Conformal Higher Spin Research Articles

Covariant action for conformal higher spin gravity

Conformal higher spin (HS) gravity is a HS extension of Weyl gravity and is a family of local HS theories, which was put forward by Segal and Tseytlin. We propose a manifestly covariant and coordinate-independent action for these theories. The result is based on an interplay between HS symmetries and deformation quantization: a locally equivalent but manifestly background-independent reformulation, known as the parent system, of the off-shell multiplet of conformal HS fields (Fradkin–Tseytlin fields) can be interpreted in terms of Fedosov deformation quantization of the underlying cotangent bundle. This brings into the game the invariant quantum trace, induced by the Feigin–Felder–Shoikhet cocycle of Weyl algebra, which extends Segal’s action into a gauge invariant and globally well-defined action functional on the space of configurations of the parent system. The same action can be understood within the worldline approach as a correlation function in the topological quantum mechanics on the circle.

Conformal Interactions Between Matter and Higher‐Spin (Super)Fields

AbstractIn even spacetime dimensions, the interacting bosonic conformal higher‐spin (CHS) theory can be realised as an induced action. The main ingredient in this definition is the model describing a complex scalar field φ coupled to an infinite set of background CHS fields h, with possessing a non‐abelian gauge symmetry. Two characteristic features of the perturbative constructions of given in the literature are: (i) the background spacetime is flat; and (ii) conformal invariance is not manifest. In the present paper we provide a new derivation of this action in four dimensions such that (i) is defined on an arbitrary conformally‐flat background; and (ii) the background conformal symmetry is manifestly realised. Next, our results are extended to the supersymmetric case. Specifically, we construct, for the first time, a model for a conformal scalar/chiral multiplet Φ coupled to an infinite set of background higher‐spin superfields H. Our action possesses a non‐abelian gauge symmetry which naturally generalises the linearised gauge transformations of conformal half‐integer superspin multiplets. The other fundamental features of this model are: (i) is defined on an arbitrary conformally‐flat superspace background; and (ii) the background superconformal symmetry is manifest. Making use of , an interacting superconformal higher‐spin theory can be defined as an induced action.

A calculation of the Weyl anomaly for 6D conformal higher spins

In this work we continue the study of the one-loop partition function for higher derivative conformal higher spin (CHS) fields in six dimensions and its holographic counterpart given by massless higher spin Fronsdal fields in seven dimensions.In going beyond the conformal class of the boundary round 6-sphere, we start by considering a Ricci-flat, but not conformally flat, boundary and the corresponding Poincaré-Einstein space-filling metric. Here we are able to match the UV logarithmic divergence of the boundary with the IR logarithmic divergence of the bulk, very much like in the known 4D/5D setting, under the assumptions of factorization of the higher derivative CHS kinetic operator and WKB-exactness of the heat kernel of the dual bulk field. A key technical ingredient in this construction is the determination of the fourth heat kernel coefficient b6 for Lichnerowicz Laplacians on both 6D and 7D Einstein manifolds. These results allow to obtain, in addition to the already known type-A Weyl anomaly, two of the three independent type-B anomaly coefficients in terms of the third, say c3 for instance.In order to gain access to c3, and thus determine the four central charges independently, we further consider a generic non Ricci-flat Einstein boundary. However, in this case we find a mismatch between boundary and bulk computations for spins higher than two. We close by discussing the nature of this discrepancy and perspectives for a possible amendment.

Generalised superconformal higher-spin multiplets

We propose generalised mathcal{N} = 1 superconformal higher-spin (SCHS) gauge multiplets of depth t, {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} , with n ≥ m ≥ 1. At the component level, for t > 2 they contain generalised conformal higher-spin (CHS) gauge fields with depths t − 1, t and t + 1. The supermultiplets with t = 1 and t = 2 include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} on conformally-flat superspace backgrounds are then derived. For the case n = m = t = 1, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for {Upsilon}_{alpha (n)overset{cdot }{alpha }(m)}^{(t)} , are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.

On the structure of the conformal higher-spin wave operators

We study conformal higher spin (CHS) fields on constant curvature backgrounds. By employing parent formulation technique in combination with tractor description of GJMS operators we find a manifestly factorized form of the CHS wave operators for symmetric fields of arbitrary integer spin s and gauge invariance of arbitrary order t ≤ s. In the case of the usual Fradkin-Tseytlin fields t = 1 this gives a systematic derivation of the factorization formulas known in the literature while for t > 1 the explicit formulas were not known. We also relate the gauge invariance of the CHS fields to the partially-fixed gauge invariance of the factors and show that the factors can be identified with (partially gauge-fixed) wave operators for (partially)-massless or special massive fields. As a byproduct, we establish a detailed relationship with the tractor approach and, in particular, derive the tractor form of the CHS equations and gauge symmetries.

