Higher Spin Gravity Theory Research Articles

Finiteness of 3D higher spin gravity Landscape

Abstract We give Swampland constraints on the three dimensional Landscape of Anti-de Sitter higher spin gravity in the Chern-Simons formulation with connection valued in various split real forms of Lie algebras. We derive the finiteness conjecture by computing the upper bound on the rank of possible gauge groups then we refine it using the AdS distance conjecture. We discuss the implications of this Swampland constraint on the spectrum of higher spin gravity theories and we compare it with the gravitational exclusion principle required from BTZ black hole consideration to excerpt a constraint on the Chern-Simons level k.

A chiral limit for Chern-Simons-matter theories

Large N quasi-fermionic Chern-Simons-matter theories have an approximate higher-spin symmetry that strongly constrains their correlation functions. In particular, the 3-point functions for generic spins are combinations of 3 structures (with specific dependence on the positions and helicities), and the coupling-dependence of the coefficient of each structure is uniquely determined. In the past few years, several relations between different structures were found. In this paper we show that all the relations between the structures follow from (or, conversely, they imply) a specific form written by Skvortsov for the vertices of the dual higher-spin gravity theory on four-dimensional anti-de Sitter space, when written in spinor-helicity variables. The dual bulk theory has a specific limit where it simplifies and becomes a “chiral higher-spin gravity theory”, and we discuss what can be said about this limit in the dual Chern-Simons-matter theories, where it involves an analytic continuation to complex couplings.

W(0,b) algebra and the dual theory of 3D asymptotically flat higher spin gravity

Bondi-Metzner-Sachs (BMS) algebra in three spacetime dimensions can be deformed into a two parameter family of algebra known as W(a,b) algebra. For a=0, we show that other than W(0,−1), no other W(0,b) algebra admits a nondegenerate bilinear and thus one cannot have a Chern-Simons gauge theory formulation with them. However, they may appear in a three-dimensional gravity description, where we also need to have a spin 2 generator, that comes from the (a=0,b=−1) sector. In the present work, we have demonstrated that the asymptotic symmetry algebra of a spin-3 gravity theory on flat spacetime has both the W(0,−1) and W(0,−2) algebras as subalgebras. We have also constructed a dual boundary field theory for this higher spin gravity theory by using the Chern-Simons/Wess-Zumino-Witten correspondence. Published by the American Physical Society 2024

Geometric tool kit for higher-spin gravity (Part I): An introduction to the geometry of differential operators

This paper provides an introduction to the algebra and geometry of differential operators and jet bundles. Its point of view is guided by the leitmotiv that higher-spin gravity theories call for higher-order generalizations of Lie derivatives and diffeomorphisms. Nevertheless, the material covered here may be of general interest to anyone working on topics where geometrical (coordinate-free, global, generic) and mathematically rigorous definitions of differential operators are required.

Late-time correlation functions in dS3/CFT2 correspondence

We compute the late-time correlation functions on three-dimensional de Sitter spacetime for a higher-spin gravity theory. For this, we elaborate on the formulation to obtain the wave functional of universe from a dual conformal field theory, which is used to compute the late-time correlation functions. We argue that the relation to direct bulk Feynman diagram computations in the in-in formulation. We furthermore provide a precise prescription to construct a higher-spin dS3 holography as an analytic continuation of Gaberdiel-Gopakumar duality for AdS3. Part of results here were already reported in an earlier letter. We explain the details of their derivations and extend the analysis to more generic cases in this paper. Previously, we have examined two- and three-point functions and a simple four-point correlator at the leading order in Newton constant. Here we also evaluate more complicated four-point correlators. Finally, we study late-time correlators in an alternative limit of dS3/CFT2 with critical level coset, such as, two-point correlator on conical defect geometry. We also examine one-loop corrections to two-point correlator on dS3.

Massless scalars and higher-spin BMS in any dimension

Starting from the asymptotic kinematics of massless scalar fields near null infinity in any spacetime dimension, we build two higher-spin extensions of the Carrollian definition of the BMS group and its generalisations. The first extension exhibits conformal properties reminiscent of the singleton in Anti-de Sitter space. The second acts on the space of radiative solutions of the d’Alembert equation, i.e. on Sachs’s representation of BMS, which we relate to the scalar massless Poincaré representation and extend to any Carrollian manifold. The corresponding enveloping algebra is a higher-spin extension of BMS that can be interpreted as the asymptotic symmetry of a putative exotic higher-spin gravity theory around Minkowski spacetime. Along the way, we provide a pedagogical introduction to Carrollian geometry and its relation to BMS.

