Higher Spin Fields Research Articles

On massive higher spin supermultiplets in d = 4

In this work we discuss the cubic interactions for massless spin 3/2 gravitino with massive higher spin supermultiplets using three superblocks (2, 3/2), (5/2, 2) and (3, 5/2) as the first non-trivial examples. We use gauge invariant formalism for the massive higher spin fields and, as is common in such cases, we face an ambiguity related with the possible field redefinitions due to the presence of Stueckelberg fields. From one hand, we show how this ambiguity can be used as one more way to classify possible cubic vertices. We also note that all these field redefinitions do not change the part of the Lagrangian which appears in the unitary gauge (where all Stueckelberg fields are set to zero) so we still have some important independent results. From the other hand, we show how using the so-called unfolded formalism one can fix these ambiguities and obtain consistent deformations for all massive field gauge invariant curvatures which is the most important step in the Fradkin-Vasiliev formalism. Unfortunately, this works for the massive fields only so the way to construct deformations for the massless field curvatures is still has to be found.

Fractional spins, unfolding, and holography. Part II. 4D higher spin gravity and 3D conformal dual

This paper completes the analysis initiated in the companion work arXiv:2403.02283 — referred to as Paper I — by showing how Vasiliev’s 4D higher-spin gravity (HSG) and 3D coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields (coloured conformal HSG, or CCHSG) emerge as consistent reductions of a common parent model. The latter is a Frobenius-Chern-Simons model with superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra, and was defined and studied in Paper I. Here, we i) realize HSG as a subcase of a more general 4D reduction, describing HSG coupled to coloured, fractional-spin matter, which we refer to as 4D fractional-spin gravity; ii) study the CCHSG model, in particular exhibiting the crucial role played by novel colour gauge fields in coupling conformal matter to conformal HSG, thereby completing the models due to Vasiliev and Nilsson; iii) extract conformal currents and composite coloured sources in an expansion of the CCHSG model around 3D Minkowskian leaves; and iv) compare our results with Vasiliev’s holography proposal of arXiv:1203.5554. The common origin of HSG and CCHSG, besides relating the two models directly, is the starting point for establishing the holographic correspondence between the two models via overlap conditions, to be presented separately.

Higher spins and Finsler geometry

Finsler geometry is a natural generalization of (pseudo-)Riemannian geometry, where the line element is not the square root of a quadratic form but a more general homogeneous function. Parameterizing this in terms of symmetric tensors suggests a possible interpretation in terms of higher-spin fields. We will see here that, at linear level in these fields, the Finsler version of the Ricci tensor leads to the curved-space Fronsdal equation for all spins, plus a Stueckelberg-like coupling. Nonlinear terms can also be systematically analyzed, suggesting a possible interacting structure. No particular choice of spacetime dimension is needed. The Stueckelberg mechanism breaks gauge transformations to a redundancy that does not change the geometry. This creates a serious issue: non-transverse modes are not eliminated, at least for the versions of Finsler dynamics examined in this paper.

Fractional spins, unfolding, and holography. Part I. Parent field equations for dual higher-spin gravity reductions

In this work and in the companion paper arXiv:2403.02301, we initiate an approach to holography based on the AKSZ formalism. As the first step, we refine Vasiliev’s holography proposal in arXiv:1203.5554 by obtaining 4D higher-spin gravity (HSG) and 3D coloured conformal higher-spin gravity (CCHSG) — i.e., coloured conformal matter fields coupled to conformal higher-spin gauge fields and colour gauge fields — as two distinct and classically consistent reductions of a single parent theory. The latter consists, on-shell, of a flat superconnection valued in a fractional-spin extension of Vasiliev’s higher-spin algebra. The HSG and CCHSG reductions are characterized by dual structure groups and two-form cohomology elements, and their embedding in a common parent model provides a rationale for deriving holographic relations from multi-dimensional AKSZ partition functions on cylinders with dual boundary conditions, to appear separately. In this work we i) construct the underlying non-commutative geometry as a metaplectic operator algebra represented in a Hermitian module of a pair of conformal particles; ii) identify a discrete modular group, arising from twisted boundary conditions of the first-quantized system, and connecting different boundary conditions of the second-quantized system; and iii) identify the holonomies, structure groups and two-form cohomology elements that characterize the HSG and CCHSG reductions, and equate the dual second Chern classes.

