Higher-spin theory

Higher Spin Theory or Higher Spin Gravity is a common name for field theories that contain massless fields of spin greater than two. Usually, the spectrum of such theories contains the graviton as a massless spin-two field, which explains the second name. Massless fields are gauge fields and the theories should be (almost) completely fixed by these higher spin symmetries. Higher spin theories are supposed to be consistent quantum theories and, for this reason, to give examples of quantum gravity. Most of the interest in the topic is due to the AdS/CFT correspondence where there is a number of conjectures relating higher spin theories to weakly coupled conformal field theories. It is important to note that only certain parts of these theories are known at present (in particular, standard action principles are not known) and not many examples have been worked out in detail except some specific toy models (such as the higher spin extension of pure Chern-Simons,[1][2] Jackiw-Teitelboim,[3] selfdual (chiral)[4][5] and Weyl gravity theories[6][7]).

Free Higher Spin Fields

Systematic study of massless arbitrary spin fields was initiated by Christian Fronsdal. A free spin-s field can be represented by a tensor gauge field.[8]

${\displaystyle \delta \Phi _{\mu _{1}\mu _{2}...\mu _{s}}=\partial _{\mu _{1}}\xi _{\mu _{2}...\mu _{s}}+{\text{permutations}}}$

This (linearised) gauge symmetry generalises that of massless spin-one (photon) ${\displaystyle \delta A_{\mu }=\partial _{\mu }\xi }$ and that of massless spin-two (graviton) ${\displaystyle \delta h_{\mu \nu }=\partial _{\mu }\xi _{\nu }+\partial _{\nu }\xi _{\mu }}$. Fronsdal also found linear equations of motion and a quadratic action that is invariant under the symmetries above. For example, the equations are

${\displaystyle \square \Phi _{\mu _{1}\mu _{2}...\mu _{s}}-\left(\partial _{\mu _{1}}\partial ^{\nu }\Phi _{\nu \mu _{2}...\mu _{s}}+{\text{ permutations}}\right)+{\frac {1}{2}}\left(\partial _{\mu _{1}}\partial _{\mu _{2}}\Phi ^{\nu }{}_{\nu \mu _{3}...\mu _{s}}+{\text{permutations}}\right)=0}$

where in the first bracket one needs ${\displaystyle s-1}$ terms more to make the expression symmetric and in the second bracket one needs ${\displaystyle s(s-1)/2-1}$ permutations. The equations are gauge invariant provided the field is double-traceless ${\displaystyle \Phi ^{\nu }{}_{\nu }{}^{\lambda }{}_{\lambda \mu _{5}\mu _{2}...\mu _{s}}=0}$ and the gauge parameter is traceless ${\displaystyle \xi ^{\nu }{}_{\nu \mu _{3}...\mu _{s-1}}=0}$.

Essentially, the higher spin problem can be stated as a problem to find a nontrivial interacting theory with at least one massless higher spin field (higher in this context usually means greater than two).

A theory for massive arbitrary higher-spin fields is proposed by C. Hagen and L. Singh.[9][10] This massive theory is important because, according to Vasiliev conjecture,[11][12] spontaneously broken gauges of higher-spins may contain an infinite tower of massive higher-spin particles on the top of the massless modes of lower spins s ≤ 2 like graviton similarly as in string theories.

The linearized version of the higher-spin supergravity gives rise to dual graviton field in first order form.[13] Interestingly, the Curtright field of such dual gravity model is of a mixed symmetry, hence the dual gravity theory can also be massive.[14] Also the chiral and nonchiral actions can be obtained from the manifestly covariant Curtright action.[15][16]

No-go Theorems

Possible interactions of massless higher spin particles with themselves and with low spin particles are (over)constrained by the basic principles of quantum field theory like Lorentz invariance. Many results in the form of no-go theorems have been obtained up to date[17]

Flat Space

Most of the no-go theorems constrain interactions in the flat space.

One of the most well-known is the Weinberg low energy theorem[18] that explains why there are no macroscopic fields corresponding to particles of spin 3 or higher. The Weinberg theorem can be interpreted in the following way: Lorentz invariance of the S-matrix is equivalent, for massless particles, to decoupling of longitudinal states. The latter is equivalent to gauge invariance under the linearised gauge symmetries above. These symmetries lead, for ${\displaystyle s>2}$, to 'too many' conservation laws that trivialise scattering so that ${\displaystyle S=1}$.

