Titre de la thèse: « Aspects of higher-spin symmetry in flat space »

Promoteur: Andrea Campoleoni

Résumé de la dissertation: From the viewpoint of relativistic field theory, general relativity describes the self-interactions of a massless particle of spin two, called graviton. This suggests applying to it the tools of quantum field theory, that allowed to build the standard model of particle physics. However, Einstein’s theory is non-renormalisable: to eliminate the divergences that plague the quantum theory one should add infinitely many counterterms to the action, thereby loosing predictive power. Many approaches have been explored to deal with this key problem and one promising direction is provided by higher-spin gravity. This name denotes all models including, besides the graviton, also gauge fields of spin greater than two. Strong indications in support of this options come from String Theory – that contains massive higher-spin excitations in its spectrum and that has been conjectured to emerge as a broken phase of a higher-spin gauge theory – and from the AdS/CFT correspondence – in which higher-spin theories constitute the gravitational duals of weakly interacting conformal field theories (CFTs). Interactions involving particles of spin greater than two are however severely constrained by various results that accumulated in the literature since the 1960’s. These show that higher-spin interactions must display at least some unconventional features: for instance, interactions typically involve more than two derivatives and an infinite number of particles of increasing spin is needed for consistency when the dimension D of spacetime is greater than three. Another common lore that persisted in the literature for several years is that, when D > 3, higher-spin interactions are possible only in the presence of a non-vanishing cosmological constant. This conclusion was first supported by the explicit construction of interacting models on Anti de Sitter (AdS) space by Vasiliev and collaborators. More recently, it was reinforced by the observation that the AdS/CFT correspondence provides a sort of indirect justification for the existence of higher-spin theories on Anti de Sitter space. More recently, a class of interacting models in Minkowski space dubbed chiral higher-spin gravity has been however proposed, and the goal of this Thesis is to show that the algebraic setup that underlies Vasiliev’s construction can be extended to Minkowski space too.

The problem of constructing a consistent interacting theory of higher-spin gravity can be reformulated in algebraic terms, by requiring the existence of a non-Abelian Lie algebra encoding the correction to the free gauge symmetry brought by cubic vertices. The existence of this algebra is not guaranteed, and in fact, starting from the free dynamics given by the Fronsdal Lagrangian in Minkowski space, one cannot find any such algebra, since the commutators of the gauge transformations induced by the known non-Abelian cubic vertices do not close. In AdS space, the setup is quite different since this algebra exists and it is essentially unique. This algebra can then be “gauged” in the sense that a complete interacting theory can be obtained along the same lines as for ordinary gauge theories such as Yang-Mills, or even general relativity in the language of Cartan. Essentially, one considers fields and curvatures taking values in the resulting higher-spin algebra and impose equations of motion using the curvatures as building blocks. Vasiliev construction involves additional technical steps, but the essential ingredients are the same as in the Cartan formulation of gravity. In the first part of this Thesis, we show that the infinite-dimensional Lie algebra encoding higher-spin symmetry and underlying the construction of Vasiliev’s fully interacting theory in AdS spacetime admits a smooth flat space limit, which is also unique under certain conditions. Moreover, we show that the gauging of this algebra reproduces the correct linearised dynamics of massless fields of arbitrary spin on flat background and provides an explanation for the origin of the obstruction to the completion of the Fronsdal Lagrangian in flat space.

Another important feature of Vasiliev’s theory is its holographic character, that is the fact that it can be described by a conformal field theory living on the co-dimension one boundary of AdS spacetime. The proposed dual of higher-spin gravity is actually very simple: in its simplesr form it is conjectured to be described by a collection of scalar fields. The flat space counterpart of the AdS/CFT correspondence is a subject of active research, and Carrollian holography – which is the conjectured duality between a gravitational theory in asymptotically Minkowski spacetime and a Carrollian Conformal Field Theory living on null infinity – provides a promising candidate. In the second part of this Thesis, we show that our proposed flat space higher-spin algebra fits within the framework of Carrollian holography, as one can understand it as an algebra of higher-differential operators of the scalar Carrollian CFT. The latter admits an extension as an algebra of asymptotic symmetries further extending the BMS algebra by higher-spin modes, which are expected to play a role in a holographic quantum theory of gravity in flat space.

Keywords: higher-spin gravity; higher-spin symmetries; flat-space holography