Partially massless higher-spin theory

Journal of High Energy Physics, Feb 2017

We study a generalization of the D-dimensional Vasiliev theory to include a tower of partially massless fields. 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 infinite 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 fixed by the massless or partially massless gauge symmetry, finding 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 specific dimensions; in particular, the theory dramatically truncates in bulk dimensions D = 3, 5 and has non-diagonalizable mixings which occur in D = 4, 7.

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Partially massless higher-spin theory

Received: December massless higher-spin theory Open Access 0 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 Open Access. 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Christopher Brust, Kurt Hinterbichler. Partially massless higher-spin theory, Journal of High Energy Physics, 2017, 86, DOI: 10.1007/JHEP02(2017)086