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Elements of Vasiliev Theory AEI-2014-006 NSF-KITP-14-008 Elements of Vasiliev theory V.E. Didenko∗ and E.D. Skvortsov∗† ∗Lebedev Institute of Physics, Moscow, Russia † Albert Einstein Institute, Potsdam, Germany Abstract We propose a self-contained description of Vasiliev higher-spin theories with the emphasis on nonlinear equations. The main sections are supplemented with some additional material, including introduction to gravity as a gauge theory; the review of the Fronsdal formulation of free higher-spin fields; Young diagrams and tensors as well as sections with advanced topics. The shortest route to Vasiliev equations covers 40 pages. The general discussion is dimension independent, while the essence of the Vasiliev formulation is discussed on the base of the four-dimensional higher-spin theory. arXiv:1401.2975v5 [hep-th] 28 Dec 2015 Three-dimensional and d-dimensional higher-spin theories follow the same logic. ∗[email protected] †[email protected] 1 Contents 1 Introduction 4 2 Metric-like formulation for free HS fields 7 2.1 Massless fields on Minkowski background . 7 2.2 Massless fields on (anti)-de Sitter background . ... 12 3 Gravity as gauge theory 15 3.1 Tetrad, Vielbein, Frame, Vierbein,.... 15 3.2 Gravityasagaugetheory ........................... 22 4 Unfolding gravity 30 5 Unfolding, spin by spin 37 5.1 s = 2 retrospectively.............................. 39 5.2 s ≥ 2 ...................................... 41 5.3 s = 0 ...................................... 48 5.4 s = 1 ...................................... 51 5.5 Zero-forms.................................... 53 5.6 Allspinstogether................................ 54 6 Unfolding 56 6.1 Basics ...................................... 57 6.2 Structureconstants ............................... 59 7 Vector-spinor dictionary 64 7.1 so(3, 1) sl(2, C)................................ 65 7.2 Dictionary....................................∼ 68 8 Free HS fields in AdS4 69 8.1 Massless scalar and HS zero-forms on Minkowski . ... 69 8.2 Spinor version of AdS4 background ...................... 72 8.3 HS zero-forms on AdS4 ............................. 74 8.4 HS gauge potentials on AdS4 ......................... 76 9 Higher-spin algebras 78 10 Vasiliev equations 89 10.1Generalities ................................... 89 10.2 Quasi derivation of Vasiliev equations . 91 10.3 Perturbationtheory............................... 101 10.4 ManifestLorentzsymmetry . 106 10.5 Higherordersandgaugefixing. 111 10.6 Topological fields and integrating flow . 116 2 11 Extras 118 11.1 Fronsdal operator on Riemannian manifolds . 118 11.2 Other gravity-like actions . 119 11.3 MacDowell-Mansouri-Stelle-West . 120 11.4 Interplay between diffeomorphisms and gauge symmetries . ......125 11.5 Chevalley-Eilenberg cohomology and interactions . 127 11.6 Universal enveloping realization of HS algebra . 129 11.7 Advanced ⋆-products: Cayleytransform . 131 11.8 PoincareLemma,Homotopyintegrals . 133 A Indices 134 B Multi-indices and symmetrization 134 C Solving for spin-connection 135 D Differential forms 136 E Young diagrams and tensors 137 E.1 Generalities ...................................137 E.2 Tensorproducts.................................144 E.3 Generatingfunctions ..............................146 F Symplectic differential calculus 147 G More on so(3, 2) 149 G.1 Restriction of so(3, 2) to so(3, 1)........................149 G.2 so(3, 2) sp(4, R)................................150 ∼ 3 1 Introduction One of the goals of quantum field theory is to explore the landscape of consistent theories. Normally, but not always, one starts with some set of free, noninteracting fields and then tries to make them interact. Given the fact that the free fields are characterized by spin and mass we can ask the following question: which sets of fields specified by their spins 3 and masses admit consistent interaction? Massless fields of spins s = 1, 2 , 2, ... being gauge fields are of particular interest because their interactions are severely constrained by gauge symmetry. The well known examples include: Yang-Mills theory as a theory of massless spin-one fields; gravity as a theory of a massless spin-two; supergravities as 3 theories of a number of spin- 2 fields, graviton and possibly some other fields required for consistency; string theory which spectrum contains infinitely many fields of all spins, mostly massive being highly degenerate by spin. There is a threshold value of spin, which is 2. Once a theory contains fields of spins not greater than two its spectrum can be finite. If there is at least one field of spin greater than two, a higher-spin field, the spectrum is necessarily infinite, containing fields of all spins. String theory is an example of such theory. The Vasiliev higher-spin theory is the missing link in the evolution from the field theories of lower spins, s 2, to string ≤ theories. The Vasiliev theory is the minimal theory whose spectrum contains higher-spin fields. Its spectrum consists of massless fields of all spins s =0, 1, 2, 3, ..., each appearing once in the minimal theory, and in this