We study a generalization of the -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 and has non-diagonalizable mixings which occur in .
Partially Massless Higher-Spin Theory
Christopher Brust111E-mail: email@example.com, Kurt Hinterbichler222E-mail: firstname.lastname@example.org
Perimeter Institute for Theoretical Physics,
31 Caroline St. N, Waterloo, Ontario, Canada, N2L 2Y5
CERCA, Department of Physics, Case Western Reserve University,
10900 Euclid Ave, Cleveland, OH 44106, USA
- 1 Introduction
- 2 Review of Partially Massless Fields
- 3 The Algebra
- 4 A Partially Massless Higher-Spin Theory
- 5 Mass Computations
- 6 Nuances in the Spectrum
- 7 Conclusions and Future Directions
- A Bilinear and Trilinear Forms in
In this paper, we explore an explicit description of a partially massless (PM) higher-spin (HS) theory, discussed previously in Bekaert:2013zya (); Basile:2014wua (); Grigoriev:2014kpa (); Alkalaev:2014nsa (); Joung:2015jza (). 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 theory333Throughout this work, we refer only to the bosonic CP-even Vasiliev theory. Vasiliev:1990en (); Vasiliev:1992av (); Vasiliev:1999ba (); Vasiliev:2003ev () (see Vasiliev:1995dn (); Vasiliev:1999ba (); Bekaert:2005vh (); Iazeolla:2008bp (); Didenko:2014dwa (); Vasiliev:2014vwa (); Giombi:2016ejx () for reviews), it contains an infinite tower of massless fields of all spins, but in addition it contains a second infinite 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 Vasiliev theory.
The theory on AdS is expected to be the holographic dual to the singlet sector of the bosonic free conformal field theory (CFT) studied in Brust:2016gjy () (see also Karananas:2015ioa (); Osborn:2016bev (); Guerrieri:2016whh (); Nakayama:2016dby (); Peli:2016gio (); Gwak:2016sma (); Gliozzi:2016ysv (); Gliozzi:2017hni ()), 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 Sezgin:2002rt (); Klebanov:2002ja (); Anninos:2011ui (). We define the bulk theory as the Vasiliev-type gauging of the CFT’s underlying global symmetry algebra, which we refer to here as . It is a part of a family of theories based on the field theory which contain towers of partially massless states. We study this theory for several reasons:
In our universe, we’ve confirmed the existence of seemingly fundamental particles with spins , and , and we have good reason to believe that gravity is described by a particle with spin . It is an interesting field-theoretic question to ask, even in principle, what spins we are allowed to have in our universe. Famous arguments, such as those reviewed in Bekaert:2010hw (); Porrati:2012rd (), would näively seem to indicate that we should not expect particles with spin greater than to be relevant to an understanding of our universe, but these no-go theorems are evaded by specific 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 fields fall into the allowed class. Partially massless fields are of interest due to a possible connection between partially massless spin-2 field and cosmology (see e.g., deRham:2013wv () and the review Schmidt-May:2015vnx ()), which has led to many studies of the properties of the linear theory and possible nonlinear extensions Zinoviev:2006im (); Hassan:2012gz (); Hassan:2012rq (); Hassan:2013pca (); Deser:2013uy (); deRham:2013wv (); Zinoviev:2014zka (); Garcia-Saenz:2014cwa (); Hinterbichler:2014xga (); Joung:2014aba (); Alexandrov:2014oda (); Hassan:2015tba (); Hinterbichler:2015nua (); Cherney:2015jxp (); Gwak:2015vfb (); Gwak:2015jdo (); Garcia-Saenz:2015mqi (); Hinterbichler:2016fgl (); Apolo:2016ort (); Apolo:2016vkn (); Gwak:2016sma (). No examples (other than non-unitary conformal gravity Maldacena:2011mk (); Deser:2012qg (); Deser:2013bs ()) of UV-complete theories in four dimensions containing an interacting partially massless field and a finite number of other fields 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 infinite 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 Strominger:2001pn (); Hull:1998vg (); Witten:2001kn (); Strominger:2001gp (); Balasubramanian:2002zh (); Maldacena:2002vr (). It was argued in Anninos:2011ui () that the future boundary correlators of the non-minimal and minimal Vasiliev higher-spin theories on dS should match the correlators of the singlet sector of free “” or Grassmann scalar field 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. PhysRevLett.60.1229 ()), 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 Chang:2012kt () appears to relate the super-Vasiliev theory and IIA superstring theory on . However, 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 infinite set of Vasiliev-like towers of ever increasing masses, and one would require an infinite number of copies of the fields in the Vasiliev theory in order to construct a fully Higgsed string spectrum. Without the aid of the 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 Stückelberg fields.
