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Aspects of higher spin symmetry in flat space

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La Rivista del Nuovo Cimento Aims and scope

Abstract

We review some aspects of higher spin symmetry, in (Anti-)de Sitter and flat space–times, aiming at closing the gap between the constantly curved and flat cases. On (Anti-)de Sitter space, non-Abelian higher spin algebras are at the core of the construction of interacting theories of higher spin gravity. By considering a suitable contraction of these algebras, we show that similar considerations can apply to Minkowski space–time. We identify a unique candidate to the role of higher spin symmetry in flat space that can also be built as a quotient of the universal enveloping algebra of the isometries of the vacuum, as in the (Anti-)de Sitter case. We then show how to recover the free dynamics from the gauging of the resulting algebra at the linear level. Finally, we show how to realise this gauge algebra as a subset of the global symmetries of a Carrollian conformal scalar field theory living on the null infinity of Minkowski space–time. This theory emerges as the limit of vanishing speed of light of a free massless relativistic scalar. The identification of the same higher spin algebra that rules the dynamics in the bulk of space–time within the global symmetries of this boundary theory paves the way to a flat counterpart of higher spin holography.

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Notes

  1. Some years later, the Bargmann–Wigner equations [30] were considered for this role. Although the latter were originally formulated as first-order equations on mixed-symmetry fields in \(D=4\,\), they were generalised to any D in Ref. [31], where it was also proven that they are equivalent to higher derivative equations in terms of symmetric fields. They were eventually abandoned due to their problematic coupling to electromagnetism.

  2. In general, self-coupling terms for fields of odd spin are problematic, even for spin one. A way out consists in introducing colour factors, like in Yang–Mills theory. However, it was found in Ref. [42] that the Berends–Burgers–van Dam spin-three vertex was necessarily inconsistent.

  3. Although one can build theories up to cubic order with only a finite number of fields, the necessity of closing of the algebra of gauge transformations at cubic level, or the consistency of interactions at quartic level, forces us to consider the full spectrum, see for instance the review [43]. The problem of an infinite spectrum is distinct from the potentially infinite number of interactions with increasing derivatives in a perturbative expansion, which is also present in the case of higher spin gravity.

  4. Higher spin cubic interactions were actually constructed earlier [48] but the analysis was performed in the light-cone gauge, thus obscuring the structure of a gauge theory.

  5. The S-matrix measures the scattering amplitudes in an asymptotically far region of Minkowski space, and has to be replaced in Anti de Sitter by correlation functions of the boundary theory, while no direct equivalent seems to exist in de Sitter space–time.

  6. It was proven, much later after the original construction of Fradkin and Vasiliev, that this algebra is the unique one that reproduces known non-Abelian cubic interactions [65] in any dimensions \(D \ge 4\,\), save for a one-parameter deformation in \(D=5\,\).

  7. The original formulation of the AdS/CFT correspondence postulated the equivalence between type IIB super-string theory on AdS\(_5 \times S^5\) and a dual \(\mathcal{N}= 4\) super-conformal Yang–Mill theory on four-dimensional Minkowski space, the boundary of AdS\(_5\,\).

  8. The name Carrollian was inspired by Lewis Carroll’s book Through the looking glass, where the protagonist Alice is puzzled by the Queen’s remark ‘it takes all the running you can do, to keep in the same place’. In Carrollian space–time, the speed of light is sent to zero, creating this sense of ‘static motion’.

  9. The ‘magnetic’ (or ‘space-like’) action, as well as a holographic realisation of both theories, will be discussed in Ref. [105].

  10. However, unconstrained and geometric theories [18, 37, 156,157,158,159,160] are placed on a different footing from the get-go, since they require traceful gauge parameters to start with.

  11. As explained in, e.g. Ref. [121], the component of \(\textrm{d}C^{a(s),b(s)}\) whose Young diagram has three rows is killed as a consequence of Eq. (2.27b), which is also the case of its trace, so that the only surviving component is a Lorentz-irreducible tensor \(C^{a(s+1),b(s)}\) with two rows of length \((s+1)\) and \(s\,\).

