Higher Spin Gauge Fields Research Articles

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.

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.

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.

Interaction of symmetric higher-spin gauge fields

Interaction of symmetric higher-spin gauge fields

On special quartic interaction of higher spin gauge fields with scalars and gauge symmetry commutator in the linear approximation

Local quartic interaction of higher-spin gauge field with a scalar field is considered. In this special case, the nontrivial task of construction of interacting Lagrangian for the higher spin field in physical gauge was solved using the full power of Noether's procedure. As a result, the linear on-field gauge transformation is obtained and the corresponding commutator of transformation is analyzed. To understand the closure of this algebra the right-hand side of this commutator is classified in respect to gauge transformations coming from cubic interactions with different higher spin symmetric tensor fields and with mixed symmetry tensor fields transformations.

Manifest form of the spin-local higher-spin vertex varUpsilon ^{eta eta }_{omega CCC}

Vasiliev generating system of higher-spin equations allowing to reconstruct nonlinear vertices of field equations for higher-spin gauge fields contains a free complex parameter eta . Solving the generating system order by order one obtains physical vertices proportional to various powers of eta and {bar{eta }}. Recently eta ^2 and {bar{eta }}^2 vertices in the zero-form sector were presented in Didenko et al. (JHEP 2012:184, 2020) in the Z-dominated form implying their spin-locality by virtue of Z-dominance Lemma of Gelfond and Vasiliev (Phys. Lett. B 786:180, 2018). However the vertex of Didenko et al. (2020) had the form of a sum of spin-local terms dependent on the auxiliary spinor variable Z in the theory modulo so-called Z-dominated terms, providing a sort of existence theorem rather than explicit form of the vertex. The aim of this paper is to elaborate an approach allowing to systematically account for the effect of Z-dominated terms on the final Z-independent form of the vertex needed for any practical analysis. Namely, in this paper we obtain explicit Z-independent spin-local form for the vertex varUpsilon ^{eta eta }_{omega CCC} for its omega CCC-ordered part where omega and C denote gauge one-form and field strength zero-form higher-spin fields valued in an arbitrary associative algebra in which case the order of product factors in the vertex matters. The developed formalism is based on the Generalized Triangle identity derived in the paper and is applicable to all other orderings of the fields in the vertex.

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 interacting higher-spin bosonic gauge fields in the BRST-antifield formalism

Abstract We examine interacting bosonic higher-spin gauge fields in the BRST-antifield formalism. Assuming that an interacting action $S$ is a deformation of the free action with a deformation parameter $g$, we solve the master equation $(S,S)=0$ from the lower orders in $g$. It is shown that, choosing a certain cubic interaction as the first-order deformation, we can solve the master equation and obtain an action containing all orders in $g$. The antighost number of the action obtained is less than or equal to two. Furthermore, we show that the action obtained is lifted to that of interacting bosonic higher-spin gauge fields on anti-de Sitter spaces.

On Cubic Interaction for Higher Spins in $$Ad{{S}_{{d + 1}}}$$

In this talk we present the result of [1] where we slightly modified prescription of the radial pullback formalism proposed previously by R. Manvelyan, R. Mkrtchyan and W. Ruhl in 2012 [2]. We succeeded to solve all necessary recurrence relations to finalize full radial pullback of the main term of cubic self-interaction for higher spin gauge fields in Fronsdal’s formulation from flat to one dimension less $$Ad{{S}_{{d + 1}}}$$ space. Nontrivial solutions of recurrence relations lead to the possibility to obtain the full set of $$Ad{{S}_{{d + 1}}}$$ dimensional interacting terms with all curvature corrections including trace and divergence terms from any interaction term in $$d + 2$$ dimensional flat space.

Restrictions for n-point vertices in higher-spin theories

We give a simple classification of the independent n-point interaction vertices for bosonic higher-spin gauge fields in d-dimensional Minkowski spacetimes. We first give a characterisation of such vertices for large dimensions, d ≥ 2n − 1, where one does not have to consider Schouten identities due to over-antisymmetrisation of spacetime indices. When the dimension is lowered, such identities have to be considered, but their appearance only leads to equivalences of large-d vertices and does not lead to new types of vertices. We consider the case of low dimensions (d < n) in detail, where a large number of Schouten identities leads to strong restrictions on independent vertices. We also comment on the generalisation of our results to the intermediate region n ≤ d ≤ 2n − 2. In all cases, the independent vertices are expressed in terms of elementary manifestly gauge-invariant quantities, suggesting that no deformations of the gauge transformations are induced.