Correction: Bychkov et al. The σ− Cohomology Analysis for Symmetric Higher-Spin Fields. Symmetry 2021, 13, 1498

There was an error in the original publication [...]

Massive spin three-half field in a constant electromagnetic background

Massive higher-spin fields are difficult to introduce consistent interactions, including electromagnetic and gravitational ones which are clearly exhibited by (non-elementary) higher-spin particles in nature. We construct an action that describes consistent interactions of massive spin three-half field with a constant electromagnetic background. We also work out the relation to the chiral approach.

Compton Amplitude for Rotating Black Hole from QFT.

We construct a candidate tree-level gravitational Compton amplitude for a rotating Kerr black hole, for any quantum spin s=0,1/2,1,…,∞, from which we extract the corresponding classical amplitude to all orders in the spin vector S^{μ}. We use multiple insights from massive higher-spin quantum field theory, such as massive gauge invariance and improved behavior in the massless limit. A chiral-field approach is particularly helpful in ensuring correct degrees of freedom, and for writing down compact off-shell interactions for general spin. The simplicity of the interactions is echoed in the structure of the spin-s Compton amplitude, for which we use homogeneous symmetric polynomials of the spin variables. Where possible, we compare to the general-relativity results in the literature, available up to eighth order in spin.

Momentum-space formulae for AdS correlators for diverse theories in diverse dimensions

In this paper, we explore correlators of a series of theories in anti-de Sitter space: we present comprehensive results for interactions involving scalars, gluons, and gravitons in multiple dimensions. One aspect of our investigation is the establishment of an intriguing connection between the kinematic factors of these theories; indeed, such a connection directly relates these theories among themselves and with other theories of higher spin fields. Besides providing several explicit results throughout the paper, we also highlight the interconnections and relationships between these different theories, providing valuable insights into their similarities and distinctions.

Bilinear Fronsdal currents in the AdS4 higher-spin theory

We analyse higher-spin theory with general coupling constant η at the second order, focusing on the gauge non-invariant vertices Υ(ω, ω), Υ(Ω, ω, C) and Υ(ω, C), that are shown to generate nontrivial currents in the Fronsdal equations. Explicit expressions for the currents are found in the frame-like formalism counterpart of the TT gauge worked out in the paper. The nonlinear higher-spin theory is shown to generate all types of Metsaev’s currents with the coupling constants manifestly expressed via the complex coupling constant η of the higher-spin theory. It is shown that all currents in the higher-spin theory are conformal in the TT gauge except for those bilinear in the higher-spin gauge fields ω.

Higher-spin self-dual General Relativity: 6d and 4d pictures, covariant vs. lightcone

We study the higher-spin extension of self-dual General Relativity (GR) with cosmological constant, proposed by Krasnov, Skvortsov and Tran. We show that this theory is actually a gauge-fixing of a 6d diffeomorphism-invariant Abelian theory, living on (4d spacetime)×(2d spinor space) modulo a finite group. On the other hand, we point out that the theory respects the 4d geometry of a self-dual GR solution, with no backreaction from the higher-spin fields. We also present a lightcone ansatz that reduces the covariant fields to one scalar field for each helicity. The field equations governing these scalars have only cubic vertices. We compare our lightcone ansatz to Metsaev’s lightcone formalism. We conclude with a new perspective on the lightcone formalism in (A)dS spacetime: not merely a complication of its Minkowski-space cousin, it has a built-in Lorentz covariance, and is closely related to Vasiliev’s concept of unfolding.

