Higher Spin Gravity Research Articles

Rectangular W-algebras, extended higher spin gravity and dual coset CFTs

We analyze the asymptotic symmetry of higher spin gravity with M × M matrix valued fields, which is given by rectangular W-algebras with su(M) symmetry. The matrix valued extension is expected to be useful for the relation between higher spin gravity and string theory. With the truncation of spin as s = 2, 3,…, n, we evaluate the central charge c of the algebra and the level k of the affine currents with finite c, k. For the simplest case with n = 2, we obtain the operator product expansions among generators by requiring their associativity. We conjecture that the symmetry is the same as that of Grassmannian-like coset based on our proposal of higher spin holography. Comparing c, k from the both theories, we obtain the map of parameters. We explicitly construct low spin generators from the coset theory, and, in particular, we reproduce the operator product expansions of the rectangular W-algebra for n = 2. We interpret the map of parameters by decomposing the algebra in the coset description.

Multi-centered higher spin solutions from {mathcal{W}}_N conformal blocks

Motivated by the question of bulk localization in holography, we study the problem of constructing multi-centered solutions in higher spin gravity which describe point particles in the interior of AdS3. In the Chern-Simons formulation these take into account the backreaction after adding Wilson line sources. We focus on chiral solutions where only the left-moving sector is excited. In that case it is possible to choose a gauge where the dynamical variables are a set of Toda fields living in the bulk. The problem then reduces to solving the {{mathcal{A}}_N}_{-1} Toda equations with delta function sources, which in turn requires solving an associated monodromy problem. We show that this monodromy problem is equivalent to the monodromy problem for a particular {mathcal{W}}_N vacuum conformal block at large central charge. Therefore, knowledge of the {mathcal{W}}_N vacuum block determines the multi-centered solution. Our calculations go beyond the heavy-light approximation by including the backreaction of all higher spin particles.

Holographic quantization of linearized higher-spin gravity in the de Sitter causal patch

We study the dS/CFT duality between minimal type-A higher-spin gravity and the free Sp(2N) vector model. We consider the bulk spacetime as “elliptic” de Sitter space dS4/ℤ2, in which antipodal points have been identified. We apply a technique from arXiv:1509.05890, which extracts the quantum-mechanical commutators (or Poisson brackets) of the linearized bulk theory in an observable patch of dS4/ℤ2 directly from the boundary 2-point function. Thus, we construct the Lorentzian commutators of the linearized bulk theory from the Euclidean CFT. In the present paper, we execute this technique for the entire higher-spin multiplet, using a higher-spin-covariant language, which provides a promising framework for the future inclusion of bulk interactions. Aside from its importance for dS/CFT, our construction of a Hamiltonian structure for a bulk causal region should be of interest within higher-spin theory itself. The price we pay is a partial symmetry breaking, from the full dS group (and its higher-spin extension) to the symmetry group of an observable patch. While the boundary field theory plays a role in our arguments, the results can be fully expressed within a boundary particle mechanics. Bulk fields arise from this boundary mechanics via a version of second quantization.

Conformal higher-spin gravity: linearized spectrum = symmetry algebra

The linearized spectrum and the algebra of global symmetries of conformal higher-spin gravity decompose into infinitely many representations of the conformal algebra. Their characters involve divergent sums over spins. We propose a suitable regularization adapted to their evaluation and observe that their characters are actually equal. This result holds in the case of type-A and type-B (and their higher-depth generalizations) theories and confirms previous observations on a remarkable rearrangement of dynamical degrees of freedom in conformal higher-spin gravity after regularization.