It is natural to suspect that there should be a smooth Higgsing process by which an infinite set of massless Vasiliev towers eat each other and become the massive spectrum of string theory Girardello:2002pp (); Bianchi:2003wx (); Bianchi:2005ze (). On AdS, there seems to be no obstruction to this, but on dS the situation is different. As we review in section (2), there is a unitarity bound for a mass , spin particle in dimensional dS space. Below this bound, particles are non-unitary and so any smooth Higgs mechanism starting from would necessarily be doomed to pass through this non-unitary region before becoming fully massive. The PM fields, 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 fields 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 fields).
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 defines 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 Vasiliev:1988sa (); Boulanger:2011dd (); Doroud:2011xs (); Boulanger:2012bj (); Boulanger:2015kfa (); Bekaert:2015tva (); Bonezzi:2016ttk (); Sleight:2016dba () for efforts 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 defined on both AdS and dS, and is essentially nonlocal on the scale of the curvature radius , though it has a local expansion in which derivatives are suppressed by the scale . Nevertheless, this theory admits a weakly-coupled description and so can be studied perturbatively in ; 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 is the gauge theory of the so-called algebra, an infinite-dimensional extension of the diffeomorphism 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 field 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 free field theory, which includes all of the generators of the algebra, and in addition “higher-order Killing tensors”, studied in 2006math…..10610E (). The representations and the bilinear form of this algebra were studied by Joung and Mkrtchyan Joung:2015jza (), and we make use of many of their results444They referred to this algebra as ; however as this algebra arises from a dual CFT, we refer to this algebra in this paper simply as the algebra.. The structure of this algebra is very rigid, and its gauging completely fixes 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 fields 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 defining 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 infinite-dimensional underlying gauge algebra 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 different particles, and without a Lagrangian we cannot directly check whether this is the case.
In the CFT, we demonstrated in Brust:2016gjy () that certain dimensions were special; in there existed what we dubbed the “finite theories”; we will show here that the PM HS theory in mimics the structure of these finite CFTs. Furthermore, in there was module mixing that took place in the CFT. We will see that this manifests as non-diagonalizability of the dual free PM HS action in . The fact that these Verma module structures mimic each other comes as no surprise, but does offer evidence that the PM HS theory is truly the AdS dual of the CFT. Furthermore, the details of the duality in these cases are new, and are not specific 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 field theory was the one-loop matching of the partition functions of the boundary and bulk theories Giombi:2013fka (); Giombi:2014iua (). 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 Maldacena:2011jn (). Therefore, when computing the one-loop correction to the inverse Newton’s constant in the Vasiliev theory, one was forced to obtain an integer multiple of the dual theory’s -type conformal anomaly (even ) or sphere free energy (odd ), which was precisely what happened. Despite the fact that the CFT is non-unitary, its 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 Brust:2016xif () we do this computation in several dimensions and find a positive result (see also Gunaydin:2016amv ()); we obtain integer multiples of the -type conformal anomaly or sphere free energy of a single real conformally coupled scalar in one dimension fewer. In particular, we obtain identical results to the Vasiliev case Giombi:2014iua (), namely for the non-minimal/ duality and for the minimal/ duality.
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 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 fields break up into unfolding fields for the physical particles. In section 5, we compute the masses of the four particles whose masses are not fixed by gauge invariance. We discuss which boundary conditions are necessary on the various fields so as to reproduce CFT expectations. In section 6, we explore what happens to the PM HS spectrum in , demonstrating 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 Brust:2016xif ().
Conventions: We use the mostly plus metric signature, and the curvature conventions of Carroll:2004st (). We (anti) symmetrize tensors with unit weight, e.g., . The notation indicates that the enclosed indices are to be symmetrized and made completely traceless. Throughout this work, we unfortunately must reference three different spacetime dimensions; the dimension of the dual CFT is denoted , the dimension of the bulk (A)dS is denoted , and the dimension of the ambient or embedding space in which the symmetry algebra is defined is denoted . They are related by . Embedding space coordinates are indexed by , and moved with the flat ambient metric . (A)dS spacetime coordinates are indexed by , and moved with the (A)dS metric . (A)dS tangent space indices are indexed by , and moved with the tangent space flat metric . The boundary CFT indices are , and are moved with the flat boundary metric . The background (A)dS space has a vielbein which relates AdS spacetime and AdS tangent space indices. refers to the AdS length scale, and 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 , , , .). We use the shorthand to denote a Young tableau with boxes in the first row, 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 is denoted where the indices to be projected should be clear from the context. The action of the projector is to first symmetrize indices in each row, and then anti-symmetrize indices in each column, with the overall normalization chosen so that . This projector does not include the subtraction of traces. Introductions to Young tableaux can be found in section 4 of Bekaert:2006py () or the book Tung:1985na ().