  12. A complete classification of cubic vertices in light-cone was performed in Ref. [52] which exhibits \((2\,s-2)\), \((2\,s+2)\) and 2-derivative couplings with gravity. The latter class can not be seen within the Fronsdal formulation and only the first leads to a deformation of gauge transformations [58, 131]

  13. A Lagrangian formulation is not strictly necessary to construct an interacting theory, but if one wants to quantise it using the Feynman rules, it is a desirable feature. However, an underlying algebra is always necessary to have a consistent theory. Indeed, in order to build quartic interactions, the cubic vertices must satisfy a consistency condition which is that the structure constants of the algebra they define satisfy the Jacobi identity.

  14. Remember that \(\mathfrak {sl}(4,{{\mathbb {R}}})\) and \(\mathfrak {su}(2,2)\) are two real forms of \(\mathfrak {sl}(4,{{\mathbb {C}}})\,\).

  15. An alternative construction, in which mixed products of the \(\mathcal{L}\) and \(\bar{\mathcal{L}}\) generators are allowed, leads to an extended algebra dubbed ‘large AdS higher-spin algebra’ in Ref. [181]. Similar extended algebras also appear in the description of partially massless fields in three dimensions [182].

  16. The lower bound on the value of \(C_2\) is related to unitarity constraints [140], and the case \(\lambda \in {{\mathbb {N}}}\) reproduces some finite-dimensional truncations.

  17. In the case of massless Poincaré representations, the Pauli–Lubanski pseudo-vector or its higher dimensional generalisation is proportional to the generator of translations, where the proportionality constant encodes the helicity. In our case, \({{\mathbb {W}}}_{a_1 \,\cdots \, a_{D-3}} \sim 0\), meaning that we are looking at a massless scalar.

  18. The requirement that the boundary field theory has exact higher spin symmetry is quite constraining. Theories with slightly broken (i.e. exact up to order 1/N in a large-N expansion) higher spin symmetry such as the quartic model with critical coupling gives more flexibility and would correspond to a situation where higher spin symmetry in the bulk is broken by quantum effects in a controlled way.

  19. Another possibility is given by the extended BMS algebra of Barnich and Troessaert [92], which enhances the group of conformal transformations of the two-sphere to transformations of the Riemann sphere by arbitrary holomorphic and anti-holomorphic functions, except in 0 and \(\infty \,\).

  20. Alternatively, one may consider higher differential symmetries, up to the quotient by the equivalence relation \(\sim \) defined by \({\hat{\mathcal{D}}}_1 \sim {\hat{\mathcal{D}}}_2\) if and only if there exists \({\hat{\mathcal{P}}}\) such that \({\hat{\mathcal{D}}}_1 - {\hat{\mathcal{D}}}_2 = \mathcal{P}\circ A\,\).

  21. The requirement that \({\hat{\mathcal{D}}}\) is non-trivial means that we can always look for \(V^{\mu _1 \,\cdots \, \mu _{s-1}}\). Indeed, if \(V^{\mu _1 \,\cdots \, \mu _{s-1}} = \eta ^{(\mu _1 \mu _2} W^{a_3 \,\cdots \, \mu _{s-1})}\,\), then the highest order of \({\hat{\mathcal{D}}}\) is trivial. This means that \({\hat{\mathcal{D}}}\) is equivalent to another differential operator of strictly lower degree, in the definition of footnote 20.

  22. In this case, it is customary to add a logarithmic branch [203] so that the holographic reconstruction procedure can be performed without obstruction, but in the case of the singleton we precisely want to keep things that way so that a truncation of the spectrum appears, corresponding to defining an ‘ultra-short’ representation of the conformal group.

  23. Even with these two requirements, it is in general non-unique [210, 213].

  24. In its original form, the Carroll group of coordinate transformations is the \(c \rightarrow 0\) contraction of the Poincaré group, and plays a dual role to the Galilei group obtained as the \(c \rightarrow \infty \) contraction.

  25. Let us also mention that the algebra of isometries of any null projective hypersurface (for instance null infinity \({\mathscr {I}}\)) is the BMS algebra [95, 209], which is a straightforward extension of the conformal Carroll algebra.

