Higher Spin Theories Research Articles

Critical Sp(N ) models in 6 − ϵ dimensions and higher spin dS/CFT

Theories of anti-commuting scalar fields are non-unitary, but they are of interest both in statistical mechanics and in studies of the higher spin de Sitter/Conformal Field Theory correspondence. We consider an Sp(N ) invariant theory of N anti-commuting scalars and one commuting scalar, which has cubic interactions and is renormalizable in 6 dimensions. For any even N we find an IR stable fixed point in 6 − ϵ dimensions at imaginary values of coupling constants. Using calculations up to three loop order, we develop ϵ expansions for several operator dimensions and for the sphere free energy F . The conjectured F -theorem is obeyed in spite of the non-unitarity of the theory. The 1/N expansion in the Sp(N ) theory is related to that in the corresponding O(N ) symmetric theory by the change of sign of N . Our results point to the existence of interacting non-unitary 5-dimensional CFTs with Sp(N ) symmetry, where operator dimensions are real. We conjecture that these CFTs are dual to the minimal higher spin theory in 6-dimensional de Sitter space with Neumann future boundary conditions on the scalar field. For N = 2 we show that the IR fixed point possesses an enhanced global symmetry given by the super-group OSp(1|2). This suggests the existence of OSp(1|2) symmetric CFTs in dimensions smaller than 6. We show that the 6 − ϵ expansions of the scaling dimensions and sphere free energy in our OSp(1|2) model are the same as in the q → 0 limit of the q-state Potts model.

Towards an exact frame formulation of conformal higher spins in three dimensions

In this note we discuss some aspects of the frame formulation of conformal higher spins in three dimensions. We give some exact formulae for the coupled spin two - spin three part of the full higher spin theory and propose a star product Lagrangian for all spins from two and up. Since there is no consistent Lagrangian formulation based on the Poisson bracket we start the construction from the field equations in this approximation of the star product. The higher spin algebra is then realized in terms of classical variables which leads to certain important simplifications that we take advantage of. The suggested structure of the all-spin Lagrangian given here is, however, obtained using an expansion of the star product beyond the Poisson bracket in terms of multi-commutators and the Lagrangian should be viewed as a starting point for the derivation of the full theory based on a star product. How to do this is explained as well as how to include the coupling to scalar fields. We also comment on the AdS/CFT relation to four dimensions.

Connection versus metric description for non-AdS solutions in higher-spin theories

We consider recently-constructed solutions of three-dimensional Chern–Simons theories with non-relativistic symmetries. Solutions of the Chern–Simons theories can generically be mapped to solutions of a gravitational theory with a higher-spin gauge symmetry. However, we will show that some of the non-relativistic solutions are not equivalent to metric solutions, as this mapping fails to be invertible. We also show that these Chern–Simons solutions always have a global symmetry. We argue that these results pose a challenge to constructing a duality relating these solutions to field theories with non-relativistic symmetries.

Some aspects of holographic W-gravity

We use the Chern-Simons formulation of higher spin theories in three dimensions to study aspects of holographic W-gravity. Concepts which were useful in studies of pure bulk gravity theories, such as the Fefferman-Graham gauge and the residual gauge transformations, which induce Weyl transformations in the boundary theory and their higher spin generalizations, are reformulated in the Chern-Simons language. Flat connections that correspond to conformal and lightcone gauges in the boundary theory are considered.

Identification of bulk coupling constant in higher spin/ABJ correspondence

We study the conjectured duality between the $\mathcal{N}=6$ Vasiliev higher spin theory on $AdS_4$ and 3d $\mathcal{N}=6$ superconformal Chern-Simons matter theory known as the ABJ theory. We discuss how the parameters in the ABJ theory should be related to the bulk coupling constant in the Vasiliev theory. For this purpose, we compute two-point function of stress tensor in the ABJ theory by using supersymmmetry localization. Our result justifies the proposal by arXiv:1504.00365 and determine the unknown coefficient in the previous work.