2 Review of Partially Massless Fields
We begin by reviewing some properties of partially massless higher-spin fields in AdS or dS Deser:1983tm (); Deser:1983mm (); Higuchi:1986py (); Brink:2000ag (); Deser:2001pe (); Deser:2001us (); Deser:2001wx (); Deser:2001xr (); Zinoviev:2001dt (); Skvortsov:2006at (); Skvortsov:2009zu (), and how they behave as we take them to the boundary, i.e. the properties of the dual CFT operators. Partially massless fields are fields with more degrees of freedom, and correspondingly less gauge symmetry, than a massless field, but fewer degrees of freedom, and correspondingly more gauge symmetry, than a fully massive field. For a given spin, the amount of gauge symmetry fixes the mass on both AdS and dS. Partially massless fields are necessarily below the unitarity bound in AdS, but are unitary in dS.
2.1 Free Massive Fields
A spin- field on dimensional (A)dS with mass is described by a symmetric -index field which satisfies the equations of motion
i.e. it is transverse, traceless, and satisfies a Klein-Gordon equation. is the curved space Laplacian.
Here is the (A)dS curvature scale, i.e. for dS, in which case is the Hubble constant, and for AdS (in which case we usually write with the usual AdS radius). The scalar curvature and cosmological constant are related to the Hubble constant as
In the AdS case, , the high spin fields are dual to symmetric tensor “single-trace” primaries . For generic , these satisfy no particular conservation conditions. Their scaling dimensions are given in terms of the mass by
Here 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 Klebanov:1999tb ().
The unitarity bound Mack:1975je () for symmetric traceless tensor operators is
For scalars, , we have
and the unitarity bound is
so for both ordinary and alternate quantizations are possible in a unitary theory. For , only the ordinary quantization is compatible with unitarity. However, in the (non-unitary) partially massless theory, we will see that we do indeed need to use the alternate quantization for certain particles with .
Solving for gives
For , we have in the bulk, and there is no analog of the Breitenlohner-Freedman bound Breitenlohner:1982bm (); Breitenlohner:1982jf ()555The Breitenlohner-Freedman bound for scalars is allowing for slightly tachyonic but stable scalars. For , as soon as the mass is negative, we generically expect instabilities owing to the theory becoming ghostly/non-unitary.
In the dS case, the unitarity bound for massive particles is not at . Instead, the bound below which the particle is generically non-unitary is the Higuchi bound Higuchi:1986py (); Higuchi:1986wu (); Higuchi:1989gz (),
Below this bound, the kinetic term for one of the Stückelberg fields 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 field is unitary at these special points. These points are the partially massless fields, and we turn to them next.
2.2 Free Partially Massless Fields
Partially massless fields occur at the special mass values
Here, is called the depth of partial masslessness. At these mass values, the system of equations (2.3) becomes invariant under a gauge symmetry,
and so counts the number of indices on the gauge parameter . Here stands for lower-derivative terms proportional to . On shell, the gauge parameter is transverse and traceless and satisfies a Klein-Gordon equation
Just as massive and massless fields carry irreducible representations of the dS group, partially massless fields also carry irreducible representations, albeit ones which have no flat space counterpart. A generic massive field has, in the massless limit, the degrees of freedom of massless fields of spin (usually called, with some abuse of terminology, helicity components). The gauge symmetry of a PM field removes some of the lower helicity components; a depth PM field has helicity components
The highest depth is , which corresponds to the usual massless field containing only helicity components . We see that on AdS, all but the highest depth PM fields have negative masses, and are non-unitary. On dS, the masses are positive, and the PM fields are unitary (despite sitting below the Higuchi bound). The lowest depth is . 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 figure 1, which shows the Higuchi bound on as well as the first few partially massless particles’ masses and spins.
PM fields are dual to multiply-conserved symmetric tensor single-trace primaries , i.e. they satisfy a conservation condition involving multiple derivatives Dolan:2001ih (),
For the massless case, , this is the usual single-derivative conservation law. More generally, , where is the degree of “conservedness” of the operator (a notation we introduced in Brust:2016gjy ()), i.e. the number of derivatives you need to dot into the operator to kill it.