Higher-Spin Symmetries and Deformed Schrödinger Algebra in Conformal Mechanics

The dynamical symmetries of 1+1-dimensional Matrix Partial Differential Equations with a Calogero potential (with/without the presence of an extra oscillatorial de Alfaro-Fubini-Furlan, DFF, term) are investigated. The first-order invariant differential operators induce several invariant algebras and superalgebras. Besides the sl(2)⊕u(1) invariance of the Calogero Conformal Mechanics, an osp2∣2 invariant superalgebra, realized by first-order and second-order differential operators, is obtained. The invariant algebras with an infinite tower of generators are given by the universal enveloping algebra of the deformed Heisenberg algebra, which is shown to be equivalent to a deformed version of the Schrödinger algebra. This vector space also gives rise to a higher-spin (gravity) superalgebra. We furthermore prove that the pure and DFF Matrix Calogero PDEs possess isomorphic dynamical symmetries, being related by a similarity transformation and a redefinition of the time variable.

On the explicit asymptotic W5 symmetry of 3D Chern-Simons higher spin AdS3 gravity

In this paper, we explicitly construct an asymptotic W5 symmetry algebra of the three-dimensional anti-de Sitter (AdS3) higher spin gravity. We use an sl(5,R)⊕sl(5,R) Lie algebra valued Chern-Simons gauge theory with a negative cosmological constant, and its asymptotic symmetry algebra is explicitly calculated as two copies of the classical W5 algebra with central charge c. Our results can be interpreted as a spin 5 extension of AdS3 gravity and a proof of how the higher spin Ward identities and the asymptotic W5 symmetry algebra is derived from the higher spin bulk field equations of motion. This higher spin asymptotic W5 symmetry algebra contains a finite number of conformal primary spin s: s = 2, 3, 4, 5. We also indicated how to introduce chemical potentials and holonomy conditions associated with these higher spin charges in AdS3 higher spin gravity in a manner that it preserves the asymptotic symmetry algebra.

Quantum Chiral Higher Spin Gravity.

An example of a higher spin gravity in four-dimensional flat space has recently been constructed by D. Ponomarev and E. D. Skvortsov, J. Phys. A, 50, 095401(2017). This theory is chiral and the action is written in the light-cone gauge. The theory has certain stringy features, e.g., admits Chan-Paton factors. We show that the theory is consistent, both at the classical and quantum level. Even though the interactions are nontrivial, due to the coupling conspiracy all tree level amplitudes vanish on shell. The loop corrections also vanish. Therefore, the full quantum S matrix is one and the theory is consistent with the numerous no-go theorems. This provides the first example of a (quantum) interacting higher spin gravity with an action. We argue that higher spin gravities in anti-de Sitter space should display the same features.

Generalized Chern–Simons higher-spin gravity theories in three dimensions

The coupling of spin-3 gauge fields to three-dimensional Maxwell and AdS-Lorentz gravity theories is presented. After showing how the usual spin-3 extensions of the AdS and the Poincaré algebras in three dimensions can be obtained as expansions of sl(3,R) algebra, the procedure is generalized so as to define new higher-spin symmetries. Remarkably, the spin-3 extension of the Maxwell symmetry allows one to introduce a novel gravity model coupled to higher-spin topological matter with vanishing cosmological constant, which in turn corresponds to a flat limit of the AdS-Lorentz case. We extend our results to define two different families of higher-spin extensions of three-dimensional Einstein gravity.

Character integral representation of zeta function in AdSd+1. Part II. Application to partially-massless higher-spin gravities

We compute the one-loop free energies of the type-Aℓ and type-Bℓ higher-spin gravities in (d + 1)-dimensional anti-de Sitter (AdSd+1) spacetime. For large d and ℓ, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper [arXiv:1805.05646] to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories (ℓ = 1) generalize to their partially-massless counterparts (arbitrary ℓ) in arbitrary dimensions.

Twisted Flato-Fronsdal theorem for higher-spin algebras

We explore the relation between the singleton and adjoint modules of higher-spin algebras via so(2, d) characters. In order to relate the tensor product of the singleton and its dual to the adjoint module, we consider a heuristic formula involving symmetrization over the variables of the character. We show that our formula reproduces correctly the adjoint-module character for type-A (and its high-order extensions) and type-B higher-spin gravity theories in any dimension. Implications and subtleties of this symmetrization prescription in other models are discussed.

