Higher Spin Gravity Research Articles

Towards a gravity dual of the unitary Fermi gas

Inspired by the method of null dimensional reduction and by the holographic correspondence between Vasiliev's higher-spin gravity and the critical O(N) model, a bulk dual of the unitary and the ideal non-relativistic Fermi gases is proposed.

Higher spin black holes from CFT

Higher spin gravity in three dimensions has explicit black holes solutions, carrying higher spin charge. We compute the free energy of a charged black hole from the holographic dual, a 2d CFT with extended conformal symmetry, and find exact agreement with the bulk thermodynamics. In the CFT, higher spin corrections to the free energy can be calculated at high temperature from correlation functions of W-algebra currents.

Towards non-AdS holography in 3-dimensional higher spin gravity

We take the first steps towards non-AdS holography in higher spin gravity. Namely, we propose a variational principle for generic 3-dimensional higher spin gravity that accommodates asymptotic backgrounds beyond AdS, like asymptotically Schrodinger, Lifshitz or warped AdS spacetimes. As examples we study in some detail the four sl(2) embeddings of spin-4 gravity and provide associated geometries, including an asymptotic Lifshitz black hole.

D = 3 bosonic vector models coupled to Chern-Simons gauge theories

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 lambda = 4 \pi N / k. For infinite N we argue (and show explicitly at two-loop order) that the theories with finite lambda are conformally invariant, and also have an exactly marginal (\phi^2)^3 deformation. For large but finite N and small 't Hooft coupling lambda, we show that there is still a line of fixed points parameterized by the 't Hooft coupling lambda. We show that, at infinite N, the interacting non-parity-invariant theory with finite lambda 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 \Delta=1; however, the correlation functions of these operators do depend on lambda. Our results suggest that there should exist a family of higher spin gravity theories, parameterized by lambda, and continuously connected to Vasiliev's theory. For finite N the higher spin currents are not conserved.

Light‐front higher‐helicity interactions

AbstractThe reformulation, using Fock space vertex operators, of the original light‐front cubic interaction terms for higher spin gauge fields is reviewed with comments on quantum higher spin gravity. The formalism is generalized to all orders in the interaction. The ensuing recursive equations for the higher order vertices, if they can be explicitly solved, will encode all interaction data into a denumerable set of rational functions of p+, the overall transverse momentum structure being fixed already at the kinematical level. A more thorough exposition can be found in the archive. This work was presented at the XVII European Workshop on String Theory, Padua, 5–9 September 2011.

The coset spin-4 Casimir operator and its three-point functions with scalars

We find the GKO coset construction of the dimension 4 Casimir operator that contains the quartic WZW currents contracted with completely symmetric SU(N) invariant tensors of ranks 4, 3, and 2. The requirements, that the operator product expansion with the diagonal current is regular and it should be primary under the coset Virasoro generator of dimension 2, fix all the coefficients in spin-4 current, up to two unknown coefficients. The operator product expansion of coset primary spin-3 field with itself fixes them completely. We compute the three-point functions with scalars for all values of the 't Hooft coupling in the large N limit. At fixed 't Hooft coupling, these three-point functions are dual to that found by Chang and Yin recently in the undeformed AdS_3 bulk theory (higher spin gravity with matter).

Aspects of three-dimensional spin-4 gravity

We discuss some interesting holographical aspects of three-dimensional higher-spin gravity with a negative cosmological constant in the framework of SL(4, R) \times SL(4, R) Chern-Simons theory. Using a recently found technique, we construct explicitly a solution that can be interpreted as spin-4 generalization of the BTZ solution, and demonstrate how W_4 symmetry and the higher-spin Ward identities arise from the bulk equations of motion coupled to spin-3 and spin-4 currents. We match the eigenvalues of a Wilson loop along the time-like direction of the BTZ to that of the spin-4 solution, and show that this yields remarkably consistent gravitational thermodynamics for the latter. This furnishes an important, concrete supporting example for a recent proposal to understand spacetime geometries in three-dimensional higher-spin gravity formulated via SL(N, R) \times SL(N, R) Chern-Simons theories.

Symmetries and currents of the ideal and unitary Fermi gases

Abstract The maximal algebra of symmetries of the free single-particle Schrödinger equation is determined and its relevance for the holographic duality in non-relativistic Fermi systems is investigated. This algebra of symmetries is an infinite dimensional extension of the Schrödinger algebra, it is isomorphic to the Weyl algebra of quantum observables, and it may be interpreted as a non-relativistic higher-spin algebra. The associated infinite collection of Noether currents bilinear in the fermions are derived from their relativistic counterparts via a light-like dimensional reduction. The minimal coupling of these currents to background sources is rewritten in a compact way by making use of Weyl quantisation. Pushing forward the similarities with the holographic correspondence between the minimal higher-spin gravity and the critical O(N ) model, a putative bulk dual of the unitary and the ideal Fermi gases is briefly discussed.

