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 new class of conformal higher spin gravities in three dimensions, which extends the one by Pope and Townsend. The main new feature is that there are infinitely many examples of the new theories with a finite number of higher spin fields, much as in the massless case. The action has the Chern-Simons form for a higher spin extension of the conformal algebra. In general, the new theories contain Fradkin-Tseytlin fields with higher derivatives in the gauge transformations, which is reminiscent of partially-massless fields. A relation of the old and new theories to the parity anomaly is pointed out.

/pdf/new-conformal-higher-spin-gravities-in-3d-271xwq81a6.pdf

New Conformal Higher Spin Gravities in $3d$

We propose non-linear equations for the formal Type-B Higher Spin Gravity that is dual to the free fermion or to the Gross-Neveu model, depending on the boundary conditions. The equations are directly obtained from the first principles: the gauge invariance of the CFT partition function on an arbitrary background for single-trace operators. We also get equations describing propagation of certain mixed-symmetry fields over higher spin flat backgrounds.

/pdf/type-b-formal-higher-spin-gravity-2h6xxqkifj.pdf

Type-B formal higher spin gravity

We construct a new covariant action for “flat” self-dual gravity in four space-time 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.

/pdf/flat-self-dual-gravity-p91sl9nefe.pdf

Flat Self-dual Gravity

We calculate the anomalous dimensions of higher spin singlet currents in the critical O(N) vector model at order 1/N-2. The results are shown to be in agreement with the four-loop perturbative computation in phi(4) theory in 4-2 epsilon dimensions. It is known that the order 1/N anomalous dimensions of higher-spin currents happen to be the same in the Gross-Neveu and the critical vector model. On the contrary, the order 1/N-2 corrections are different. The results can also be interpreted as a prediction for the two-loop computation in the dual higher-spin gravity.

/pdf/higher-spin-currents-in-the-critical-o-n-vector-model-at-1-n-1q6rvjiwfe.pdf

Higher spin currents in the critical O(N) vector model at 1/N-2

We compute the parity violating three point amplitudes with one scalar leg in higher spin gravity and compare results with those of Chern-Simons matter theories. The three-point correlators of the free boson, free fermion, critical vector model and Gross-Neveu model are reproduced including the dependence on the Chern-Simons coupling. We also perform a simple test of the modified higher spin equations proposed in arXiv:1605.02662 [hep-th] and find that the results are consistent with the AdS/CFT correspondence.

Evgeny Skvortsov

Papers

New Conformal Higher Spin Gravities in $3d$

Type-B formal higher spin gravity

Flat Self-dual Gravity

Higher spin currents in the critical O(N) vector model at 1/N-2

Chern-Simons Matter Theories and Higher Spin Gravity