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 review recent progress in the construction of black holes in three dimensional higher spin gravity theories. Starting from spin-3 gravity and working our way toward the theory of an infinite tower of higher spins coupled to matter, we show how to harness higher spin gauge invariance to consistently generalize familiar notions of black holes. We review the construction of black holes with conserved higher spin charges and the computation of their partition functions to leading asymptotic order. In view of the anti-de Sitter/conformal field theory (CFT) correspondence as applied to certain vector-like conformal field theories with extended conformal symmetry, we successfully compare to CFT calculations in a generalized Cardy regime. A brief recollection of pertinent aspects of ordinary gravity is also given.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Higher spin theories and holography’.

Black holes in three dimensional higher spin gravity: a review

In these lectures I give an introduction to the quantum physics of black holes, including recent developments based on quantum information theory such as the rewall paradox and its various cousins. I also give an introduction to holography and the AdS/CFT correspondence, focusing on those aspects which are relevant for the black hole information problem.

Jerusalem Lectures on Black Holes and Quantum Information

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 study three-dimensional conformal field theories described by U(N) Chern–Simons theory at level k coupled to massless fermions in the fundamental representation. By solving a Schwinger–Dyson equation in light-cone gauge, we compute the exact planar free energy of the theory at finite temperature on ℝ2 as a function of the ’t Hooft coupling λ=N/k. Employing a dimensional reduction regularization scheme, we find that the free energy vanishes at |λ|=1; the conformal theory does not exist for |λ|>1. We analyze the operator spectrum via the anomalous conservation relation for higher spin currents, and in particular show that the higher spin currents do not develop anomalous dimensions at leading order in 1/N. We present an integral equation whose solution in principle determines all correlators of these currents at leading order in 1/N and present explicit perturbative results for all three-point functions up to two loops. We also discuss a light-cone Hamiltonian formulation of this theory where a W
 ∞ algebra arises. The maximally supersymmetric version of our theory is ABJ model with one gauge group taken to be U(1), demonstrating that a pure higher spin gauge theory arises as a limit of string theory.

http://strings.physics.du.ac.in/NSM-Talks/chern-simons.pdf

Chern–Simons theory with vector fermion matter

We study the thermal properties of the O(N) vector-like scalar theory in the singlet sector in 2+1 dimensions. This theory is conjectured to be the AdS/CFT dual of Vasiliev higher spin gravity. We find that a large N transition occurs but only at a very high temperature of order √ N. This corresponds to the bulk Planck energy. The transition signals a decrease in the number of degrees of freedom from that expected in the simple higher spin gas, due to relations among the O(N) bilinear invariants.

Xi Yin

Papers

Vector Models in the Singlet Sector at Finite Temperature