W3i
hW1W3i hW1W2W3i
15 (D 6)(D 4)(D
256 (D 1)D(D+1)(D+3)(D+5) hW1W3i hW1W2W3W3i
128 (D 1)D(D+1)(D+3)(D+5) hW1W1W3W3ihW1W2W3W3i
128 (D 1)D(D+1)(D+3)(D+5) hW1W1W3W3W2W1W3W3i
4096 (D 1)(D+1)(D+3)(D+5) hW1W2i
15 (D 6)(D 4)(D
512 (D 1)D(D+1)(D+3)(D+5) hW1W2i hW1W1W2W2i
512 (D 1)D(D+1)(D+3)(D+5) hW1W1W2W2i
512 (D 1)D(D+1)(D+3)(D+5) hW1W1W2W2W1W1W2W2i
1024 (D 1)(D+1)(D+3)(D+5) hW1W3i hW1W2W3i
45 (D 6)(D 4)(D
1024 (D 1)D(D+1)(D+3)(D+5) hW1W3i hW1W1W2W3W3i
2045 (D 1)(D+1)(D+3)(D+5)(D+7) hW1W3i hW2W3ihW1W1W3W3i
1024 (D 1)(D+1)(D+3)(D+5)(D+7) hW1W3i hW1W2W3W3i
2048 (D 1)D(D+1)(D+3)(D+5)(D+7) hW1W2ihW1W1W2W2i
2048 (D 1)D(D+1)(D+3)(D+5)(D+7) hW1W2ihW1W1W2W2W1W1W2W2i
256 (D 1)D(D+1)(D+3)(D+5) hW1W1W3W3ihW1W1W2W3W3i
15 (D 6)(D 4)(D
512 (D 1)D(D+1)(D+3)(D+5) hW1W3ihW1W2W3ihW1W1W3W3i
15 (D 6)(D 4)(D
512 (D 1)D(D+1)(D+3)(D+5) hW1W3ihW1W1W3W3W1W2W3i
256 (D 1)D(D+1)(D+3)(D+5) hW1W1W2W3W3W1W1W3W3i
16384 (D 1)(D+1)(D+3)(D+5)(D+7) hW1W3i hW2W3i
16384 (D 1)(D+1)(D+3)(D+5)(D+7) hW1W2i
2048 (D 1)(D+1)(D+3)(D+5)(D+7) hW1W2i hW1W1W2W2i
2048 (D 1)D(D+1)(D+3)(D+5)(D+7) hW2W3ihW1W1W3W3i
2048 (D 1)D(D+1)(D+3)(D+5)(D+7) hW2W3ihW1W1W3W3W1W1W3W3i
512 (D 1)D(D+1)(D+3)(D+5)(D+7) hW1W3ihW1W1W3W3ihW1W2W3W3i
512 (D 1)D(D+1)(D+3)(D+5)(D+7) hW1W3ihW1W1W3W3W2W1W3W3i
8192 (D 1)(D+1)(D+3)(D+5)(D+7)
1024 (D 1)D(D+1)(D+3)(D+5)(D+7)
4096 (D 1)(D+1)(D+3)(D+5)(D+7)
4096 (D 1)(D+1)(D+3)(D+5)(D+7)
2048 (D 1)D(D+1)(D+3)(D+5)(D+7)
2048 (D 1)D(D+1)(D+3)(D+5)(D+7)
2048 (D 1)D(D+1)(D+3)(D+5)(D+7)
2048 (D 1)D(D+1)(D+3)(D+5)(D+7)
1024 (D 1)D(D+1)(D+3)(D+5)(D+7)
hW1W3i hW1W1W2W3W3i
hW1W1W3W3ihW1W1W3W3W1W2W3i
hW1W3i2hW1W2W3)hW1W1W3W3i
hW1W3i hW1W1W3W3W1W2W3i
hW1W2W3ihW1W1W3W3i
hW1W2W3ihW1W1W3W3W1W1W3W3i
hW1W3ihW1W1W3W3ihW1W1W2W3W3i
hW1W3ihW1W1W2W3W3W1W1W3W3i
hW1W1W3W3W1W1W3W3W1W2W3i
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