On AdS, the mass-scaling dimension relation (2.5) (with the positive root) tells us that the dimension of these partially conserved currents should be666As a check, one can see that the general form for the two-point correlation functions, (2.16) become conserved, doubly conserved, etc. precisely at these values, e.g. the expression satisfies (2.17)
The second equality shows that these operators violate the CFT unitarity bound (2.6) except for the conserved operator with , which saturates it.
3 The Algebra
We now discuss the symmetry algebra, , which we will ultimately gauge in order to obtain a partially massless higher spin theory. In the linearized partially massless higher-spin theory, there will be two “master” fields (a gauge field and a field strength), and a “master” gauge parameter, which are valued in the algebra. The Vasiliev equations themselves are also valued in the algebra.
There is a multi-linear form which is defined 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 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 Vasiliev:2003ev (); Bekaert:2005vh (); Joung:2014qya (); Joung:2015jza (). 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 different tensor structures which emerge and greatly simplifies calculations.
3.1 Generalities About the Algebra
The algebra is realized as the algebra of global symmetries of a conformal field theory 2006math…..10610E (); Bekaert:2013zya (); Basile:2014wua (); Grigoriev:2014kpa (); Alkalaev:2014nsa (); Joung:2015jza (); Brust:2016gjy (), the CFT described by the action
The CFT contains as its underlying linearly realized777These are not to be confused with the non-linearly realized higher shift symmetries of Hinterbichler:2014cwa (); Griffin:2014bta (), which are also present. symmetry algebra precisely the algebra . The spectrum of operators and conserved currents form a representation of this algebra.
We first discuss this algebra abstractly. can be abstractly defined as a quotient of the universal enveloping algebra (UEA), , of the dimensional888This construction is independent of the signature. embedding space Lorentz algebra , by a particular ideal. The abstract generators of transform in the adjoint representation of the algebra,
The commutation relations for are
where is the invariant metric tensor.
The universal enveloping algebra, and then , will be described as successive quotients of the algebra of all formal products of the ’s. First, we consider the tensor product algebra formed from the ’s. We can label the elements of the tensor product algebra by the irrep under , which we display as a tableau, as well as by the number of powers of they came from, which we indicate using a subscript on the tableau, and which we’ll refer to as the “level”. For example, we may decompose the product of two ’s as
The scalar is the quadratic Casimir , and the antisymmetric tensor is the commutator.
In the top line of (3.4) are terms which are symmetric in the interchange of the two s, whereas the bottom line contains terms which are antisymmetric in the interchange of the two 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 ’s, leaving only the symmetric parts in the top line (see the Poincaré-Birkhoff-Witt theorem).
To pass to the algebra, we quotient by a further ideal. The generators , , and finally generate an ideal of the UEA (here comes in at level 4, e.g. from the tensor product ). We quotient the UEA to the algebra by replacing , , and .
Those generators which remain in the resulting quotient define the generators of the algebra, and consist of the representations:
The first line are generators which are in the same representations as the generators of the massless algebra, which we will call at level , whereas the second line are generators new to , which we will call at level . The old generators correspond to Killing tensors of AdS and conformal Killing tensors of the CFT, whereas the new generators correspond to so-called order three Killing tensors in AdS and order three conformal Killing tensors in the CFT, as reviewed in Joung:2015jza (), Brust:2016gjy (), and in the appendix. They are associated with multiply conserved currents in the CFT, and are gauged by partially massless fields in AdS. It is noteworthy that contains as a sub-vector space. However, as the values of the Casimirs do not match, it is not, strictly speaking, a subalgebra999We thank Evgeny Skvortsov for discussions of this point..
In the process of taking the quotient, all of the Casimirs are fixed to specific values:
All of the generators in the algebra are traceless two-row Young tableaux, which we generically call with . These generators can be written as elements of the UEA in the appropriate representations
where is the normalized projector onto the tableau (this definition fixes the normalization of the generators). These generators carry indices in the fully traceless tableau of shape . We use the anti-symmetric convention, which means that they are anti-symmetric in any pair, vanishes if we try to anti-symmetrize any pair with any third index to the right of the pair. For we have only the constants, and for the original generators . In the original algebra, all the generators have . In , as shown schematically in equation (3.5), we have generators with , which we referred to as , as well as generators with , which we referred to as .
A general algebra element is a linear combination of the above generators,
where the coefficient tensors have the symmetry of a traceless tableau and the coefficient tensors have the symmetry of a traceless tableau.