Higher spins in the symmetric orbifold of K3

The symmetric orbifold of K3 is believed to be the conformal field theory (CFT) dual of string theory on ${\mathrm{AdS}}_{3}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{3}\ifmmode\times\else\texttimes\fi{}\mathrm{K}3$ at the tensionless point. For the case when the K3 is described by the orbifold ${\mathbb{T}}^{4}/{\mathbb{Z}}_{2}$, we identify a subsector of the symmetric orbifold theory that is dual to a higher spin theory on ${\mathrm{AdS}}_{3}$. We analyze how the Bogomol'nyi-Prasad-Sommerfield (BPS) spectrum of string theory can be described from the higher spin perspective and determine which single-particle BPS states are accounted for by the perturbative higher spin theory.

New unfolded higher spin systems in AdS3

We investigate the unfolded formulation of bosonic Lorentz tensor fields of arbitrary spin in containing a parity breaking mass parameter. They include deformations of the linearizations of the Prokushkin–Vasiliev higher spin theory around its critical points. They also provide unfolded formulations of linearized topologically massive higher spin fields including their critical versions. The gauge invariant degrees of freedom are captured by infinite towers of zero forms. We also introduce two inequivalent sets of gauge potentials given by trace constrained Fronsdal fields and trace unconstrained metric-like fields.

Gauge theory formulations for continuous and higher spin fields

We consider a gauge theory action for continuous spin particles formulated in a spacetime enlarged by an extra coordinate recently proposed by Schuster and Toro. It requires one scalar gauge field and has two local symmetries. We show that the local symmetries are reducible in the sense that the parameters also have a local symmetry. Using reducibility we get an action which has two scalar gauge fields and a reducible but simpler local symmetry. We then show how this action and equations of motion are related to previously proposed formulations for continuous spin particles and higher spin theories. We also discuss the physical contents of each formulation to reveal the physical degrees of freedom.

On higher spin partition functions

We observe that the partition function of the set of all free massless higher spins s = 0, 1, 2, 3,... in flat space is equal to one: the ghost determinants cancel against the ‘physical’ ones or, equivalently, the (regularized) total number of degrees of freedom vanishes. This reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory. The Z = 1 property extends also to the AdS background, i.e. the 1-loop vacuum partition function of Vasiliev theory is equal to 1 (assuming a particular regularization of the sum over spins); this was noticed earlier as a consistency requirement for the vectorial AdS/CFT duality. We find that Z = 1 is true also in the conformal higher spin theory (with higher-derivative kinetic terms) expanded near flat or conformally flat S4 background. We also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2-derivative kinetic term but only scalar gauge invariance in flat 4d space. This non-unitary theory has Weyl-invariant action in curved background and it corresponds to ‘partially massless’ field in AdS5. We discuss in detail the special case of s = 2 (or ‘conformal graviton’), compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions.

Star-product functions in higher-spin theory and locality

Properties of the functional classes of star-product elements associated with higher-pin gauge fields and gauge parameters are elaborated. Cohomological interpretation of the nonlinear higher-spin equations is given. An algebra $$ \mathrm{\mathscr{H}} $$ , where solutions of the nonlinear higher-spin equations are valued, is found. A conjecture on the classes of star-product functions underlying (non)local maps and gauge transformations in the nonlinear higher-spin theory is proposed.

On classical solutions of 4d supersymmetric higher spin theory

We present a simple construction of solutions to the supersymmetric higher spin theory based on solutions to bosonic theories. We illustrate this for the case of the Didenko-Vasiliev solution in arXiv:0906.3898, for which we have found a striking simplification where the higher-spin connection takes the vacuum value. Studying these solutions further, we check under which conditions they preserve some supersymmetry in the bulk, and when they are compatible with the boundary conditions conjectured to be dual to certain 3d SUSY Chern-Simons-matter theories. We perform the analysis for a variety of theories with $\mathcal{N}$ = 2, $\mathcal{N}$ = 3, $\mathcal{N}$ = 4 and $\mathcal{N}$ = 6 and find a rich spectrum of $1/4$, $1/3$ and $1/2$-BPS solutions.