Scattering of conformal higher spin fields

We develop a formalism for describing the most general notion of tree-level scattering amplitudes in 4d conformal higher spin theory. As conformal higher spin fields obey higher-derivative equations of motion, there are many distinct on-shell external states which may contribute to their scattering, some of which grow polynomially with time, leading to ill-defined amplitudes. We characterize the set of admissible scattering states which produce finite tree amplitudes, noting that there are more such states than just standard massless higher spins obeying two-derivative equations of motion. We use conformal gravity as a prime example, where the set of scattering states includes the usual Einstein graviton and a ‘ghost’ massless spin 1 particle. An extension of the usual spinor helicity formalism allows us to encode these scattering states efficiently in terms of ‘twistor-spinors’. This leads to compact momentum space expressions for all finite tree-level 3-point amplitudes of conformal higher spin theory. While some of these 3-point amplitudes vanish (including all those with only standard two-derivative higher spin external states), there are many others which are non-vanishing. We also comment on the generalization to scattering of conformal higher spins in AdS4.

On the Weyl anomaly of 4D conformal higher spins: a holographic approach

We present a first attempt to derive the full (type-A and type-B) Weyl anomaly of four dimensional conformal higher spin (CHS) fields in a holographic way. We obtain the type-A and type-B Weyl anomaly coefficients for the whole family of 4D CHS fields from the one-loop effective action for massless higher spin (MHS) Fronsdal fields evaluated on a 5D bulk Poincaré-Einstein metric with an Einstein metric on its conformal boundary. To gain access to the type-B anomaly coefficient we assume, for practical reasons, a Lichnerowicz-type coupling of the bulk Fronsdal fields with the bulk background Weyl tensor. Remarkably enough, our holographic findings under this simplifying assumption are certainly not unknown: they match the results previously found on the boundary counterpart under the assumption of factorization of the CHS higher-derivative kinetic operator into Laplacians of “partially massless” higher spins on Einstein backgrounds.

Induced Action for Conformal Higher Spins from Worldline Path Integrals

Conformal higher spin (CHS) fields, yet being non unitary, provide a remarkable example of a consistent interacting higher spin theory in flat space background, that is local to all orders. The non-linear action is defined as the logarithmically UV divergent part of a one-loop scalar effective action. In this paper we take a particle model, that describes the interaction of a scalar particle to the CHS background, and compute its path integral on the circle. We thus provide a worldline representation for the CHS action, and rederive its quadratic part. We plan to come back to the subject, to compute cubic and higher vertices, in a future work.

Conformal higher spin scattering amplitudes from twistor space

We use the formulation of conformal higher spin (CHS) theories in twistor space to study their tree-level scattering amplitudes, finding expressions for all three-point overline{mathrm{MHV}} amplitudes and all MHV amplitudes involving positive helicity conformal gravity particles and two negative helicity higher spins. This provides the on-shell analogue for the covariant coupling of CHS fields to a conformal gravity background. We discuss the restriction of the theory to a ghost-free unitary subsector, analogous to restricting conformal gravity to general relativity with a cosmological constant. We study the flat-space limit and show that the restricted amplitudes vanish, supporting the conjecture that in the unitary sector the S-matrix of CHS theories is trivial. However, by appropriately rescaling the amplitudes we find non-vanishing results which we compare with chiral flat-space higher spin theories.

Bosonic Fradkin-Tseytlin equations unfolded

We test infinite-dimensional extension of algebra $$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$ proposed by Fradkin and Linetsky as the candidate for conformal higher spin algebra. Adjoint and twistedadjoint representations of $$ \mathfrak{s}\mathfrak{u}\left(k,\;k\right) $$ on the space of this algebra are carefully explored. For k = 2 corresponding unfolded system is analyzed and it is shown to encode Fradkin-Tseytlin equations for the set of all integer spins 1, 2, . . . with infinite multiplicity.

On triviality of S-matrix in conformal higher spin theory

We consider the conformal higher spin (CHS) theory in d=4 that contains the s=1 Maxwell vector, s=2 Weyl graviton and their higher spin s=3,4,... counterparts with higher-derivative \box^s kinetic terms. The interacting action for such theory can be found as the coefficient of the logarithmically divergent part in the induced action for sources coupled to higher spin currents in a free complex scalar field model. We explicitly determine some cubic and quartic interaction vertices in the CHS action from scalar loop integrals. We then compute the simplest tree-level 4-particle scattering amplitudes 11 -> 11, 22 -> 22 and 11 -> 22 and find that after summing up all the intermediate CHS exchanges they vanish. This generalises the vanishing of the scattering amplitude for external conformal scalars interacting via the exchange of all CHS fields found earlier in arXiv:1512.08896. This vanishing should generalise to all scattering amplitudes in the CHS theory and as in the conformal scalar scattering case should be a consequence of the underlying infinite dimensional higher spin symmetry that extends the standard conformal symmetry.