On BF-type higher-spin actions in two dimensions

We propose a non-abelian higher-spin theory in two dimensions for an infinite multiplet of massive scalar fields and infinitely many topological higher-spin gauge fields together with their dilaton-like partners. The spectrum includes local degrees of freedom although the field equations take the form of flatness and covariant constancy conditions because fields take values in a suitable extension of the infinite-dimensional higher-spin algebra \U0001d525\U0001d530[λ]. The corresponding action functional is of BF-type and generalizes the known topological higher-spin Jackiw-Teitelboim gravity.

Supersymmetric continuous spin gauge theory

Taking into account the Schuster-Toro action and its fermionic analogue discovered by us, we supersymmetrize unconstrained formulation of the continuous spin gauge field theory. Afterwards, building on the Metsaev actions, we supersymmetrize constrained formulation of the theory. In each formulation, we provide supersymmetry transformations for the mathcal{N} = 1 supermultiplet in four-dimensional flat space-time, in which continuous spin particle (CSP) is considered to be a complex scalar continuous spin field, and its super- partner which can be called “ CSPino ” is considered to be a Dirac continuous spin field. It is shown that the algebra of these supersymmetry transformations are closed on-shell. Furthermore, we investigate whether obtained supersymmetry transformations reproduce the known result of the higher spin gauge field theory in the helicity limit. Finally, we illustrate how these two separate set of obtained transformations are related to each other.

A note on the double dual graviton

The (free) graviton admits, in addition to the standard Pauli-Fierz description by means of a rank-two symmetric tensor, a description in which one dualizes the corresponding -curvature tensor on one column to get a -tensor, where D is the spacetime dimension. This tensor derives from a gauge field with mixed Yound symmetry called the ‘dual graviton’ field. The dual graviton field is related non-locally to the Pauli-Fierz field (even on-shell), in much the same way as a p -form potential and its dual -form potential are related in the theory of an abelian p -form. Since the Pauli-Fierz field has a Young tableau with two columns (of one box each), one can contemplate a double dual description in which one dualizes on both columns and not just on one. The double dual curvature is now a -tensor and derives from a gauge field with mixed Young symmetry, the ‘double dual graviton’ field. We show, however, that the double dual graviton field is algebraically and locally related to the original Pauli-Fierz field and, so, does not provide a truly new description of the graviton. From this point of view, it plays a very different role from the dual graviton field obtained through a single dualization. We also show that these equations can be obtained from a variational principle in which the variables to be varied in the action are (all) the components of the double-dual field as well as an auxiliary field with Young symmetry. By gauge fixing the shift symmetries of this action principle, one recovers the Pauli-Fierz action. Our approach differs from the interesting approach based on parent actions and covers only the free, sourceless theory. Similar results are argued to hold for higher spin gauge fields.

Cubic interaction for higher spins in AdSd+1 space in the explicit covariant form

We present a slightly modified prescription of the radial pullback formalism proposed previously by R. Manvelyan, R. Mkrtchyan and W. Rühl in 2012, where authors investigated possibility to connect the main term of higher spin interaction in flat d+2 dimensional space to the main term of interaction in AdSd+1 space ignoring all trace and divergent terms but expressed directly through the AdS covariant derivatives and including some curvature corrections. In this paper we succeeded to solve all necessary recurrence relations to finalize full radial pullback of the main term of cubic self-interaction for higher spin gauge fields in Fronsdal's formulation from flat to one dimension less AdSd+1 space. Nontrivial solutions of recurrence relations lead to the possibility to obtain the full set of AdSd+1 dimensional interacting terms with all curvature corrections including trace and divergence terms from any interaction term in d+2 dimensional flat space.

The uniqueness of hypergravity

We show that consistent interactions of a spin-2 and a higher-spin Majorana fermion gauge fields in 30 fiat space lead uniquely to Aragone-Deser hypergravity or its generalization. Our analysis employs the ERST-cohomological techniques, and works in the metric-like formulation under the assumptions of locality, parity and Poincaré invariance. Local hypersymmetry shows up as the unique consistent deformation of the gauge transformations. An extension of the theory with fermion flavors does not change these features, while a cosmological deformation becomes obstructed unless we allow for some new degrees of freedom and/or non-locality.