Deformed twistors and higher spin conformal (super-)algebras in four dimensions

Massless conformal scalar field in six dimensions corresponds to the minimal unitary representation (minrep) of the conformal group SO(6,2). This minrep admits a family of deformations labelled by the spin t of an SU(2)_T group, which is the 6d analog of helicity in four dimensions. These deformations of the minrep of SO(6,2) describe massless conformal fields that are symmetric tensors in the spinorial representation of the 6d Lorentz group. The minrep and its deformations were obtained by quantization of the nonlinear realization of SO(6,2) as a quasiconformal group in arXiv:1005.3580. We give a novel reformulation of the generators of SO(6,2) for these representations as bilinears of deformed twistorial oscillators which transform nonlinearly under the Lorentz group SO(5,1) and apply them to define higher spin algebras and superalgebras in AdS_7. The higher spin (HS) algebra of Fradkin-Vasiliev type in AdS_7 is simply the enveloping algebra of SO(6,2) quotiented by a two-sided ideal (Joseph ideal) which annihilates the minrep. We show that the Joseph ideal vanishes identically for the quasiconformal realization of the minrep and its enveloping algebra leads directly to the HS algebra in AdS_7. Furthermore, the enveloping algebras of the deformations of the minrep define a discrete infinite family of HS algebras in AdS_7 for which certain 6d Lorentz covariant deformations of the Joseph ideal vanish identically. These results extend to superconformal algebras OSp(8*|2N) and we find a discrete infinite family of HS superalgebras as enveloping algebras of the minimal unitary supermultiplet and its deformations. Our results suggest the existence of a discrete family of (supersymmetric) HS theories in AdS_7 which are dual to free (super)conformal field theories (CFTs) or to interacting but integrable (supersymmetric) CFTs in 6d.

Unravelling holographic entanglement entropy in higher spin theories

There are two proposals that compute holographic entanglement entropy in AdS3 higher spin theories based on SL(N, ℝ) Chern-Simons theory. We show explicitly that these two proposals are equivalent. We also designed two methods that solve systematically the equations for arbitrary N. For finite charge backgrounds in AdS3, we find exact agreement between our expressions and the short interval correction of the entanglement entropy for an excited state in a CFT2.

Exact renormalization group and higher-spin holography

In this paper, we revisit scalar field theories in $d$ space-time dimensions possessing $U(N)$ global symmetry. Following our recent work arXiv:1402.1430v2, we consider the generating function of correlation functions of all $U(N)$-invariant, single-trace operators at the free fixed point. The exact renormalization group equations are cast as Hamilton equations of radial evolution in a model space-time of one higher dimension, in this case $AdS_{d+1}$. The geometry associated with the RG equations is seen to emerge naturally out of the infinite jet bundle corresponding to the field theory, and suggests their interpretation as higher-spin equations of motion. While the higher-spin equations we obtain are remarkably simple, they are non-local in an essential way. Nevertheless, solving these bulk equations of motion in terms of a boundary source, we derive the on-shell action and demonstrate that it correctly encodes all of the correlation functions of the field theory, written as `Witten diagrams'. Since the model space-time has the isometries of the fixed point, it is possible to construct new higher spin theories defined in terms of geometric structures over other model space-times. We illustrate this by explicitly constructing the higher spin RG equations corresponding to the $z=2$ non-relativistic free field theory in $D$ spatial dimensions. In this case, the model space-time is the Schr\"odinger space-time, $Schr_{D+3}$.

