Table of contents

The principles of quantum mechanics and relativity impose rigid constraints on theories of massless particles with nonzero spin. Indeed, Yang–Mills theory and General Relativity are the unique solution in the case of spin-1 and spin-2. In asymptotically flat spacetime, there are fundamental obstacles to formulating fully consistent interacting theories of particles of spin greater than 2. However, indications are that such theories are just barely possible in asymptotically anti-de Sitter or de Sitter spacetimes, where the non-existence of an S-matrix provides an escape from the theorems restricting theories in Minkowski spacetime. These higher spin gravity theories are therefore of great intrinsic interest, since they, along with supergravity, provide the only known field theories generalizing the local invariance principles of Yang–Mills theory and General Relativity.

While work on higher spin gravity goes back several decades, the subject has gained broader appeal in recent years due to its appearance in the AdS/CFT correspondence. In three and four spacetime dimensions, there exist duality proposals linking higher spin gravity theories to specific conformal field theories living in two and three dimensions respectively. The enlarged symmetry algebra of the conformal field theories renders them exactly soluble, which makes them excellent laboratories for understanding in detail the holographic mechanism behind AdS/CFT duality.

Steady progress is also being made on better understanding the space of possible higher spin gravity theories and their physical content. This work includes classifying the possible field multiplets and their interactions, constructing exact solutions of the nonlinear field equations, and relating higher spin theories to string theory. A full understanding of these theories will involve coming to grips with the novel symmetry principles that enlarge those of General Relativity and Yang–Mills theory, and one can hope that this will provide some insight into the unification of gravity with the other forces of nature.

Per Kraus and Simon F Ross Guest Editors

We consider CFTs conjectured to be dual to higher spin theories of gravity in AdS₃ and AdS₄. Two-dimensional CFTs with $\mathcal {W}_N$ symmetry are considered in the λ = 0 (k → ∞) limit where they are conjectured to be described by continuous orbifolds. The torus partition function is computed, using reasonable assumptions, and equals that of a free-field theory. We find no phase transition at temperatures of order 1; the usual Hawking–Page phase transition is removed by the highly degenerate light states associated with conical defect states in the bulk. Three-dimensional Chern–Simons matter CFTs with vector-like matter are considered on T³, where the dynamics is described by an effective theory for the eigenvalues of the holonomies. Likewise, we find no evidence for a Hawking–Page phase transition at a large level k.

A recently conjectured microscopic realization of the dS₄/CFT₃ correspondence relating Vasiliev's higher-spin gravity on dS₄ to a Euclidean Sp(N) CFT₃ is used to illuminate some previously inaccessible aspects of the dS/CFT dictionary. In particular it is argued that states of the boundary CFT₃ on S² are holographically dual to bulk states on geodesically complete, spacelike R³ slices which terminate on an S² at future infinity. The dictionary is described in detail for the case of free scalar excitations. The ground states of the free or critical Sp(N) model are dual to dS-invariant plane-wave type vacua, while the bulk Euclidean vacuum is dual to a certain mixed state in the CFT₃. CFT₃ states created by operator insertions are found to be dual to (anti) quasinormal modes in the bulk. A norm is defined on the R³ bulk Hilbert space and shown for the scalar case to be equivalent to both the Zamolodchikov and pseudounitary C-norm of the Sp(N) CFT₃.

We consider three-dimensional conformal field theories that have a higher spin symmetry that is slightly broken. The theories have a large-N limit in the sense that the operators separate into single trace and multitrace and obey the usual large-N factorization properties. We assume that the spectrum of single trace operators is similar to the one that one obtains in the Vasiliev theories. Namely the only single trace operators are the higher spin currents plus an additional scalar. The anomalous dimensions of the higher spin currents are of the order 1/N. Using the slightly broken higher spin symmetry, we constrain the three-point functions of the theories to a leading order in N. We show that there are two families of solutions. One family can be realized as a theory of N fermions with an O(N) Chern–Simons gauge field, the other as an N bosons plus the Chern–Simons gauge field. The family of solutions is parametrized by the 't Hooft coupling. At special parity preserving points, we obtain the critical O(N) models: the Wilson–Fisher one and the Gross–Neveu one. Our analysis also fixes the on-shell three-point functions of Vasiliev's theory on AdS₄ or dS₄.

Higher spin gravity in three dimensions is efficiently formulated as a Chern–Simons gauge-theory, typically with gauge algebra sl(N)⊕sl(N). The classical and quantum properties of the higher spin theory depend crucially on the embedding into the full gauge algebra of the sl(2)⊕sl(2) factor associated with gravity. It has been argued previously that non-principal embeddings do not allow for a semi-classical limit (large values of the central charge) consistent with unitarity. In this work we show that it is possible to circumvent these conclusions. Based upon the Feigin–Semikhatov generalization of the Polyakov–Bershadsky algebra, we construct infinite families of unitary higher spin gravity theories at certain rational values of the Chern–Simons level that allow arbitrarily large values of the central charge up to $c = N/4 - 1/8 - \mathcal{O}(1/N)$, thereby confirming a recent speculation by us (Afshar et al 2012 arXiv:1209.2860).

Following the work of Maldacena and Zhiboedov, we study the implementation of the Coleman–Mandula theorem in the free O(N)/higher spin correspondence. In the bi-local framework we first define an S-matrix for scattering of collective dipoles. Its evaluation in the case of free UV fixed point theory leads to the result S = 1 stated in the title. We also present an appropriate field transformation that is seen to transform away all the nonlinear 1/N interactions of this theory. A change of boundary conditions and/or external potentials results in a nontrivial theory.

Multiparticle extension of a higher spin algebra l is introduced as the Lie superalgebra associated with the universal enveloping algebra U(l). While conventional higher spin symmetry does not mix n-particle states with different n, multiparticle symmetries do so. Quotients of multiparticle algebras are considered, that act on the space of n-particle states with 0 ⩽ n ⩽ k analogous to the space of first k Regge trajectories of string theory. Original higher spin algebra is reproduced at k = 1. Full multiparticle algebras are conjectured to describe vacuum symmetries of string-like extensions of higher spin gauge theories. The relation of the multiparticle algebras with 3D current operator algebras is described. The central charge parameter, to be related to the parameter ${\mathcal {N}}$ in AdS/CFT correspondence, enters via the definition of supertrace. Extension to higher p-brane-like symmetries is introduced inductively.