Supersymmetric localization on dS: sum over topologies

We find the exact quantum gravity partition function on the static patch of 3d de Sitter spacetime. We have worked in the Chern Simons formulation of 3d gravity. To obtain a non-perturbative result, we supersymmetrized the Chern Simons action and used the technique of supersymmetric localization. We have obtained an exact non-perturbative result for the spin-2 gravity case. We comment on the divergences present in the theory. We also comment on higher spin gravity theories and analyse the nature of divergences present in such theories.

Towards higher-spin AdS2/CFT1 holography

We aim at formulating a higher-spin gravity theory around AdS2 relevant for holography. As a first step, we investigate its kinematics by identifying the low-dimensional cousins of the standard higher-dimensional structures in higher-spin gravity such as the singleton, the higher-spin symmetry algebra, the higher-rank gauge and matter fields, etc. In particular, the higher-spin algebra is given here by [λ] and parameterized by a real parameter λ. The singleton is defined to be a Verma module of the AdS2 isometry subalgebra so (2, 1) ⊂ [λ] with conformal weight Delta =frac{1pm lambda }{2}. On the one hand, the spectrum of local modes is determined by the Flato-Fronsdal theorem for the tensor product of two such singletons. It is given by an infinite tower of massive scalar fields in AdS2 with ascending masses expressed in terms of λ. On the other hand, the higher-spin fields arising through the gauging of [λ] algebra do not propagate local degrees of freedom. Our analysis of the spectrum suggests that AdS2 higher-spin gravity is a theory of an infinite collection of massive scalars with fine-tuned masses, interacting with infinitely many topological gauge fields. Finally, we discuss the holographic CFT1 duals of the kinematical structures identified in the bulk.

Constraints for three-dimensional higher-spin interactions and conformal correlators

In the context of higher-spin holography, we compare the classification of cubic interaction vertices for higher-spin gravity theories in three dimensions to the possible three-point correlation functions of conserved higher-spin currents in two-dimensional conformal field theories. In both cases, the allowed structures are governed by triangle inequalities for the involved spins. It is established that higher-order correlators satisfy similar polygon inequalities and that the same inequalities are valid for higher-order continuations of cubic vertices in the three-dimensional higher-spin gravity.

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.

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.

Frobenius–Chern–Simons gauge theory

Given a set of differential forms on an odd-dimensional noncommutative manifold valued in an internal associative algebra , we show that the most general cubic covariant Hamiltonian action, without mass terms, is controlled by an -graded associative algebra with a graded symmetric nondegenerate bilinear form. The resulting class of models provide a natural generalization of the Frobenius–Chern–Simons model (FCS) that was proposed in (arXiv:1505.04957) as an off-shell formulation of the minimal bosonic four-dimensional higher spin gravity theory. If is unital and the -grading is induced from a Klein operator that is outer to a proper Frobenius subalgebra, then the action can be written on a form akin to topological open string field theory in terms of a superconnection valued in . We give a new model of this type based on a twisting of , which leads to self-dual complexified gauge fields on AdS4. If is 3-graded, the FCS model can be truncated consistently as to contain no zero-form constraints on-shell. Two examples thereof are a twisting of that yields the original model, and the Clifford algebra which provides an FCS formulation of the bosonic Konstein–Vasiliev model with gauge algebra .

Flato-Fronsdal theorem for higher-order singletons

We prove a generalised Flato-Fronsdal theorem for higher-order, scalar and spinor, singletons. In the resulting infinite tower of bulk higher-spin fields, we point out the occurrence of partially-massless fields of all depths. This leads us to conjecture a holographic duality between a higher-spin gravity theory around $AdS_{d+1}$ with the aforementioned spectrum of fields, and a free $CFT_d$ of fundamental (higher-order) scalar and spinor singletons. As a first check of this conjecture, we find that the total Casimir energy vanishes at one loop.

Spin-5 Casimir operator its three-point functions with two scalars

By calculating the second-order pole in the operator product expansion (OPE) between the spin-3 Casimir operator and the spin-4 Casimir operator known previously, the spin-5 Casimir operator is obtained in the coset model based on (A_{N-1}^{(1)} \oplus A_{N-1}^{(1)}, A_{N-1}^{(1)}) at level (k,1). This spin-5 Casimir operator consisted of the quintic, quartic (with one derivative) and cubic (with two derivatives) WZW currents contracted with SU(N) invariant tensors. The three-point functions with two scalars for all values of 't Hooft coupling in the large N limit were obtained by analyzing the zero-mode eigenvalue equations carefully. These three-point functions were dual to those in AdS_3 higher spin gravity theory with matter. Furthermore, the exact three-point functions that hold for any finite N and k are obtained. The zero mode eigenvalue equations for the spin-5 current in CFT coincided with those of the spin-5 field in asymptotic symmetry algebra of the higher spin theory on the AdS_3. This paper also describes the structure constant appearing in the spin-4 Casimir operator from the OPE between the spin-3 Casimir operator and itself for N=4, 5 in the more general coset minimal model with two arbitrary levels (k_1, k_2).