Higher spin Chern–Simons theory and the super Boussinesq hierarchy

In this paper, we construct a map between a solution of supersymmetric Chern–Simons higher spin gravity based on the superalgebra [Formula: see text] with Lifshitz scaling and the [Formula: see text] super Boussinesq hierarchy. We show that under this map the time evolution equations of both theories coincide. In addition, we identify the Poisson structure of the Chern–Simons theory induced by gauge transformation with the second Hamiltonian structure of the super Boussinesq hierarchy.

Type-B formal higher spin gravity

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.

FRW and domain walls in higher spin gravity

We present exact solutions to Vasiliev’s bosonic higher spin gravity equations in four dimensions with positive and negative cosmological constant that admit an interpretation in terms of domain walls, quasi-instantons and Friedman-Robertson-Walker (FRW) backgrounds. Their isometry algebras are infinite dimensional higher-spin extensions of spacetime isometries generated by six Killing vectors. The solutions presented are obtained by using a method of holomorphic factorization in noncommutative twistor space and gauge functions. In interpreting the solutions in terms of Fronsdal-type fields in space-time, a field-dependent higher spin transformation is required, which is implemented at leading order. To this order, the scalar field solves Klein-Gordon equation with conformal mass in (A)dS4. We interpret the FRW solution with de Sitter asymptotics in the context of inflationary cosmology and we expect that the domain wall and FRW solutions are associated with spontaneously broken scaling symmetries in their holographic description. We observe that the factorization method provides a convenient framework for setting up a perturbation theory around the exact solutions, and we propose that the nonlinear completion of particle excitations over FRW and domain wall solutions requires black hole-like states.

Black holes from large N singlet models

The emergent nature of spacetime geometry and black holes can be directly probed in simple holographic duals of higher spin gravity and tensionless string theory. To this end, we study time dependent thermal correlation functions of gauge invariant observables in suitably chosen free large N gauge theories. At low temperature and on short time scales the correlation functions encode propagation through an approximate AdS spacetime while interesting departures emerge at high temperature and on longer time scales. This includes the existence of evanescent modes and the exponential decay of time dependent boundary correlations, both of which are well known indicators of bulk black holes in AdS/CFT. In addition, a new time scale emerges after which the correlation functions return to a bulk thermal AdS form up to an overall temperature dependent normalization. A corresponding length scale was seen in equal time correlation functions in the same models in our earlier work.

Quantizing higher-spin gravity in free-field variables

We study the formulation of massless higher-spin gravity on AdS3 in a gauge in which the fundamental variables satisfy free field Poisson brackets. This gauge choice leaves a small portion of the gauge freedom unfixed, which should be further quotiented out. We show that doing so leads to a bulk version of the Coulomb gas formalism for WN CFT’s: the generators of the residual gauge symmetries are the classical limits of screening charges, while the gauge-invariant observables are classical WN charges.Quantization in these variables can be carried out using standard techniques and makes manifest a remnant of the triality symmetry of W∞[λ]. This symmetry can be used to argue that the theory should be supplemented with additional matter content which is precisely that of the Prokushkin-Vasiliev theory. As a further application, we use our formulation to quantize a class of conical surplus solutions and confirm the conjecture that these are dual to specific degenerate WN primaries, to all orders in the large central charge expansion.

Supersymmetric dS/CFT

We put forward new explicit realisations of dS/CFT that relate mathcal{N} = 2 supersymmetric Euclidean vector models with reversed spin-statistics in three dimensions to specific supersymmetric Vasiliev theories in four-dimensional de Sitter space. The partition function of the free supersymmetric vector model deformed by a range of low spin deformations that preserve supersymmetry appears to specify a well-defined wave function with asymptotic de Sitter boundary conditions in the bulk. In particular we find the wave function is globally peaked at undeformed de Sitter space, with a low amplitude for strong deformations. This suggests that supersymmetric de Sitter space is stable in higher-spin gravity and in particular free from ghosts. We speculate this is a limiting case of the de Sitter realizations in exotic string theories.

