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 give a complete classification of dynamical invariants in 3d and 4d Higher Spin Gravity models, with some comments on arbitrary d. These include holographic correlation functions, interaction vertices, on-shell actions, conserved currents, surface charges, and some others. Surprisingly, there are a good many conserved p-form currents with various p. The last fact, being in tension with ‘no nontrivial conserved currents in quantum gravity’ and similar statements, gives an indication of hidden integrability of the models. Our results rely on a systematic computation of Hochschild, cyclic, and Chevalley-Eilenberg cohomology for the corresponding higher spin algebras. A new invariant in Chern-Simons theory with the Weyl algebra as gauge algebra is also presented.

/pdf/characteristic-cohomology-and-observables-in-higher-spin-1swc7gx37q.pdf

Characteristic cohomology and observables in higher spin gravity

A new approach to the construction of formal deformations of associative algebras is proposed. It exploits the machinery of injective resolutions of an associative algebra $A$ in the category of $A$-bimodules. Specifically, we show that certain first-order deformations of $A$ extend to all orders and we derive explicit recurrent formulas determining this extension. In physical terms, this may be regarded as the deformation quantization of noncommutative Poisson structures on $A$.

Noncommutative deformation quantization via injective resolutions

We propose a simple injective resolution for the Hochschild complex of the Weyl algebra. By making use of this resolution, we derive explicit expressions for nontrivial cocycles of the Weyl algebra with coefficients in twisted bimodules as well as for the smash products of the Weyl algebra and a finite group of linear symplectic transformations. A relationship with the higher-spin field theory is briefly discussed.

Hochschild cohomology of the Weyl algebra and Vasiliev's equations

https://link.springer.com/content/pdf/10.1007/JHEP03(2023)204.pdf

On (spinor)-helicity and bosonization in AdS_4/CFT_3

We construct a new covariant action for "flat" self-dual gravity in four spacetime dimensions. The action has just one term, but when expanded around an appropriate background gives rise to a kinetic term and a cubic interaction. Upon imposing the light-cone gauge, the action reproduces the expected chiral interaction of Siegel. The new action is in many ways analogous to the known covariant action for self-dual Yang-Mills theory. There is also a sense in which the new self-dual gravity action exhibits the double copy of self-dual Yang-Mills structure.

Evgeny Skvortsov

Papers

Characteristic cohomology and observables in higher spin gravity

Noncommutative deformation quantization via injective resolutions

Hochschild cohomology of the Weyl algebra and Vasiliev's equations

On (spinor)-helicity and bosonization in AdS_4/CFT_3

Flat Self-dual Gravity