Another well-known result is the Coleman-Mandula theorem.[19] that, under certain assumptions, states that any symmetry group of S-matrix is necessarily locally isomorphic to the direct product of an internal symmetry group and the Poincaré group. This means that there cannot be any symmetry generators transforming as tensors of the Lorentz group - S-matrix cannot have symmetries that would be associated with higher spin charges.

Massless higher spin particles also cannot consistently couple to nontrivial gravitational backgrounds.[20] An attempt to simply replace partial derivatives with the covariant ones turns out to be inconsistent with gauge invariance.

Other no-go results include a direct analysis of possible interactions[21][22] and show, for example, that the gauge symmetries cannot be deformed in a consistent way so that they form an algebra.

Anti-de Sitter Space

In anti-de Sitter space many of the flat space no-go results are invalid. In particular, it was shown by Fradkin and Vasiliev[23] that one can consistently couple massless higher spin fields to gravity at the first non-trivial order.

Nevertheless, an analog of the Coleman-Mandula theorem was obtained by Maldacena and Zhiboedov.[24] AdS/CFT correspondence replaces the flat space S-matrix with the holographic correlation functions. It then can be shown that the asymptotic higher spin symmetry in anti-de Sitter space implies that the holographic correlation functions are those of the singlet sector a free vector model conformal field theory (see also Higher Spin AdS/CFT Correspondence below). Let us stress that all n-point correlation functions are not vanishing so this statement is not exactly the analogue of the triviality of the S-matrix (which would be that the conformal field theory is a generalized free field one).

Various Approaches to Higher Spin Theories

The existence of many higher spin theories is well-justified on the basis of AdS/Correspondence, but none of these hypothetical theories is known in full detail. Most of the common approaches to the higher spin problem are described below.

Conformal Higher Spin Theories

Usual massless higher spin symmetries generalise the action of the linearised diffeomorphisms from the metric tensor to higher spin fields. In the context of gravity one may also be interested in Conformal gravity that enlarges diffeomorphisms with Weyl transformations ${\displaystyle g_{\mu \nu }\rightarrow \Omega ^{2}(x)g_{\mu \nu }}$ where ${\displaystyle \Omega (x)}$ is an arbitrary function. The simplest example of a conformal gravity is in four dimensions

${\displaystyle {\mathcal {S}}=\int \mathrm {d} ^{4}x{\sqrt {-g}}C_{\mu \nu \lambda \rho }C^{\mu \nu \lambda \rho }}$

One can try to generalise this idea to higher spin fields by postulating the linearised gauge transformations of the form

${\displaystyle \delta \Phi _{\mu _{1}\mu _{2}...\mu _{s}}=\partial _{\mu _{1}}\xi _{\mu _{2}...\mu _{s}}+g_{\mu _{1}\mu _{2}}\zeta _{\mu _{3}...\mu _{s}}+{\text{permutations}}}$

where ${\displaystyle \zeta _{\mu _{1}...\mu _{s-2}}}$ is a higher spin generalisation of the Weyl symmetry. As different from massless higher spin fields, conformal higher spin fields are much more tractable: they can propagate on nontrivial gravitational background and admit interactions in flat space. In particular, the action of conformal higher spin theories is known to some extent[6][7] - it can be obtained as an effective action for a free conformal field theory coupled to the conformal higher spin background.

Collective Dipole

The idea is conceptually similar to the reconstruction approach just described, but performs a complete reconstruction in some sense. One begins with the free ${\displaystyle O(N)}$ model partition function and performs a change of variables by passing from the ${\displaystyle O(N)}$ scalar fields ${\displaystyle \phi ^{i}(x)}$, ${\displaystyle i=1,...,N}$ to a new bi-local variable ${\displaystyle \Psi (x,y)=\sum _{i}\phi ^{i}(x)\phi ^{i}(y)}$. In the limit of large ${\displaystyle N}$ this change of variables is well-defined, but has a nontrivial Jacobian. The same partition function can then be rewritten as a path integral over bi-local ${\displaystyle \Psi (x,y)}$. It can also be shown that in the free approximation the bi-local variables describe free massless fields of all spins ${\displaystyle s=0,1,2,3,....}$ in anti-de Sitter space. Therefore, the action in term of the bi-local ${\displaystyle \Psi (x,y)}$ is a candidate for the action of a higher spin theory[25]