Metric- and frame-like higher-spin gauge theories in three dimensions

We study the relation between the frame-like and metric-like formulation of higher-spin gauge theories in three space–time dimensions. We concentrate on the theory that is described by an Chern–Simons theory in the frame-like formulation. The metric-like theory is obtained by eliminating the generalized spin connection by its equation of motion, and by expressing everything in terms of the metric and a spin-3 Fronsdal field. We give an exact map between fields and gauge parameters in both formulations. To work out the gauge transformations explicitly in terms of metric-like variables, we have to make a perturbative expansion in the spin-3 field. We describe an algorithm for how to do this systematically, and we work out the gauge transformations to cubic order in the spin-3 field. We use these results to determine the gauge algebra to this order, and explain why the commutator of two spin-3 transformations only closes on-shell.

Uniformizing higher-spin equations

Vasilievʼs higher-spin (HS) theories in various dimensions are uniformly represented as a simple system of equations. These equations and their gauge invariances are based on two superalgebras and have a transparent algebraic meaning. For a given HS theory these algebras can be inferred from the vacuum HS symmetries. The proposed system of equations admits a concise AKSZ formulation. We also discuss novel HS systems including partially-massless and massive fields in AdS, as well as conformal and massless off-shell fields.

Strings vs. spins on the null orbifold

Abstract We study the null orbifold singularity in 2+1 d flat space higher spin theory as well as string theory. Using the Chern-Simons formulation of 2+1 d Einstein gravity, we first observe that despite the singular nature of this geometry, the eigenvalues of its Chern-Simons holonomy are trivial. Next, we construct a resolution of the singularity in higher spin theory: a Kundt spacetime with vanishing scalar curvature invariants. We also point out that the UV divergences previously observed in the 2-to-2 tachyon tree level string amplitude on the null orbifold do not arise in the α′ → ∞ limit. We find all the divergences of the amplitude and demonstrate that the ones remaining in the tensionless limit are physical IR-type divergences. We conclude with a discussion on the meaning and limitations of higher spin (cosmological) singularity resolution and its potential connection to string theory.

Extended supersymmetry in AdS3 higher spin theories

We determine the asymptotic symmetry algebra (for fields of low spin) of the M × M matrix extended Vasiliev theories on AdS3 and find that it agrees with the $$ \mathcal{W} $$ -algebra of their proposed coset duals. Previously it was noticed that for M = 2 the supersymmetry increases from $$ \mathcal{N}=2 $$ to $$ \mathcal{N}=4 $$ . We study more systematically this type of supersymmetry enhancements and find that, although the higher spin algebra has extended supersymmetry for all M ≥ 2, the corresponding asymptotic symmetry algebra fails to be superconformal except for M = 2, when it has large $$ \mathcal{N}=4 $$ superconformal symmetry. Moreover, we find that the Vasiliev theories based on $$ \mathfrak{s}\mathfrak{h}{\mathfrak{s}}^E\left(\mathcal{N}\Big|2,\mathrm{\mathbb{R}}\right) $$ are special cases of the matrix extended higher spin theories, and hence have the same supersymmetry properties.

A higher-spin Chern-Simons theory of anyons

We propose Chern-Simons models of fractional-spin fields interacting with ordinary tensorial higher-spin fields and internal color gauge fields. For integer and half-integer values of the fractional spins, the model reduces to finite sets of fields modulo infinite-dimensional ideals. We present the model on-shell using Fock-space representations of the underlying deformed-oscillator algebra.

A new spin on entanglement entropy

We argue that the usual notions of thermodynamic and entanglement entropy have novel analogs in the context of higher spin theories. In particular, the Wald and Ryu-Takayanagi formulas have natural higher spin extensions that we work out and study. On the CFT side, just as standard entanglement entropy in CFT_2 can be computed from twist field correlators, we demonstrate that by introducing corresponding operators carrying higher spin charge we can precisely reproduce our results from the bulk. We also show that the first law for entanglement entropy implies the linearized field equations for the metric and higher spin fields, generalizing recent work on deriving the linearized Einstein equations from the first law.