On Tensorial Spaces and BCFW Recursion Relations for Higher Spin Fields

In this short review we briefly consider two topics in the higher spin gauge theory: the method of "tensorial (super) spaces" and application of BCFW recursion relations to higher spin fields.Comment: 26 pages, Invited Review, discussion improved, references adde

BMS Modules in Three Dimensions

We build unitary representations of the BMS algebra and its higher-spin extensions in three dimensions, using induced representations as a guide. Our prescription naturally emerges from an ultrarelativistic limit of highest-weight representations of Virasoro and W algebras, which is to be contrasted with non-relativistic limits that typically give non-unitary representations. To support this dichotomy, we also point out that the ultrarelativistic and non-relativistic limits of generic W algebras differ in the structure of their non-linear terms.Comment: 27 pages; contribution to the proceedings of the workshop "Higher Spin Gauge Theories", Institute for Advanced Studies, NTU, Singapore, November 4-6, 201

New Recursion Relations and a Flat Space Limit for AdS/CFT Correlators

We consider correlation functions of the stress-tensor or a conserved current in AdS_{d+1}/CFT_d computed using the Hilbert or the Yang-Mills action in the bulk. We introduce new recursion relations to compute these correlators at tree level. These relations have an advantage over the BCFW-like relations described in arXiv:1102.4724 and arXiv:1011.0780 because they can be used in all dimensions including d=3. We also introduce a new method of extracting flat-space S-matrix elements from AdS/CFT correlators in momentum space. We show that the (d+1)-dimensional flat-space amplitude of gravitons or gluons can be obtained as the coefficient of a particular singularity of the d-dimensional correlator of the stress-tensor or a conserved current; this technique is valid even at loop-level in the bulk. Finally, we show that our recursion relations automatically generate correlators that are consistent with this observation: they have the expected singularity and the flat-space gluon or graviton amplitude appears as its coefficient.Comment: 22+6 pages (v2) typos fixe

Quantizing higher-spin gravity in free-field variables

We study the formulation of massless higher-spin gravity on AdS$_3$ in a gauge in which the fundamental variables satisfy free field Poisson brackets. This gauge choice leaves a small portion of the gauge freedom unfixed, which should be further quotiented out. We show that doing so leads to a bulk version of the Coulomb gas formalism for $W_N$ CFT's: the generators of the residual gauge symmetries are the classical limits of screening charges, while the gauge-invariant observables are classical $W_N$ charges. Quantization in these variables can be carried out using standard techniques and makes manifest a remnant of the triality symmetry of $W_\infty[\lambda]$. This symmetry can be used to argue that the theory should be supplemented with additional matter content which is precisely that of the Prokushkin-Vasiliev theory. As a further application, we use our formulation to quantize a class of conical surplus solutions and confirm the conjecture that these are dual to specific degenerate $W_N$ primaries, to all orders in the large central charge expansion.Comment: 31 pages + appendices. V2: typos corrected, reference adde

The action for higher spin black holes in three dimensions

In the context of (2+1)--dimensional Chern-Simons SL(N,R)\times SL(N,R) gauge fields and spin N black holes we compute the on-shell action and show that it generates sensible and consistent thermodynamics. In particular, the Chern-Simons action solves the integrability conditions recently considered in the literature.Comment: Paper shortened and generalized. Main results unchanged. 25 pages, Latex, no figure

On the relation between local and geometric Lagrangians for higher spins

Equations of motion for free higher-spin gauge fields of any symmetry can be formulated in terms of linearised curvatures. On the other hand, gauge invariance alone does not fix the form of the corresponding actions which, in addition, either contain higher derivatives or involve inverse powers of the d'Alembertian operator, thus introducing possible subtleties in degrees of freedom count. We suggest a path to avoid ambiguities, starting from local, unconstrained Lagrangians previously proposed, and integrating out the auxiliary fields from the functional integral, thus generating a unique non-local theory expressed in terms of curvatures.Comment: 14 pages. Contribution to the proceedings of the 1st Mediterranean Conference on Classical and Quantum Gravity, Kolymbary (Crete, Greece) September 14-18 200