Holographic RG Flow

The idea is that the equations of the exact renormalization group can be reinterpreted as equations of motions with the RG energy scale playing the role of the radial coordinate in anti-de Sitter space. This idea can be applied to the conjectural duals of higher spin theories, for example, to the free ${\displaystyle O(N)}$ model.[26][27]

Noether Procedure

Noether procedure is a canonical perturbative method to introduce interactions. One begins with a sum of free (quadratic) actions ${\displaystyle S_{2}}$ and linearised gauge symmetries ${\displaystyle \delta _{0}}$, which are given by Fronsdal Lagrangian and by the gauge transformations above. The idea is to add all possible corrections that are cubic in the fields ${\displaystyle S_{3}}$ and, at the same time, allow for field-dependent deformations ${\displaystyle \delta _{1}}$ of the gauge transformations. One then requires the full action to be gauge invariant

${\displaystyle 0=\delta S=\delta _{0}S_{2}+\delta _{0}S_{3}+\delta _{1}S_{2}+...}$

and solves this constraint at the first nontrivial order in the weak-field expansion (note that ${\displaystyle \delta _{0}S_{2}=0}$ because the free action is gauge invariant). Therefore, the first condition is ${\displaystyle \delta _{0}S_{3}+\delta _{1}S_{2}=0}$. One has to mod out by the trivial solutions that result from nonlinear field redefinitions in the free action. The deformation procedure may not stop at this order and one may have to add quartic terms ${\displaystyle S_{4}}$ and further corrections ${\displaystyle \delta _{2}}$ to the gauge transformations that are quadratic in the fields and so on. The systematic approach is via BV-BRST techniques.[28] Unfortunately, the Noether procedure approach has not given yet any complete example of a higher spin theory, the difficulties being not only in the technicalities but also in the conceptual understanding of locality in higher-spin theories. Unless locality is imposed one can always find a solution to the Noether procedure (for example, by inverting the kinetic operator in ${\displaystyle \delta _{0}S_{3}+\delta _{1}S_{2}=0}$ that results from the second term) or, the same time, by performing a suitable nonlocal redefinition one can remove any interaction. At present, it seems that higher spin theories cannot be fully understood as field theories due to quite non-local interactions they have.[29]

Reconstruction

The Higher Spin AdS/CFT Correspondence can be used in the reverse order - one can attempt to build the interaction vertices of the higher spin theory in such a way that they reproduce the correlation functions of a given conjectural CFT dual.[30] This approach takes advantage of the fact that the kinematics of AdS theories is, to some extent, equivalent to the kinematics of conformal field theories in one dimension lower - one has exactly the same number of independent structures on both sides. In particular, the cubic part of the action of the Type-A higher spin theory was found[31] by inverting the three-point functions of the higher spin currents in the free scalar CFT. Some quartic vertices have been reconstructed too.[32]

Three dimensions and Chern-Simons

In three dimensions neither gravity nor massless higher spin fields have any propagating degrees of freedom. It is known[33] that the Einstein-Hilbert action with negative cosmological constant can be rewritten in the Chern-Simons form for ${\displaystyle SL(2,\mathbb {R} )\oplus SL(2,\mathbb {R} )}$

${\displaystyle S=S_{CS}(A)-S_{CS}({\bar {A}})\qquad \qquad S_{CS}(A)={\frac {k}{4\pi }}\int \mathrm {tr} (A\wedge dA+{\frac {2}{3}}A\wedge A\wedge A)\,,}$