CFT driven cosmology and conformal higher spin fields

Conformal higher spin (CHS) field theory, which is a solid part of recent advanced checks of AdS/CFT correspondence, finds applications in cosmology. Hidden sector of weakly interacting CHS fields suggests a resolution of the hierarchy problem in the model of initial conditions for inflationary cosmology driven by a conformal field theory. These initial conditions are set by thermal garland type cosmological instantons in the sub-planckian energy range for the model of CHS fields with a large positive coefficient $\beta$ of the Gauss-Bonnet term in their total conformal anomaly and a large number of their polarizations $\mathbb{N}$. The upper bound of this range $M_P/\sqrt\beta$ is shown to be much lower than the gravitational cutoff $M_P/\sqrt\mathbb{N}$ which is defined by the requirement of smallness of the perturbatively nonrenormalizable graviton loop contributions. In this way we justify the approximation scheme in which the nonrenormalizable graviton sector is subject to effective field theory under this cutoff, whereas the renormalizable sector of multiple CHS fields is treated beyond perturbation theory and dynamically generates the bound on the inflation scale of the CFT cosmology $M_P/\sqrt\beta\ll M_P/\sqrt\mathbb{N}$. This confirms recent predictions for the origin of the Starobinsky $R^2$ and Higgs inflation models from the CHS cosmology, which occurs at the energy scale three or four orders of magnitude below the gravitational cutoff, $\sqrt{\mathbb{N}/\beta}\sim 10^{-3}\div 10^{-4}$. We also consider cosmological models dominated by fermionic CHS fields with a negative $\beta$ and anomaly free models of infinite towers of CHS fields with $\beta=0$ and briefly discuss the status of unitarity in CHS models.

Partition function of free conformal higher spin theory

We compute the canonical partition function Z of non-interacting conformal higher spin (CHS) theory viewed as a collection of free spin s CFT's in R^d. We discuss in detail the 4-dimensional case (where s=1 is the standard Maxwell vector, s=2 is the Weyl graviton, etc.), but also present a generalization for all even dimensions d. Z may be found by counting the numbers of conformal operators and their descendants (modulo gauge identities and equations of motion) weighted by scaling dimensions. This conformal operator counting method requires a careful analysis of the structure of characters of relevant (conserved current, shadow field and conformal Killing tensor) representations of the conformal algebra so(d,2). There is also a close relation to massless higher spin partition functions with alternative boundary conditions in AdS_{d+1}. The same partition function Z may also be computed from the CHS path integral on a curved S^1 x S^{d-1} background. This allows us to determine a simple factorized form of the CHS kinetic operator on this conformally flat background. Summing the individual conformal spin contributions Z_s over all spins we obtain the total partition function of the CHS theory. We also find the corresponding Casimir energy and show that it vanishes if one uses the same regularization prescription that implies the cancellation of the total conformal anomaly a-coefficient. This happens to be true in all even dimensions d >= 2.

On partition function and Weyl anomaly of conformal higher-spin fields

We study 4-dimensional higher-derivative conformal higher-spin (CHS) fields generalizing Weyl graviton and conformal gravitino. They appear, in particular, as “induced” theories in the AdS/CFT context. We consider their partition function on curved Einstein-space backgrounds like (A)dS or sphere and Ricci-flat spaces. Remarkably, the bosonic (integer spin s) CHS partition function appears to be given by a product of partition functions of the standard 2nd-derivative “partially massless” spin s fields, generalizing the previously known expression for the 1-loop Weyl graviton (s=2) partition function. We compute the corresponding spin s Weyl anomaly coefficients as and cs. Our result for as reproduces the expression found recently in arXiv:1306.5242 by an indirect method implied by AdS/CFT (which relates the partition function of a CHS field on S4 to a ratio of known partition functions of massless higher-spin field in AdS5 with alternate boundary conditions). We also obtain similar results for the fermionic CHS fields. In the half-integer s case the CHS partition function on (A)dS background is given by the product of squares of “partially massless” spin s partition functions and one extra factor corresponding to a special massive conformally invariant spin s field. It was noticed in arXiv:1306.5242 that the sum of the bosonic as coefficients over all s is zero when computed using the ζ-function regularization, and we observe that the same property is true also in the fermionic case.

Unfolded dynamics and conformal higher spin gauge theory

Unfolded dynamics approach is illustrated by the analysis of conformal gauge fields of generic symmetry type. For the case of general free fields it is shown how unfolded field equations can be reformulated in the BRST form.