Black holes and singularity resolution in higher spin gravity

We investigate higher spin theories of gravity in three dimensions based on the gauge group SL(N,R)*SL(N,R). In these theories the usual diffeomorphism symmetry is enhanced to include higher spin gauge transformations under which traditional geometric notions of curvature and causality are no longer invariant. This implies, for example, that apparently singular geometries can be rendered smooth by a gauge transformation, much like the resolution of orbifold singularities in string theory. The classical solutions, including the recently constructed higher spin black hole, are characterized by their holonomies around the non-contractible cycles of space-time. The black hole solutions are shown to be gauge equivalent to a BTZ black hole which is charged under a set of U(1) Chern-Simons fields. Nevertheless, depending on the choice of embedding of the gravitational gauge group, the space-time geometry may be non-trivial. We study in detail the N=3 example, where this observation allows us to find a gauge where the black hole geometry takes a simple form and the thermodynamic properties can be studied.

One loop partition function for topologically massive higher spin gravity

We calculate the one loop partition function for topologically massive higher spin gravity (TMHSG) for arbitrary spin by taking the spin-3 TMHSG action constructed in arXiv:1107.0915 and subsequently generalising it for an arbitrary spin. We find that the final result can be put into a product form which cannot be holomorphically factorised. This is consistent with the expectation of a logarithmic CFT dual for TMHSG with boundary conditions that retain all normalisable modes in the bulk.

Families of exact solutions to Vasiliev’s 4D equations with spherical, cylindrical and biaxial symmetry

We provide Vasiliev's four-dimensional bosonic higher-spin gravities with six families of exact solutions admitting two commuting Killing vectors. Each family contains a subset of generalized Petrov Type-D solutions in which one of the two so(2) symmetries enhances to either so(3) or so(2,1). In particular, the spherically symmetric solutions are static and we expect one of them to be gauge-equivalent to the extremal Didenko-Vasiliev solution given in arXiv:0906.3898. The solutions activate all spins and can be characterized either via generalized electric and magnetic charges defined asymptotically in weak-field regions or via the values of fully higher-spin gauge-invariant observables given by on-shell closed zero-forms. The solutions are obtained by combining the gauge-function method with separation of variables in twistor space via expansion of the Weyl zero-form in Di-Rac supersingleton projectors times deformation parameters in a fashion that is suggestive of a generalized electromagnetic duality.

An action principle for Vasiliev’s four-dimensional higher spin gravity

We provide Vasiliev’s fully nonlinear equations of motion for bosonic higher spin gauge fields in four spacetime dimensions with an action principle. We first extend Vasiliev’s original system with differential forms in degrees higher than 1. We then derive the resulting duality-extended equations of motion from a variational principle based on a generalized Hamiltonian sigma-model action. The generalized Hamiltonian contains two types of interaction freedoms: one, a set of functions that appears in the Q-structure of the generalized curvatures of the odd forms in the duality-extended system; and the other, a set depending on the Lagrange multipliers, encoding a generalized Poisson structure, i.e. a set of polyvector fields of rank 2 or higher in target space. We find that at least one of the two sets of interaction-freedom functions must be linear in order to ensure gauge invariance. We discuss consistent truncations to the minimal type A and B models (with only even spins), spectral flows on-shell and provide boundary conditions on fields and gauge parameters that are compatible with the variational principle and that make the duality-extended system equivalent, on-shell, to Vasiliev’s original system.

Collective dipole model of AdS/CFT and higher spin gravity

We formulate a first quantized construction of the AdSd + 1/CFTd correspondence using the bi-local representation of the free d-dimensional large-N vector model. The earlier reconstruction of AdS4 higher spin gravity provides a scheme where the AdS spacetime (and higher spin fields) is given by the composite bi-local fields. The underlying first quantized, world-sheet picture is extracted in the present work and generalized to any dimension. A higher spin AdS particle model is shown to emerge from the collective bi-particle system of Minkowski particles through a canonical transformation. As such, this construction provides a simple explicit mechanism of the AdS/CFT correspondence.