where there are two independent ${\displaystyle sl(2,\mathbb {R} )}$-connections, ${\displaystyle A}$ and ${\displaystyle {\bar {A}}}$. Due to isomorphisms ${\displaystyle so(2,2)\sim sl(2,\mathbb {R} )\oplus sl(2,\mathbb {R} )}$ and ${\displaystyle sl(2,\mathbb {R} )\sim so(2,1)}$ the algebra ${\displaystyle sl(2,\mathbb {R} )}$ can be understood as the Lorentz algebra in three dimensions. These two connections are related to vielbein ${\displaystyle e_{\mu }^{a}}$ and spin-connection ${\displaystyle \omega _{\mu }^{a,b}}$ (Note that in three dimensions, the spin-connection, being anti-symmetric in ${\displaystyle a,b}$ is equivalent to an ${\displaystyle so(2,1)}$ vector via ${\displaystyle {\tilde {\omega }}_{\mu }^{a}=\epsilon ^{a}{}_{bc}\omega _{\mu }^{b,c}}$, where ${\displaystyle \epsilon ^{abc}}$ is the totally anti-symmetric Levi-Civita symbol). Higher-spin extensions are straightforward to construct:[34] instead of ${\displaystyle sl(2,\mathbb {R} )\oplus sl(2,\mathbb {R} )}$ connection one can take a connection of ${\displaystyle {\mathfrak {g}}\oplus {\mathfrak {g}}}$, where ${\displaystyle {\mathfrak {g}}}$ is any Lie algebra containing the 'gravitational' ${\displaystyle sl(2,\mathbb {R} )}$ subalgebra. Such theories have been extensively studied[2][1] due their relation to AdS/CFT and W-algebras as asymptotic symmetries.

Vasiliev Equations

Vasiliev equations are formally consistent gauge invariant nonlinear equations whose linearization over a specific vacuum solution describes free massless higher-spin fields on anti-de Sitter space. The Vasiliev equations are classical equations and no Lagrangian is known that starts from canonical two-derivative Fronsdal Lagrangian and is completed by interactions terms. There is a number of variations of Vasiliev equations that work in three, four and arbitrary number of space-time dimensions. Vasiliev's equations admit supersymmetric extensions with any number of super-symmetries and allow for Yang-Mills gaugings. Vasiliev's equations are background independent, the simplest exact solution being anti-de Sitter space. However, locality has not been an assumption used in the derivation and, for this reason, some of the results obtained from the equations are inconsistent with higher spin theories and AdS/CFT duality. Locality issues remain to be clarified.

Higher Spin AdS/CFT Correspondence

Higher spin theories are of interest as models of AdS/CFT correspondence.

Klebanov-Polyakov Conjecture

In 2002 Klebanov and Polyakov put forward a conjecture[35] that the free and critical ${\displaystyle O(N)}$ vector models, as conformal field theories in three dimensions, should be dual to a theory in four-dimensional anti-de Sitter space with infinite number of massless higher-spin gauge fields. This conjecture was further extended and generalised to Gross-Neveu and super-symmetric models.[36][37] The most interesting extension is to a class of Chern-Simons matter theories.[38]

The rationale for the conjectures is that there are some conformal field theories that, in addition to the stress-tensor, have an infinite number of conserved tensors ${\displaystyle \partial ^{c}j_{ca_{2}...a_{s}}=0}$, where spin runs over all positive integers ${\displaystyle s=1,2,3,...}$ (in the ${\displaystyle O(N)}$ model the spin is even). The stress-tensor corresponds to the ${\displaystyle s=2}$ case. By the standard AdS/CFT lore, the fields that are dual to conserved currents have to be gauge fields. For example, the stress-tensor is dual to the spin-two graviton field. A generic example of a conformal field theory with higher spin currents is any free CFT. For instance, the free ${\displaystyle O(N)}$ model is defined by

${\displaystyle S={\frac {1}{2}}\int d^{d}x\,\partial _{m}\phi ^{i}\partial ^{m}\phi ^{j}\delta _{ij},}$

where ${\displaystyle i,j=1,...,N}$. It can be shown that there exist an infinite number of quasi-primary operators

${\displaystyle j_{a_{1}a_{2}...a_{s}}=\partial _{a_{1}}...\partial _{a_{s}}\phi ^{i}\phi ^{j}\delta _{ij}+{\text{plus terms with different arrangement of derivatives and minus traces}}}$

that are conserved. Under certain assumptions it was shown by Maldacena and Zhiboedov[24] that conformal field theories with higher spin currents are free ones. Therefore, higher spin theories are generic duals of free conformal field theories. A theory that is dual to the free scalar CFT is called Type-A in the literature and the theory that is dual to the free fermion CFT is called Type-B.