Towards causal patch physics in dS/CFT

This contribution is a status report on a research program aimed at obtaining quantum-gravitational physics inside a cosmological horizon through dS/CFT, i.e. through a holographic description at past/future infinity of de Sitter space. The program aims to bring together two main elements. The first is the observation by Anninos, Hartman and Strominger that Vasiliev’s higher-spin gravity provides a working model for dS/CFT in 3+1 dimensions. The second is the proposal by Parikh, Savonije and Verlinde that dS/CFT may prove more tractable if one works in so-called “elliptic” de Sitter space – a folded-in-half version of global de Sitter where antipodal points have been identified. We review some relevant progress concerning quantum field theory on elliptic de Sitter space, higher-spin gravity and its holographic duality with a free vector model. We present our reasons for optimism that the approach outlined here will lead to a full holographic description of quantum (higher-spin) gravity in the causal patch of a de Sitter observer.

The holographic dual of the Penrose transform

We consider the holographic duality between type-A higher-spin gravity in AdS4 and the free U(N) vector model. In the bulk, linearized solutions can be translated into twistor functions via the Penrose transform. We propose a holographic dual to this transform, which translates between twistor functions and CFT sources and operators. We present a twistorial expression for the partition function, which makes global higher-spin symmetry manifest, and appears to automatically include all necessary contact terms. In this picture, twistor space provides a fully nonlocal, gauge-invariant description underlying both bulk and boundary spacetime pictures. While the bulk theory is handled at the linear level, our formula for the partition function includes the effects of bulk interactions. Thus, the CFT is used to solve the bulk, with twistors as a language common to both. A key ingredient in our result is the study of ordinary spacetime symmetries within the fundamental representation of higher-spin algebra. The object that makes these “square root” spacetime symmetries manifest becomes the kernel of our boundary/twistor transform, while the original Penrose transform is identified as a “square root” of CPT.

Noncommutative Wilson lines in higher-spin theory and correlation functions of conserved currents for free conformal fields

We first prove that, in Vasiliev’s theory, the zero-form charges studied in Sezgin E and Sundell P 2011 (arXiv:1103.2360 [hep-th]) and Colombo N and Sundell P 20 (arXiv:1208.3880 [hep-th]) are twisted open Wilson lines in the noncommutative Z space. This is shown by mapping Vasiliev’s higher-spin model on noncommutative Yang–Mills theory. We then prove that, prior to Bose-symmetrising, the cyclically-symmetric higher-spin invariants given by the leading order of these n-point zero-form charges are equal to corresponding cyclically-invariant building blocks of n-point correlation functions of bilinear operators in free conformal field theories (CFT) in three dimensions. On the higher spin gravity side, our computation reproduces the results of Didenko V and Skvortsov E 2013 J. High Energy Phys. JHEP04(2013)158 using an alternative method amenable to the computation of subleading corrections obtained by perturbation theory in normal order. On the free CFT side, our proof involves the explicit computation of the separate cyclic building blocks of the correlation functions of n conserved currents in arbitrary dimension using polarization vectors, which is an original result. It is shown to agree, for , with the results obtained in Gelfond O A and Vasiliev M A 2013 Nucl. Phys. B 876 871–917 in various dimensions and where polarization spinors were used.

Higher Spin Matrix Models

We propose a hybrid class of theories for higher spin gravity and matrix models, i.e., which handle simultaneously higher spin gravity fields and matrix models. The construction is similar to Vasiliev’s higher spin gravity, but part of the equations of motion are provided by the action principle of a matrix model. In particular, we construct a higher spin (gravity) matrix model related to type IIB matrix models/string theory that have a well defined classical limit, and which is compatible with higher spin gravity in A d S space. As it has been suggested that higher spin gravity should be related to string theory in a high energy (tensionless) regime, and, therefore to M-Theory, we expect that our construction will be useful to explore concrete connections.