Partition functions of higher spin black holes and their CFT duals

We find black hole solutions of D=3 higher-spin gravity in the hs[\lambda] + hs[\lambda] Chern-Simons formulation. These solutions have a spin-3 chemical potential, and carry nonzero values for an infinite number of charges of the asymptotic W_{\infty}[\lambda] symmetry. Applying a previously developed set of rules for ensuring smooth solutions, we compute the black hole partition function perturbatively in the chemical potential. At \lambda =0, 1 we compare our result against boundary CFT computations involving free bosons and fermions, and find perfect agreement. For generic \lambda\ we expect that our gravity result will match the partition function of the coset CFTs conjectured by Gaberdiel and Gopakumar to be dual to these bulk theories.

Twistor space observables and quasi-amplitudes in 4D higher-spin gravity

Vasiliev equations facilitate globally defined formulations of higher-spin gravity in various correspondence spaces associated with different phases of the theory. In the four-dimensional case this induces a map from a generally covariant formulation in spacetime with higher-derivative interactions to a formulation in terms of a deformed symplectic structure on a noncommutative doubled twistor space, sending spacetime boundary conditions to various sectors of an associative star-product algebra. We look at observables given by integrals over twistor space defining composite zero-forms in spacetime that do not break any local symmetries and that are closed on shell. They can be evaluated locally in spacetime and interpreted as building blocks for dual amplitudes. To regularize potential twistor-space divergencies arising in their curvature expansion, we propose a closed-contour prescription that respects associativity and hence higher-spin gauge symmetry. As a sample calculation, we examine next-to-leading corrections to quasi-amplitudes for twistor-space plane waves, and find cancellations that we interpret using transgression properties in twistor space.

Spacetime geometry in higher spin gravity

Higher spin gravity is an interesting toy model of stringy geometry. Particularly intriguing is the presence of higher spin gauge transformations that redefine notions of invariance in gravity: the existence of event horizons and singularities in the metric become gauge dependent. In previous work, solutions of spin-3 gravity in the SL(3,R) × SL(3,R) Chern-Simons formulation were found, and were proposed to play the role of black holes. However, in the gauge employed there, the spacetime metric describes a traversable wormhole connecting two asymptotic regions, rather than a black hole. In this paper, we show explicitly that under a higher spin gauge transformation these solutions can be transformed to describe black holes with manifestly smooth event horizons, thereby changing the spacetime causal structure. A related aspect is that the Chern-Simons theory admits two distinct AdS3 vacua with different asymptotic W-algebra symmetries and central charges. We show that these vacua are connected by an explicit, Lorentz symmetry-breaking RG flow, of which our solutions represent finite temperature generalizations. These features will be present in any SL(N,R) × SL(N,R) Chern-Simons theory of higher spins.

Topologically massive higher spin gravity

We look at the generalisation of topologically massive gravity (TMG) to higher spins, specifically spin-3. We find a special "chiral" point for the spin-three, analogous to the spin-two example, which actually coincides with the usual spin-two chiral point. But in contrast to usual TMG, there is the presence of a non-trivial trace and its logarithmic partner at the chiral point. The trace modes carry energy opposite in sign to the traceless modes. The logarithmic partner of the traceless mode carries negative energy indicating an instability at the chiral point. We make several comments on the asymptotic symmetry and its possible deformations at this chiral point and speculate on the higher spin generalisation of LCFT2 dual to the spin-3 massive gravity at the chiral point.

Partition functions of holographic minimal models

The partition function of the W_N minimal model CFT is computed in the large N 't Hooft limit and compared to the spectrum of the proposed holographic dual, a 3d higher spin gravity theory coupled to massive scalar fields. At finite N, the CFT contains additional light states that are not visible in the perturbative gravity theory. We carefully define the large N limit, and give evidence that, at N = infinity, the additional states become null and decouple from all correlation functions. The surviving states are shown to match precisely (for all values of the 't Hooft coupling) with the spectrum of the higher spin gravity theory. The agreement between bulk and boundary is partially explained by symmetry considerations involving the conjectured equivalence between the W_N algebra in the large N limit and the higher spin algebra of the Vasiliev theory.

Minimal model holography for SO(2N)

A duality between the large N 't Hooft limit of the WD_N minimal model CFTs and a higher spin gravity theory on AdS3 is proposed. The gravity theory has massless spin fields of all even spins s=2,4,6,..., as well as two real scalar fields whose mass is determined by the 't Hooft parameter of the CFT. We show that, to leading order in the large N limit, the 1-loop partition function of the higher spin theory matches precisely with the CFT partition function.

Higher spin black holes

We study classical solutions of three dimensional higher spin gravity in the Chern-Simons formulation. We find solutions that generalize the BTZ black hole and carry spin-3 charge. The black hole entropy formula yields a result for the asymptotic growth of the partition function at finite spin-3 chemical potential. Along the way, we develop technology for computing AdS/CFT correlation functions involving higher spin operators.