Electric-Magnetic Duality in Gravity and Higher-Spin Fields

Over the past two decades, electric-magnetic duality has made significant progress in linearized gravity and higher spin gauge fields in arbitrary dimensions. By analogy with Maxwell theory, the Dirac quantization condition has been generalized to both the conserved electric-type and magnetic-type sources associated with gravitational fields and higher spin fields. The linearized Einstein equations in $D$ dimensions, which are expressed in terms of the Pauli-Fierz field of the \textit{graviton} described by a 2nd-rank symmetric tensor, can be dual to the linearized field equations of the \textit{dual graviton} described by a Young symmetry $(D-3, 1)$ tensor. Hence, the dual formulations of linearized gravity are written by a 2nd-rank symmetric tensor describing the Pauli-Fierz field of the dual graviton in $D=4$, while we have the Curtright field with Young symmetry type $(2,1)$ in $D=5$. The equations of motion of spin-$s$ fields ($s>2$) described by the generalized Fronsdal action can also be dualized to the equations of motion of dual spin-$s$ fields. In this review, we focus on dual formulations of gravity and higher spin fields in the linearized theory, and study their SO(2) electric-magnetic duality invariance, twisted self-duality conditions, harmonic conditions for wave solutions, and their configurations with electric-type and magnetic-type sources. Furthermore, we briefly discuss the latest developments in their interacting theories.

Local action for fermionic unconstrained higher spin gauge fields in AdS and dS spacetimes

The local free action principle for bosonic unconstrained higher spin gauge fields in $d$-dimensional (A)dS$_d$ spacetime has been already established by Segal. However, later on, Schuster and Toro, in 4-dimensional flat spacetime, extracted such an action from their bosonic continuous spin gauge theory, when the continuous spin parameter vanishes. On the other hand, similar to the Schuster-Toro's action, we found the fermionic continuous spin gauge theory in 4-dimensional flat spacetime. Thus it is noteworthy to take out its fermionic higher spin formulation in a limit, and generalize it to $d$-dimensional (A)dS$_d$ spacetime. Therefore, in this paper, we will present a similar action principle \`a la Segal for fermions in $d$-dimensional (A)dS$_d$ spacetime. Moreover, at the level of equations of motion, we will demonstrate how the Fronsdal and the Fang-Fronsdal equations in $d$-dimensional (A)dS$_d$ spacetime are related, respectively, to the Euler-Lagrange equations of the bosonic and fermionic higher spin actions mentioned above.

Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields

We introduce prepotentials for fermionic higher-spin gauge fields in four space-time dimensions, generalizing earlier work on bosonic fields. To that end, we first develop tools for handling conformal fermionic higher-spin gauge fields in three dimensions. This is necessary because the prepotentials turn out to be three-dimensional fields that enjoy both “higher-spin diffeomorphism” and “higher-spin Weyl” gauge symmetries. We discuss a number of the key properties of the relevant Cotton tensors. The reformulation of the equations of motion as “twisted self-duality conditions” is then exhibited. We show next how the Hamiltonian constraints can be explicitly solved in terms of appropriate prepotentials and show that the action takes then the same remarkable form for all spins.

Higher-Spin Gauge Theories and Bulk Locality.

We present a no-go result on consistent Noether interactions among higher-spin gauge fields on anti-de Sitter space-times. We show that there is a nonlocal obstruction at the classical level to consistent interacting field theory descriptions of massless higher-spin particles that are described in the free limit by the free Fronsdal action, under the assumption that such theories arise from the gauging of a global higher-spin symmetry. Our result suggests that the Fronsdal program for introducing interactions among higher-spin gauge fields cannot be completed without introducing new guiding principles, which could potentially lie beyond the framework of classical field theory.

Erratum: Twisted self-duality for higher spin gauge fields and prepotentials [Phys. Rev. D 94 , 105027 (2016)

Erratum: Twisted self-duality for higher spin gauge fields and prepotentials [Phys. Rev. D <b>94</b> , 105027 (2016)