Another example is the critical vector model, which is a theory with action

${\displaystyle S=\int d^{3}x\,{\frac {1}{2}}\partial _{m}\phi ^{i}\partial ^{m}\phi ^{j}\delta _{ij}+{\frac {\lambda }{4}}(\phi ^{i}\phi ^{j}\delta _{ij})^{2}}$

taken at the fixed point. This theory is interacting and does not have conserved higher spin currents. However, in the large N limit it can be shown to have 'almost' conserved higher spin currents and the conservation is broken by ${\displaystyle 1/N}$ effects.

Gaberdiel-Gopakumar Conjecture

The conjecture put forward by Gaberdiel and Gopakumar[39] is an extension of the Klebanov-Polyakov conjecture to ${\displaystyle AdS_{3}/CFT^{2}}$. It states that the ${\displaystyle W_{N}}$ minimal models in the large ${\displaystyle N}$ limit should be dual to theories with massless higher spin fields and two scalar fields. Massless higher spin fields do not propagate in three dimensions, but can be described, as is discussed above, by the Chern-Simons action. However, it is not known to extend this action as to include the matter fields required by the duality.

References

1. Henneaux, Marc; Rey, Soo-Jong (1 December 2010). "Nonlinear W∞ as asymptotic symmetry of three-dimensional higher spin AdS gravity". Journal of High Energy Physics. 2010 (12): 7. arXiv:1008.4579. Bibcode:2010JHEP...12..007H. doi:10.1007/JHEP12(2010)007.
2. Campoleoni, A.; Fredenhagen, S.; Pfenninger, S.; Theisen, S. (4 November 2010). "Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields". Journal of High Energy Physics. 2010 (11): 7. arXiv:1008.4744. Bibcode:2010JHEP...11..007C. doi:10.1007/JHEP11(2010)007.
3. Alkalaev, K B (12 September 2014). "On higher spin extension of the Jackiw–Teitelboim gravity model". Journal of Physics A: Mathematical and Theoretical. 47 (36): 365401. arXiv:1311.5119. Bibcode:2014JPhA...47J5401A. doi:10.1088/1751-8113/47/36/365401.
4. METSAEV, R.R. (10 February 1991). "Poincaré-Invariant Dynamics of Massless Higher Spins—Fourth-Order Analysis on Mass Shell". Modern Physics Letters A. 06 (4): 359–367. Bibcode:1991MPLA....6..359M. doi:10.1142/S0217732391000348.
5. Ponomarev, Dmitry; Skvortsov, Evgeny (3 March 2017). "Light-front higher-spin theories in flat space". Journal of Physics A: Mathematical and Theoretical. 50 (9): 095401. arXiv:1609.04655. Bibcode:2017JPhA...50i5401P. doi:10.1088/1751-8121/aa56e7.
6. Tseytlin, A. A. (2002). "Limiting Cases of a Superstring in AdS5×S5". Theoretical and Mathematical Physics. 133 (1): 1376–1389. arXiv:hep-th/0201112. doi:10.1023/A:1020646014240.
7. Segal, Arkady Y. (August 2003). "Conformal higher spin theory". Nuclear Physics B. 664 (1–2): 59–130. arXiv:hep-th/0207212. Bibcode:2003NuPhB.664...59S. doi:10.1016/S0550-3213(03)00368-7.
8. Fronsdal, Christian (15 November 1978). "Massless fields with integer spin". Physical Review D. 18 (10): 3624–3629. Bibcode:1978PhRvD..18.3624F. doi:10.1103/PhysRevD.18.3624.
9. Singh, L. P. S.; Hagen, C. R. (1974-02-15). "Lagrangian formulation for arbitrary spin. I. The boson case". Physical Review D. 9 (4): 898–909. doi:10.1103/PhysRevD.9.898.
10. Singh, L. P. S.; Hagen, C. R. (1974-02-15). "Lagrangian formulation for arbitrary spin. II. The fermion case". Physical Review D. 9 (4): 910–920. doi:10.1103/PhysRevD.9.910.