Low spin solutions of higher spin gravity: BPST instanton

Higher spin gravities do not have a low energy limit where higher-spin fields decouple from gravity. Nevertheless, it is possible to construct fine-tuned exact solutions that activate low-spin fields without sourcing the higher-spin fields. We show that BPST (Belavin-Polyakov-Schwartz-Tyupkin) instanton is an exact solution of Chiral Higher Spin Gravity, i.e. it is also a solution of the holographic dual of Chern-Simons matter theories. This gives an example of a low-spin solution. The instanton sources the opposite helicity spin-one field and a scalar field. We derive an Effective Field Theory that describes the coupling between an instanton and the other two fields, whose action starts with the Chalmers-Siegel action and has certain higher derivative couplings.

Manifest color-kinematics duality for point particles interacting with self-dual fields

We find that point particles interacting with a self-dual Yang-Mills field and self-dual gravity manifestly satisfy color-kinematics duality at the level of action. In a similar way color-kinematics duality also holds for a scalar field minimally coupled to a self-dual Yang-Mills field and self-dual gravity. By applying the appropriate limiting procedure to these scalar field theories we reproduce point particle theories we started from. This allows us to connect worldline color-kinematics duality to amplitude color-kinematics duality in field theory. Considering that point particles act as sources of classical solutions, our results may be regarded as a step towards establishing a precise relation between the amplitude and the classical double copies in the self-dual sector. Finally, we briefly mention that the extension of this discussion to the higher-spin case suggests that scalar point particles cannot interact with chiral higher-spin fields.

Off-shell fields and conserved currents

We study interactions of higher-spin massless fields φ with conserved currents multilinear in the off-shell matter fields ϕ. Specifically, we focus on the 3d case where a slight modification of the σ−-cohomology technique developed earlier is directly applicable to control nontriviality of the interaction vertices at the convention that the vertices removable by a local field redefinition of a higher-spin field or having schematic form F(φ)G(ϕ), where F(φ) is a gauge invariant field strength of a free higher-spin field, are called deformationally trivial. It is demonstrated how the σ−-cohomology approach can be applied to the analysis of nonlinear vertices. Generally, deformationally trivial vertices are not σ−-closed while the deformationally non-trivial ones must be in H(σ−). It is shown that, at least in the 3d case, the relevant cohomology group H1(σ−)=0 and, hence, no deformationally non-trivial off-shell vertices exist. On the other hand, there exists an infinite class of deformationally trivial vertices, that includes the vertex recently proposed for spin three. Our analysis goes beyond the higher-spin vertices allowing to show that, at least in three dimensions, nonlinear combinations of the off-shell scalar fields and their derivatives cannot obey non-trivial equations.

Real time holography for higher spin theories

Real time holography is studied in the context of the embedding space formalism. In the first part of this paper, we present matching conditions for on-shell integer spin fields when going from Euclidean to Lorentzian signature on AdS background. Using the BTZ black hole as an example, we discuss various ways of lifting the physical solution from the AdS surface to the whole embedding space. The BTZ propagator for higher spin field is expressed elegantly in terms of the embedding coordinates. In the second part of the paper, we develop the proposed duality between higher spin theory and vector models. We obtain a specific map between the field configurations of these two theories in real time, so called Lorentzian AdS/CFT map. We conclude by exploring the matching conditions for higher spin fields satisfying the proposed bulk quadratic action. The physical and ghost modes can be treated independently during the Wick rotation; only physical modes are considered to be external modes.

Cubic action for spinning black holes from massive higher-spin gauge symmetry

Scattering of two Kerr Black Holes emitting gravitational waves can be captured by an effective theory of a massive higher-spin field interacting with the gravitational field. While other compact objects should activate a multitude of non-minimal interactions it is the black holes that should be captured by the simplest minimal interaction. Implementing massive higher-spin symmetry via a string-inspired BRST approach we construct an action that reproduces the correct cubic amplitude of Arkani-Hamed-Huang-Huang. The same is achieved for the root-Kerr theory, i.e. for the minimal electromagnetic interaction of a massive higher-spin field.