STRINGS AND MEMBRANES FROM EINSTEIN GRAVITY, MATRIX MODELS AND W∞ GAUGE THEORIES AS PATHS TO QUANTUM GRAVITY

It is shown how w∞, w1+∞ gauge field theory actions in 2D emerge directly from 4D gravity. Strings and membranes actions in 2D and 3D originate as well from 4D Einstein gravity after recurring to the nonlinear connection formalism of Lagrange–Finsler and Hamilton–Cartan spaces. Quantum gravity in 3D can be described by a W∞ matrix model in D = 1 that can be solved exactly via the collective field theory method. We describe why a quantization of 4D gravity could be attained via a 2D quantum W∞ gauge theory coupled to an infinite-component scalar-multiplet. A proof that noncritical W∞ (super)strings are devoid of BRST anomalies in dimensions D = 27(D = 11), respectively, follows and which coincide with the critical (super)membrane dimensions D = 27(D = 11). We establish the correspondence between the states associated with the quasifinite highest weights irreducible representations of W∞, [Formula: see text] algebras and the quantum states of the continuous Toda molecule. Schrödinger-like quantum mechanics wave functional equations are derived and solutions are found in the zeroth-order approximation. Since higher-conformal spin W∞ symmetries are very relevant in the study of 2DW∞ gravity, the quantum Hall effect, large N QCD, strings, membranes, … it is warranted to explore further the interplay among all these theories.

The gluon Green's function in [formula omitted] supersymmetric Yang–Mills theory

The high energy limit of scattering amplitudes in the N = 4 supersymmetric Yang–Mills theory is studied by solving the corresponding BFKL equation in the next-to-leading approximation. The gluon Green's function is analysed using a newly proposed method suitable for investigating the contribution from higher conformal spins. From this new approach complete agreement is obtained with the results of Kotikov and Lipatov on conformal spins and angular dependence.

POINT PARTICLE IN GENERAL BACKGROUND FIELDS AND FREE GAUGE THEORIES OF TRACELESS SYMMETRIC TENSORS

Point particle may interact with traceless symmetric tensors of arbitrary rank. Free gauge theories of traceless symmetric tensors are constructed, which provides a possibility for a new type of interactions, when particles exchange by those gauge fields. The gauge theories are parametrized by the particle's mass m and otherwise are unique for each rank s. For m=0, they are local gauge models with actions of 2s th order in derivatives, known in d=4 as "pure spin," or "conformal higher spin" actions by Fradkin and Tseytlin. For m≠0, each rank-s model undergoes a unique nonlocal deformation which entangles fields of all ranks, starting from s. There exists a nonlocal transform which maps m≠0 theories onto m=0 ones, however, this map degenerates at some m≠0 fields whose polarizations are determined by zeros of Bessel functions. Conformal covariance properties of the m=0 models are analyzed, the space of gauge fields is shown to admit an action of an infinite-dimensional "conformal higher spin" Lie algebra which leaves gauge transformations intact.

POINT PARTICLE–SYMMETRIC TENSORS INTERACTION AND GENERALIZED GAUGE PRINCIPLE

The model of a point particle in the background of external symmetric tensor fields is analyzed from the higher spin theory perspective. It is proposed that the gauge transformations of the infinite collection of symmetric tensor fields may be read off from the covariance properties of the point particle action w.r.t. general canonical transformations. The gauge group turns out to be a semidirect product of all phase space canonical transformations to an Abelian ideal of "hyperWeyl" transformations and includes U(1) and general coordinate symmetries as a subgroup. A general configuration of external fields includes rank-0,1,2 symmetric tensors, so the whole system may be truncated to ordinary particle in Einstein–Maxwell backgrounds by switching off the higher-rank symmetric tensors. When otherwise all the higher rank tensors are switched on, the full gauge group provides a huge gauge symmetry acting on the whole infinite collection of symmetric tensors. We analyze this gauge symmetry and show that the symmetric tensors which couple to the point particle should not be interpreted as Fronsdal gauge fields, but rather as gauge fields of some conformal higher spin theories. It is shown that the Fronsdal fields system possesses twice as many symmetric tensor fields as is contained in the general background of the point particle. Besides, the particle action in general backgrounds is shown to reproduce De Wit–Freedman point particle–symmetric tensors first order interaction suggested many years ago, and extends their result to all orders in interaction, while the generalized equivalence principle completes the first order covariance transformations found in their paper, in all orders.

HIGHER-SPIN SYMMETRY IN ONE AND TWO DIMENSIONS (II)

Constructed are conformal higher spin superalgebras in one and two dimensions, which contain the Virasoro algebra as a subalgebra. The general structure of these superalgebras is investigated.