  26. This operator is distinct from the Laplace–Yamabe operator on the sphere \(S^d\) with round metric \(\gamma \,\).

  27. This definition is well suited for the boundary description of the simpleton, where bulk rigid translations are represented on the boundary by the vector fields \(f_a({\textbf{x}})\,\partial _u\), with \(f_a({\textbf{x}})\) a function of the celestial sphere verifying the good-cut equation defined in (3.58).

  28. The Newman–Unti group at level k is generated by vector fields X preserving the metric up to a conformal factor, i.e. \({\mathcal {L}}_X g = \lambda g\) and such that \({\mathcal {L}}_\xi {}^k X = 0\) where \(\xi \) is the fundamental Carrollian vector field.

  29. In Ref. [212], the realisation of a higher spin extension of the BMS\(_{d+2}\) algebra on the asymptotic data of a free Minkowski scalar field was investigated, and it was proposed that the algebra realised for the value of the scaling dimension \(\Delta = \frac{d}{2} - 1\) gave rise to a candidate asymptotic symmetry algebra for unconstrained higher spin theories in Minkowski space–time.

  30. Such non-localities are also known to appear in the standard unfolded form of the equations of motion in AdS, from quartic order in the interactions [229,230,231]. It is a subject of intense research to determine whether there exists a weaker notion of locality under which higher spin interactions are local at each arbitrary order in perturbation or spin [232, 233].

  31. Recall that the maximally balanced representation corresponds to a field \(\Phi ^{A(s),\dot{A}(s)}\) with as many dotted as there are undotted indices, and coincides with a completely symmetric tensor \(\varphi _{\mu (s)}\) in a Lorentz-irreducible representation using Pauli matrices, while the maximally unbalanced representation possesses only dotted \(\Phi ^{\dot{A}(2s)}\) or undotted \(\Phi ^{A(2s)}\) indices and coincides with the self-dual or anti-self-dual part of a rectangular tensor \(C_{\mu (s),\nu (s)}\,\). The former is naturally identified with the traceless part of the Fronsdal field and the latter with the higher spin Weyl curvature, equal on-shell to s derivatives of the Fronsdal field. Representations between the maximally balanced or unbalanced ones exist, corresponding to tensor of mixed-symmetry.

  32. Some recent progress in chiral higher spin gravity [240, 241] indicates that there exists another framework in which one can translate most of the results of higher spin holography into flat space. One can prove for instance the Flato–Fronsdal theorem, but one must be ready to accept some unconventional features as well. For instance, the gauge algebra relevant for flat self-dual higher spin gravity is not based on the Poincaré algebra [235, 242], but on a different contraction of the AdS\(_4\) isometry algebra. This last statement is not restricted to higher spins, since it is also the case for flat self-dual gravity [243].

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Acknowledgements

This review draws its origin from the author’s PhD thesis, defended at the University of Mons under the supervision of A. Campoleoni. I would like to express my gratitude to him for the enthusiasm he put in this project, but also for his trust, his many advises and his insights. I would also like to thank X. Bekaert, M. Henneaux, A. Pérez and P. Salgado-Rebolledo for collaboration, N. Boulanger, E. Skvortsov, I. Basile, I. Ahlouche Lahlali, J. O’Connor, A. Delfante and the Physique de l’Univers, Champs et Gravitation group in Mons for providing a pleasant and stimulating work environment, J.M. Figueroa-O’Farrill, J. Hartong and S. Prohazka and the University of Edinburgh for their warm hospitality, and finally M. Petropoulos, M. Vilatte, D. Rivera-Betancour and the rest of Centre de Physique Théorique—CPHT at École polytechnique for enlightening discussions.

Funding

The work of the author was supported by the Fonds de la Recherche Scientifique – FNRS under Grant No. FC.36447, as well as the SofinaBoël Fund for Education and Talent and the Fonds Friedmann run by the Fondation de l’École polytechnique.

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  1. Centre de Physique Théorique – CPHT, École polytechnique, CNRS, Institut Polytechnique de Paris, 91120, Palaiseau Cedex, France

    Simon Alexandre Pekar

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The author contributed in full to the writing of this review and thanks A. Campoleoni, as well as the referee, for careful rereading of the manuscript.

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Pekar, S.A. Aspects of higher spin symmetry in flat space. Riv. Nuovo Cim. (2024). https://doi.org/10.1007/s40766-024-00051-2

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