11. Prokushkin, Sergey; Vasiliev, Mikhail (April 1999). "Higher-Spin Gauge Interactions for Massive Matter Fields in 3D AdS Space-Time". Nuclear Physics B. 545 (1–3): 385–433. doi:10.1016/S0550-3213(98)00839-6.
12. Vasiliev, Mikhail (July 2000). "Higher Spin Gauge Theories: Star-Product and AdS Space". arXiv:hep-th/9910096: 533–610. doi:10.1142/9789812793850_0030.
13. Bossard, Guillaume; Kleinschmidt, Axel; Palmkvist, Jakob; Pope, Christopher N.; Sezgin, Ergin (May 2017). "Beyond E 11". Journal of High Energy Physics. 2017 (5): 20. arXiv:1703.01305. Bibcode:2017JHEP...05..020B. doi:10.1007/JHEP05(2017)020. ISSN 1029-8479.
14. Alshal, H.; Curtright, T. L. (September 2019). "Massive dual gravity in N spacetime dimensions". Journal of High Energy Physics. 2019 (9): 63. arXiv:1907.11537. Bibcode:2019JHEP...09..063A. doi:10.1007/JHEP09(2019)063. ISSN 1029-8479.
15. Boulanger, N; Cnockaert, S (2004-03-11). "Consistent deformations of [p,p]-type gauge field theories". Journal of High Energy Physics. 2004 (3): 031. arXiv:hep-th/0402180. Bibcode:2004JHEP...03..031B. doi:10.1088/1126-6708/2004/03/031. ISSN 1029-8479.
16. Henneaux, Marc; Lekeu, Victor; Leonard, Amaury (2017-04-24). "Chiral tensors of mixed Young symmetry". Physical Review D. 95 (8): 084040. arXiv:1612.02772. Bibcode:2017PhRvD..95h4040H. doi:10.1103/PhysRevD.95.084040. ISSN 2470-0010.
17. Bekaert, Xavier; Boulanger, Nicolas; Sundell, Per A. (3 July 2012). "How higher-spin gravity surpasses the spin-two barrier". Reviews of Modern Physics. 84 (3): 987–1009. arXiv:1007.0435. Bibcode:2012RvMP...84..987B. doi:10.1103/RevModPhys.84.987.
18. Weinberg, Steven (24 August 1964). "Photons and Gravitons in Matrix Theory: Derivation of Charge Conservation and Equality of Gravitational and Inertial Mass". Physical Review. 135 (4B): B1049–B1056. Bibcode:1964PhRv..135.1049W. doi:10.1103/PhysRev.135.B1049.
19. Coleman, Sidney; Mandula, Jeffrey (25 July 1967). "All Possible Symmetries of the Matrix". Physical Review. 159 (5): 1251–1256. Bibcode:1967PhRv..159.1251C. doi:10.1103/PhysRev.159.1251.
20. Aragone, C.; Deser, S. (September 1979). "Consistency problems of hypergravity". Physics Letters B. 86 (2): 161–163. Bibcode:1979PhLB...86..161A. doi:10.1016/0370-2693(79)90808-6.
21. Berends, F.A.; Burgers, G.J.H.; Van Dam, H. (October 1985). "On the theoretical problems in constructing interactions involving higher-spin massless particles". Nuclear Physics B. 260 (2): 295–322. Bibcode:1985NuPhB.260..295B. doi:10.1016/0550-3213(85)90074-4.
22. Bekaert, Xavier; Boulanger, Nicolas; Leclercq, Serge (7 May 2010). "Strong obstruction of the Berends–Burgers–van Dam spin-3 vertex". Journal of Physics A: Mathematical and Theoretical. 43 (18): 185401. arXiv:1002.0289. Bibcode:2010JPhA...43r5401B. doi:10.1088/1751-8113/43/18/185401.
23. Fradkin, E.S.; Vasiliev, M.A. (January 1987). "Cubic interaction in extended theories of massless higher-spin fields". Nuclear Physics B. 291: 141–171. Bibcode:1987NuPhB.291..141F. doi:10.1016/0550-3213(87)90469-X.
24. Maldacena, Juan; Zhiboedov, Alexander (31 May 2013). "Constraining conformal field theories with a higher spin symmetry". Journal of Physics A: Mathematical and Theoretical. 46 (21): 214011. arXiv:1112.1016. Bibcode:2013JPhA...46u4011M. doi:10.1088/1751-8113/46/21/214011.
25. de Mello Koch, Robert; Jevicki, Antal; Jin, Kewang; Rodrigues, João P. (10 January 2011). "construction from collective fields". Physical Review D. 83 (2): 025006. arXiv:1008.0633. Bibcode:2011PhRvD..83b5006D. doi:10.1103/PhysRevD.83.025006.