Metric-like formalism for matter fields coupled to 3D higher spin gravity

The action integral for a matter system composed of 0- and 2-forms, C and Bμν, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field C satisfies an equation of motion, , where Aμ and are the Chern–Simons gauge fields. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of Bμν, say Bϕr, remains non-vanishing and satisfies . These equations are the same as those for 3D (free) Vasiliev scalars, C and . The spin connection is eliminated by solving the equation of motion for the total action, and it is shown that in the resulting metric-like formalism, (BC)2 interaction terms are induced because of the torsion. The world-volume components of the matter field, C0, Cμ and C(μν), are introduced by contracting the local-frame index of C with those of the inverse vielbeins, and , which were defined by the present authors in (2013 Class. Quantum Grav. 30 035003). 3D higher spin gravity theory contains various metric-like fields. These metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric gμν and the spin-3 field ϕμνλ by means of the ϕ-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to , obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of gμν, ϕμνρ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields. It is also found that the action of the matter-coupled theory in the metric-like formalism has larger symmetry than that of the pure spin-3 gravity. The matter-coupled theory in the metric-like formalism is invariant under the ordinary diffeomorphisms, because the vielbein and the spin connection are covariant vectors of diffeomorphisms. They are also gauge fields for the local translation. In the pure spin-3 gravity this symmetry provides the ordinary diffeomorphism and the spin-3 transformation in the metric-like formalism. When the matter is coupled, the local translation yields a new symmetry in the metric-like formalism, which does not contain diffeomorphism.

Extended higher spin holography and Grassmannian models

We propose holographic dualities between higher spin gravity theories extended with Chan-Paton factor on AdS3 and a large N limit of two dimensional Grassmannian models with and without supersymmetry. These proposals are natural extensions of the duality without Chan-Paton factor, and the extensions are motivated by a higher dimensional version of the duality, which implies a possible relation to superstring theory via ABJ theory. As evidence for the proposals, we show that the free limit of the Grassmannian models have the higher spin symmetry expected from the dual gravity theory. Furthermore, we construct currents in the ’t Hooft limit of the supersymmetric Grassmannian model and compare them with the currents from the bulk theory. One-loop partition function of the supergravity theory is reproduced by the ’t Hooft limit of the Grassmannian model after decoupling a gauge sector.

Semi-classical unitarity in three-dimensional higher spin gravity for non-principal embeddings

Higher spin gravity in three dimensions is efficiently formulated as a Chern–Simons gauge-theory, typically with gauge algebra sl(N)⊕sl(N). The classical and quantum properties of the higher spin theory depend crucially on the embedding into the full gauge algebra of the sl(2)⊕sl(2) factor associated with gravity. It has been argued previously that non-principal embeddings do not allow for a semi-classical limit (large values of the central charge) consistent with unitarity. In this work we show that it is possible to circumvent these conclusions. Based upon the Feigin–Semikhatov generalization of the Polyakov–Bershadsky algebra, we construct infinite families of unitary higher spin gravity theories at certain rational values of the Chern–Simons level that allow arbitrarily large values of the central charge up to , thereby confirming a recent speculation by us (Afshar et al 2012 arXiv:1209.2860).

Exploring three-dimensional higher-spin supergravity based on sl(N|N − 1) Chern-Simons theories

We investigate various aspects of higher-spin anti-de Sitter supergravity in three dimensions as described by Chern-Simons theory based on the finite-dimensional superalgebra sl(N |N - 1), with the particular case of N = 3 as our prime example. This class of theories serves as a natural supersymmetrization of the higher-spin gravity theory based on sl(N) Chern-Simons theories. We demonstrate explicitly that the asymptotic symmetry algebra contains the N = 2 superconformal algebra in each sector. The appropriate Killing spinor equations are derived and used to classify existing and new classical solutions. We also discuss holonomy conditions, higher-spin black holes and conical defect spacetimes in this class of theories.

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.

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.