We study the constraints imposed by the existence of a single higher spin conserved current on a three-dimensional conformal field theory (CFT). A single higher spin conserved current implies the existence of an infinite number of higher spin conserved currents. The correlation functions of the stress tensor and the conserved currents are then shown to be equal to those of a free field theory. Namely a theory of N free bosons or free fermions. This is an extension of the Coleman–Mandula theorem to CFT’s, which do not have a conventional S-matrix. We also briefly discuss the case where the higher spin symmetries are ‘slightly’ broken.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

https://iopscience.iop.org/article/10.1088/1751-8113/46/21/214011/pdf

Constraining conformal field theories with a higher spin symmetry

We study three dimensional O(N)
 k
 and U(N)
 k
 Chern-Simons theories coupled to a scalar field in the fundamental representation, in the large N limit. For infinite k this is just the singlet sector of the O(N) (U(N)) vector model, which is conjectured to be dual to Vasiliev’s higher spin gravity theory on AdS
 4. For large k and N we obtain a parity-breaking deformation of this theory, controlled by the ’t Hooft coupling λ = 4πN/k. For infinite N we argue (and show explicitly at two-loop order) that the theories with finite λ are conformally invariant, and also have an exactly marginal (ϕ
 2)3 deformation. For large but finite N and small ’t Hooft coupling λ, we show that there is still a line of fixed points parameterized by the ’t Hooft coupling λ. We show that, at infinite N, the interacting non-parity-invariant theory with finite λ has the same spectrum of primary operators as the free theory, consisting of an infinite tower of conserved higher-spin currents and a scalar operator with scaling dimension Δ = 1; however, the correlation functions of these operators do depend on λ. Our results suggest that there should exist a family of higher spin gravity theories, parameterized by λ, and continuously connected to Vasiliev’s theory. For finite N the higher spin currents are not conserved.

d = 3 bosonic vector models coupled to Chern-Simons gauge theories

We propose a way to classify the local form of all bosonic supersymmetric configurations of D = 11 supergravity, using the differential forms that can be constructed as bi-linears from the Killing spinors. We show that the most general bosonic geometries either have a privileged SU(5) or a (Spin(7)8) × structure, depending on whether the Killing vector constructed from the Killing spinor is timelike or null, respectively. In the time-like case we derive the general local form of the geometry and show that it is almost completely determined by a certain SU(5) structure on the ten-dimensional space orthogonal to the orbits of the Killing vector. We also deduce what further conditions must be imposed in order that the equations of motion are satisfied. We illustrate the formalism with some known solutions and also present some new solutions including a rotating generalisation of the resolved membrane solutions and generalisations of the recently constructed D = 11 Godel solution. We also prove some general vanishing theorems for compactifications with flux.

The geometry of D = 11 Killing spinors

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.

https://iopscience.iop.org/article/10.1088/0264-9381/30/10/104003/pdf

Constraining conformal field theories with a slightly broken higher spin symmetry

We present supergravity solutions for $1/8$-supersymmetric black supertubes with three charges and three dipoles. Their reduction to five dimensions yields supersymmetric black rings with regular horizons and two independent angular momenta. The general solution contains seven independent parameters and provides the first example of nonuniqueness of supersymmetric black holes. In ten dimensions, the solutions can be realized as D1-D5-P black supertubes. We also present a worldvolume construction of a supertube that exhibits three dipoles explicitly. This description allows an arbitrary cross section but captures only one of the angular momenta.

/pdf/supersymmetric-black-rings-and-three-charge-supertubes-12gnbxk3nn.pdf

Supersymmetric black rings and three-charge supertubes

The Many Faces of the Superworld

Symmetries of the interacting gauge-covariant bosonic string

Low-energy models with two supersymmetries

We construct a covariant bound on the energy-momentum of the M-fivebrane which is saturated by all supersymmetric configurations. This leads to a generalised notion of a calibrated geometry for M-fivebranes when the worldvolume gauge field is non-zero. The generalisation relevant for Dp-branes is also given.

Peter C. West

Papers

The Many Faces of the Superworld

Symmetries of the interacting gauge-covariant bosonic string

Low-energy models with two supersymmetries

A Calibration Bound for the M-Theory Fivebrane