We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model, cascading gravity and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

Massive Gravity

We consider several aspects of unitary higher-dimensional conformal field theories (CFTs). We first study massive deformations that trigger a flow to a gapped phase. Deep inelastic scattering in the gapped phase leads to a convexity property of dimensions of spinning operators of the original CFT. We further investigate the dimensions of spinning operators via the crossing equations in the light-cone limit. We find that, in a sense, CFTs become free at large spin and 1/s is a weak coupling parameter. The spectrum of CFTs enjoys additivity: if two twists τ
 1, τ
 2 appear in the spectrum, there are operators whose twists are arbitrarily close to τ
 1 + τ
 2. We characterize how τ
 1 + τ
 2 is approached at large spin by solving the crossing equations analytically. We find the precise form of the leading correction, including the prefactor. We compare with examples where these observables were computed in perturbation theory, or via gauge-gravity duality, and find complete agreement. The crossing equations show that certain operators have a convex spectrum in twist space. We also observe a connection between convexity and the ratio of dimension to charge. Applications include the 3d Ising model, theories with a gravity dual, SCFTs, and patterns of higher spin symmetry breaking.

Convexity and Liberation at Large Spin

We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins $J \leq 2$. But, it can be fixed by adding an infinite tower of extra massive particles with higher spins, $J > 2$. In AdS theories this implies a constraint on the conformal anomaly coefficients $\left|{a - c \over c} \right| \lesssim {1 \over \Delta_{gap}^2}$ in terms of $\Delta_{gap}$, the dimension of the lightest single particle operator with spin $J > 2$. For inflation, or de Sitter-like solutions, it indicates the existence of massive higher spin particles if the gravity wave non-gaussianity deviates significantly from the one computed in the Einstein theory.

/pdf/causality-constraints-on-corrections-to-the-graviton-three-4pr7s5ztot.pdf

Causality Constraints on Corrections to the Graviton Three-Point Coupling

Aiming at nonexperts, the key mechanisms of higher-spin extensions of ordinary gravities in four dimensions and higher are explained. An overview of various no-go theorems for low-energy scattering of massless particles in flat spacetime is given. In doing so, a connection between the $S$-matrix and the Lagrangian approaches is made, exhibiting their relative advantages and weaknesses, after which potential loopholes for nontrivial massless dynamics are highlighted. Positive yes-go results for non-Abelian cubic higher-derivative vertices in constantly curved backgrounds are reviewed. Finally, how higher-spin symmetry can be reconciled with the equivalence principle in the presence of a cosmological constant leading to the Fradkin-Vasiliev vertices and Vasiliev's higher-spin gravity with its double perturbative expansion (in terms of numbers of fields and derivatives) is outlined.

How higher-spin gravity surpasses the spin-two barrier

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 propose a self-contained description of Vasiliev higher-spin theories with the emphasis on nonlinear equations. The main sections are supplemented with some additional material, including introduction to gravity as a gauge theory; the review of the Fronsdal formulation of free higher-spin fields; Young diagrams and tensors as well as sections with advanced topics. The shortest route to Vasiliev equations covers 40 pages. The general discussion is dimension independent, while the essence of the Vasiliev formulation is discussed on the base of the four-dimensional higher-spin theory. Three-dimensional and d-dimensional higher-spin theories follow the same logic.

Elements of Vasiliev theory

All correlation functions of conserved currents of the CFT that is dual to unbroken Vasiliev theory are found as invariants of higher-spin symmetry in the bulk of AdS. The conformal and higher-spin symmetry of the correlators as well as the conservation of currents are manifest, which also provides a direct link between the Maldacena-Zhiboedov result and higher-spin symmetries. Our method is in the spirit of AdS/CFT, though we never take any boundary limit or compute any bulk integrals. Boundary-to-bulk propagators are shown to exhibit an algebraic structure, living at the boundary of SpH(4), semidirect product of Sp(4) and the Heisenberg group. N-point correlation function is given by a product of N elements.

/pdf/exact-higher-spin-symmetry-in-cft-all-correlators-in-v748800b3y.pdf

Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory

We study the uniqueness of higher-spin algebras which are at the core of higher-spin theories in AdS and of CFTs with exact higher-spin symmetry, i.e. conserved tensors of rank greater than two. The Jacobi identity for the gauge algebra is the simplest consistency test that appears at the quartic order for a gauge theory. Similarly, the algebra of charges in a CFT must also obey the Jacobi identity. These algebras are essentially the same. Solving the Jacobi identity under some simplifying assumptions spelled out, we obtain that the Eastwood-Vasiliev algebra is the unique solution for d = 4 and d > 7. In 5d there is a one-parameter family of algebras that was known before. In particular, we show that the introduction of a single higher-spin gauge field/current automatically requires the infinite tower of higher-spin gauge fields/currents. The result implies that from all the admissible non-Abelian cubic vertices in AdSd, that have been recently classified for totally symmetric higher-spin gauge fields, only one vertex can pass the Jacobi consistency test. This cubic vertex is associated with a gauge deformation that is the germ of the Eastwood-Vasiliev’s higher-spin algebra.

On the uniqueness of higher-spin symmetries in AdS and CFT

We revisit the problem of interactions of higher-spin fields in flat space. We argue that all no-go theorems can be avoided by the light-cone approach, which results in more interaction vertices as compared to the usual covariant approaches. It is stressed that there exist two-derivative gravitational couplings of higher-spin fields. We show that some reincarnation of the equivalence principle still holds for higher-spin fields-the strength of gravitational interaction does not depend on spin. Moreover, it follows from the results by Metsaev that there exists a complete chiral higher-spin theory in four dimensions. We give a simple derivation of this theory and show that the four-point scattering amplitude vanishes. Also, we reconstruct the quartic vertex of the scalar field in the unitary higher-spin theory, which turns out to be perturbatively local.

https://iopscience.iop.org/article/10.1088/1751-8121/aa56e7/pdf

Evgeny Skvortsov

Papers

Elements of Vasiliev theory

Exact higher-spin symmetry in CFT: all correlators in unbroken Vasiliev theory

On the uniqueness of higher-spin symmetries in AdS and CFT

Light-front higher-spin theories in flat space

Geometric formulation for partially massless fields