An AdS_3 Dual for Minimal Model CFTs

We propose a duality between the 2d W_N minimal models in the large N 't Hooft limit, and a family of higher spin theories on AdS_3. The 2d CFTs can be described as WZW coset models, and include, for N=2, the usual Virasoro unitary series. The dual bulk theory contains, in addition to the massless higher spin fields, two complex scalars (of equal mass). The mass is directly related to the 't Hooft coupling constant of the dual CFT. We give convincing evidence that the spectra of the two theories match precisely for all values of the 't Hooft coupling. We also show that the RG flows in the 2d CFT agree exactly with the usual AdS/CFT prediction of the gravity theory. Our proposal is in many ways analogous to the Klebanov-Polyakov conjecture for an AdS_4 dual for the singlet sector of large N vector models.Comment: 33 pages LaTe

Thermodynamics of Higher Spin Black Holes in AdS3_3

We discuss the thermodynamics of recently constructed three-dimensional higher spin black holes in SL(N,R)\times SL(N,R) Chern-Simons theory with generalized asymptotically-anti-de Sitter boundary conditions. From a holographic perspective, these bulk theories are dual to two-dimensional CFTs with W_N symmetry algebras, and the black hole solutions are dual to thermal states with higher spin chemical potentials and charges turned on. Because the notion of horizon area is not gauge-invariant in the higher spin theory, the traditional approaches to the computation of black hole entropy must be reconsidered. One possibility, explored in the recent literature, involves demanding the existence of a partition function in the CFT, and consistency with the first law of thermodynamics. This approach is not free from ambiguities, however, and in particular different definitions of energy result in different expressions for the entropy. In the present work we show that there are natural definitions of the thermodynamically conjugate variables that follow from careful examination of the variational principle, and moreover agree with those obtained via canonical methods. Building on this intuition, we derive general expressions for the higher spin black hole entropy and free energy which are written entirely in terms of the Chern-Simons connections, and are valid for both static and rotating solutions. We compare our results to other proposals in the literature, and provide a new and efficient way to determine the generalization of the Cardy formula to a situation with higher spin charges.Comment: 30 pages, PDFLaTeX; v2: typos corrected, explicit expressions for the free energy adde

Rotating Higher Spin Partition Functions and Extended BMS Symmetries

We evaluate one-loop partition functions of higher-spin fields in thermal flat space with angular potentials; this computation is performed in arbitrary space-time dimension, and the result is a simple combination of Poincar\'e characters. We then focus on dimension three, showing that suitable products of one-loop partition functions coincide with vacuum characters of higher-spin asymptotic symmetry algebras at null infinity. These are extensions of the bms_3 algebra that emerges in pure gravity, and we propose a way to build their unitary representations and to compute the associated characters. We also extend our investigations to supergravity and to a class of gauge theories involving higher-spin fermionic fields.Comment: 58 pages; clarifications and references added; version to be published in JHE

Quantum W-symmetry in AdS_3

It has recently been argued that, classically, massless higher spin theories in AdS_3 have an enlarged W_N-symmetry as the algebra of asymptotic isometries. In this note we provide evidence that this symmetry is realised (perturbatively) in the quantum theory. We perform a one loop computation of the fluctuations for a massless spin $s$ field around a thermal AdS_3 background. The resulting determinants are evaluated using the heat kernel techniques of arXiv:0911.5085. The answer factorises holomorphically, and the contributions from the various spin $s$ fields organise themselves into vacuum characters of the W_N symmetry. For the case of the hs(1,1) theory consisting of an infinite tower of massless higher spin particles, the resulting answer can be simply expressed in terms of (two copies of) the MacMahon function.Comment: 23 pages; v2: References adde