Consistent Lagrangians for Irreducible Interacting Higher-Spin Fields with Holonomic Constraints

Consistent Lagrangians for Irreducible Interacting Higher-Spin Fields with Holonomic Constraints

Covariant Cubic Interacting Vertices for Massless and Massive Integer Higher Spin Fields

We develop the BRST approach to construct the general off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher spin fields on d-dimensional Minkowski space. We consider two different cases for interacting higher spin fields: with one massive and two massless; two massive, both with coinciding and with different masses and one massless field of spins s1,s2,s3. Unlike the previous results on cubic vertices we extend our earlier result in (Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470) for massless fields and employ the complete BRST operator, including the trace constraints, which is used to formulate an irreducible representation with definite integer spin. We generalize the cubic vertices proposed for reducible higher spin fields in (Metsaev, R.R. Phys. Lett. B 2013, 720, 237) in the form of multiplicative and non-multiplicative BRST-closed constituents and calculate the new contributions to the vertex, which contains the additional terms with a smaller number of space-time derivatives. We prove that without traceless conditions for the cubic vertices in (Metsaev, R.R. Phys. Lett. B 2013, 720, 237) it is impossible to provide the noncontradictory Lagrangian dynamics and find explicit traceless solution for these vertices. As the examples, we explicitly construct the interacting Lagrangians for the massive spin of the s field and the massless scalars, both with and without auxiliary fields. The interacting models with different combinations of triples higher spin fields: massive spin s with massless scalar and vector fields and with two vector fields; massless helicity λ with massless scalar and massive vector fields; two massive fields of spins s, 0 and massless scalar is also considered.

Gauge-Invariant Lagrangian Formulations for Mixed-Symmetry Higher-Spin Bosonic Fields in AdS Spaces

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group—subject to a Young tableaux Y(s1,…,sk) with k≥2 rows—in a d-dimensional anti-de Sitter space. Auxiliary representations for a deformed non-linear HS symmetry algebra in terms of a generalized Verma module, as applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a k=2 Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic sp(2k) algebra used for mixed-symmetry HS fields in flat spaces [Buchbinder, I.L.; et al. Nucl. Phys. B 2012, 862, 270–326]. Polynomial deformations of the su(1,1) algebra related to the Bethe ansatz are studied as a byproduct. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints for Y(s1,s2) is found, with non-vanishing terms (resolving the Jacobi identities) of the third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin (s1,s2) is deduced. Following the results of [Buchbinder, I.L.; et al. Phys. Lett. B 2021, 820, 136470.; Buchbinder, I.L.; et al. arXiv 2022, arXiv:2212.07097], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields (being candidates for massive particles in the Dark Matter problem). A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin (1,1) is obtained.

Interaction of symmetric higher-spin gauge fields

Interaction of symmetric higher-spin gauge fields

Lagrangian formulation for free 6D infinite spin field

We construct a Lagrangian that describes the dynamics of a six-dimensional free infinite (continuous) spin field in 6D Minkowski space. The Lagrangian is formulated in the framework of the BRST approach to higher spin field theory and is based on a system of constraints defining an irreducible representation of the corresponding Poincaré group. The field realization of generators in the 6D Poincaré algebra and the second-, fourth-, and sixth-order Casimir operators are obtained in explicit form using additional spinor coordinates. Specific aspects of such a realization in six dimensions are discussed. We derive the conditions that determine the irreducible representation 6D infinite spin field and reformulate them as operators in the Fock space forming a first-class algebra in terms of commutators. These operators are used to construct the BRST charge and the corresponding Lagrangian. We prove that the conditions of the irreducible representation are reproduced as the consequence of the Lagrangian equations of motion, which finally provides the correctness of the results obtained.