The product on the 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 and , is
and it gives the algebra the structure of a Lie algebra which is isomorphic to the Lie algebra of linearly realized global symmetries of the CFT.
There is a natural trace on the algebra which projects onto a singlet, defined simply as
and a multi-linear form can be defined using this trace as
Note that the bilinear form is diagonal in the degree , because the product of a rank generator and a rank generator only contains a zero component if . But there can be mixing between algebra elements with the same degree but corresponding to different Young diagrams, which we will have to worry about later.
3.2 Oscillators and Star Products
Although in principle the previous subsection contains all of the ingredients necessary to define the algebra, it is incredibly cumbersome to use those definitions directly to compute anything in the algebra. In this section we review an oscillator construction of the algebra, as introduced in Vasiliev:2003ev (); Bekaert:2005vh (); Joung:2015jza (). 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 simplification 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 , called oscillator variables, which carry an index101010This is the Howe dual algebra to the , see e.g. the review Bekaert:2005vh (). in addition to an index . (For us, this is a completely auxiliary structure useful for defining 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 . The invariant tensor for is which is anti-symmetric,
Suppose we have two arbitrary polynomials in the variables, and . We may define an oscillator star product, , between them. (Note that the oscillator star product is a priori different from the product which we defined 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 defined to be
Like , is bi-linear and associative. Our goal is to understand how we can use this easy-to-evaluate product to evaluate the desired product .
With the star product we define the star commutator
The star products and commutators among the basic variables are
It should be noted that there are consequently two products available to the ; an ordinary product and a star product. The ’s commute as ordinary products, despite not commuting as star products. When we write polynomials in , we mean that they are polynomials in the ordinary product sense.
We define antisymmetric and symmetric generators as
We may use the above star product to evaluate the star commutators of (3.18), and these reproduce the commutation relations of decoupled and algebras,
To each element of the algebra , we may associate a polynomial in the ’s by replacing the generators with a product of ’s
We would like to be able to use the product on in place of the product on , but there is an obstruction in that, in general, we still have nontrivial Casimir elements in the polynomial , which must be fixed to particular numbers. We may force all of the Casimir-type elements to be set to the values required by the algebra by introducing a quasiprojector111111This is referred to as a quasiprojector rather than a projector because the explicit form does not satisfy ; rather, its square doesn’t converge Vasiliev:2004cm (). This is not a problem at the level of working to any fixed order in the algebra, as we do., , which will be useful for setting the Casimirs to their proper values, and extracting from a general polynomial an element of when working within a trace:
To extract the trace, we merely take the component of . This can be formally obtained by simply setting . Therefore
Once we have the quasiprojector, we can compute multi-linear forms using the product:
We now need to know what is. It should implement the modding out by the ideals, including replacing the Casimir with the appropriate number,
and likewise with all higher powers.
A useful form for the quasi-projector was found in Joung:2015jza (),
with a normalization factor,
3.3 Coadjoint Orbits
In order to conveniently deal with the tensor structures which emerge, it is useful to introduce, following Joung:2015jza (), the technology of coadjoint orbits. The coadjoint orbit method allows us to replace the coefficient tensors , of a general algebra element (3.8) with products of a single antisymmetric tensor , called a coadjoint orbit, which we write in a script font,
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 valued fields. These are in one-to-one correspondence with the different 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 fields.
The coadjoint orbits satisfy what we will call here the coadjoint orbit conditions:
These two together serve to enforce that products of copies of in (3.29) have the symmetry properties of either a trace-ful or trace-less tableau. (We often view as a matrix in what follows, and use to denote a matrix trace.) To see this, the first can be shown to imply the conditions , and the second can be contracted with a second coadjoint orbit to show
Note that this identity also implies that . Therefore, if we consider the quantity , then it is in the 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 indices puts the resulting tensor in the representation, which is automatically traceless.
In the computations we will do, we will have several different fields present in each multi-linear form, so we’ll introduce several different, independent coadjoint orbits, one for each field, each satisfying their own coadjoint orbit conditions (and each with the script version of the letter associated to the particular field).
As mentioned, we must subtract the single traces manually from the fields. There are no traces to subtract at level 0 or 1 in the algebra; we must first 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 coefficients, and demanding that the resulting tensor is in the representation and is totally traceless given the coadjoint orbit conditions. The results of this procedure for are:
There are no such subtleties with the tensors, which are already traceless given the coadjoint orbit conditions, so we may simply replace