respect it is much simpler than string theory. The important difference of Vasiliev theory in comparison with string theory is that while the latter has dimensionful parameter α′, the former does not being the theory of gauge fields based on the maximal space-time symmetry. This higher-spin symmetry is an ultimate symmetry in a sense that it cannot result from spontaneous symmetry breaking. The Vasiliev theory therefore has no energy scale and can be thought of as a toy model of the fundamental theory beyond Planck scale. In this note we would like to give a self-contained review on some aspects of higher- spin theory. The subject dates back to the work of Fronsdal, [1], who has first found the equations of motion and the action principle for free massless fields of arbitrary spin. His equations naturally generalize those of Maxwell for s = 1 and the linearized Einstein equations for s = 2. At the same time, as spin gets larger than two (in bosonic case) the Fronsdal fields reveal certain new features. It is a subject of many no-go theorems stating that interacting higher-spin fields cannot propagate in Minkowski space. As a yes-go result is available thanks to Fradkin and Vasiliev, [2–4] we are not going to discus these theorems in any detail referring to excellent review [5]. The crucial idea that allowed one to overcome all no-go theorems was to replace flat Minkowski background space with the anti-de Sitter one, i.e. to turn on the cosmological constant. That higher-spin theory seems to be ill-defined on the Minkowski background is not surprising from the point of view of absence of the dimensionful parameter. In other words the interaction vertices that carry space-time derivatives would be of different dimension in that case. The AdS background simply introduces such a dimensionful parameter, the cosmological constant. A canonical way of constructing interacting theories is order by order. In the realm of perturbative interaction scheme one begins with a sum of quadratic Lagrangians for a given spectrum of spins and masses (these are zero since we consider gauge fields) and 4 tries to deform them by some cubic terms while maintaining the gauge invariance, then by some quartic terms, etc. The gauge transformations perturbatively get deformed as well. If a nontrivial solution to cubic deformations is found we are said to have cubic interaction vertices. A lot is known about cubic interactions both in the metric-like approach of Fronsdal [6–23] and in the frame-like approach of Vasiliev and Fradkin-Vasiliev [2–4, 24–30]. The cubic level, however, is insensitive to the spectrum of fields, i.e. given a set of consistent cubic couplings among various fields one can simply sum up all cubic deforma- tions into a single Lagrangian, which is again consistent up to the cubic level. For the simplest case of a number of spin-one fields one finds that the cubic consistency relies on some structure constants fabc being antisymmetric. The Jacobi identity telling us that there is a Lie algebra behind the Yang-Mills theory arises at the quartic level only. The technical difficulty of a theory with at least one higher-spin field is that it is not even possible to consistently consider an interaction of a finite amount of fields, [6,17,31,32]. Altogether, this makes Fronsdal program (construction of interacting theory for higher-spin gauge fields) extremely difficult to implement. At present, the traditional methods of metric-like approach has led to little progress in this direction, basically their efficiency stops at the level of cubic interaction, see however [31,33]. This state of affairs made it clear that some other tools were really relevant to push the problem forward. A very fruitful direction initiated by Fradkin and Vasiliev and then largely extended by Vasiliev that eventually foster the appearance of complete nonlinear system for higher- spin fields, [34–39], is the so called unfolded approach, [40, 41], to dynamical systems. It rests on the frame-like concept rather than the metric-like and the differential form language which makes the whole formalism explicitly diffeomorphism invariant. This is the branch of higher-spin theory that we want to discuss in these lectures. As we will see the concept of gauge symmetry intrinsically resides in the unfolded approach and, therefore, it suits perfectly for the analysis of gauge systems. The other advantage is that being applied to a free system it reveals all its symmetries and the spectrum of (auxiliary) fields these symmetries act linearly on. Particularly, the higher-spin algebra is something that one can discover from free field theory analysis using the unfolded machinery. This is one of the corner stones towards the nonlinear higher-spin system. Nevertheless, the unfolded approach is just a tool that controls gauge symmetries, degrees of freedom and coordinate independence while containing no extra physical input. It turned out to be extremely efficient for higher-spin problem providing us with explicit nonlinear equations, still it gives no clue for their physical origin.
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