26. Douglas, Michael R.; Mazzucato, Luca; Razamat, Shlomo S. (28 April 2011). "Holographic dual of free field theory". Physical Review D. 83 (7): 071701. arXiv:1011.4926. Bibcode:2011PhRvD..83g1701D. doi:10.1103/PhysRevD.83.071701.
27. Leigh, Robert G.; Parrikar, Onkar; Weiss, Alexander B. (6 January 2015). "Exact renormalization group and higher-spin holography". Physical Review D. 91 (2): 026002. arXiv:1407.4574. Bibcode:2015PhRvD..91b6002L. doi:10.1103/PhysRevD.91.026002.
28. Barnich, Glenn; Brandt, Friedemann; Henneaux, Marc (November 1995). "Local BRST cohomology in the antifield formalism: I. General theorems". Communications in Mathematical Physics. 174 (1): 57–91. arXiv:hep-th/9405109. Bibcode:1995CMaPh.174...57B. doi:10.1007/BF02099464.
29. Sleight, Charlotte; Taronna, Massimo (2018). "Higher spin gauge theories and bulk locality: a no-go result". Phys. Rev. Lett. 121: 171604. arXiv:1704.07859. doi:10.1103/PhysRevLett.121.171604.
30. Witten, Edward. "Spacetime reconstruction".
31. Sleight, Charlotte; Taronna, Massimo (2 May 2016). "Higher Spin Interactions from Conformal Field Theory: The Complete Cubic Couplings". Physical Review Letters. 116 (18): 181602. arXiv:1603.00022. Bibcode:2016PhRvL.116r1602S. doi:10.1103/PhysRevLett.116.181602. PMID 27203314.
32. Bekaert, X.; Erdmenger, J.; Ponomarev, D.; Sleight, C. (23 November 2015). "Quartic AdS interactions in higher-spin gravity from Conformal Field Theory". Journal of High Energy Physics. 2015 (11): 149. arXiv:1508.04292. Bibcode:2015JHEP...11..149B. doi:10.1007/JHEP11(2015)149.
33. Witten, Edward (December 1988). "2 + 1 dimensional gravity as an exactly soluble system". Nuclear Physics B. 311 (1): 46–78. Bibcode:1988NuPhB.311...46W. doi:10.1016/0550-3213(88)90143-5. hdl:10338.dmlcz/143077.
34. Blencowe, M P (1 April 1989). "A consistent interacting massless higher-spin field theory in D=2+1". Classical and Quantum Gravity. 6 (4): 443–452. Bibcode:1989CQGra...6..443B. doi:10.1088/0264-9381/6/4/005.
35. Klebanov, I.R; Polyakov, A.M (December 2002). "AdS dual of the critical O(N) vector model". Physics Letters B. 550 (3–4): 213–219. arXiv:hep-th/0210114. Bibcode:2002PhLB..550..213K. doi:10.1016/S0370-2693(02)02980-5.
36. Leigh, Robert G; Petkou, Anastasios C (10 June 2003). "Holography of the Script N = 1 higher-spin theory on AdS". Journal of High Energy Physics. 2003 (6): 011. arXiv:hep-th/0304217. Bibcode:2003JHEP...06..011L. doi:10.1088/1126-6708/2003/06/011.
37. Sezgin, Ergin; Sundell, Pear (19 July 2005). "Holography in 4D (super) higher spin theories and a test via cubic scalar couplings". Journal of High Energy Physics. 2005 (7): 044. arXiv:hep-th/0305040. Bibcode:2005JHEP...07..044S. doi:10.1088/1126-6708/2005/07/044.
38. Giombi, Simone; Minwalla, Shiraz; Prakash, Shiroman; Trivedi, Sandip P.; Wadia, Spenta R.; Yin, Xi (25 August 2012). "Chern–Simons theory with vector fermion matter". The European Physical Journal C. 72 (8): 2112. arXiv:1110.4386. Bibcode:2012EPJC...72.2112G. doi:10.1140/epjc/s10052-012-2112-0.
39. Gaberdiel, Matthias R.; Gopakumar, Rajesh (8 March 2011). "An dual for minimal model CFTs". Physical Review D. 83 (6): 066007. arXiv:1011.2986. Bibcode:2011PhRvD..83f6007G. doi:10.1103/PhysRevD.83.066007.