Andrea Campoleoni

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• Ph.D.
• FNRS Research Associate at University of Mons

These lecture notes provide an introduction to higher-spin gauge theories in three spacetime dimensions, with a focus on their asymptotic symmetries, their holographic description in terms of conformal field theories with W-symmetries as well as on their couplings to scalar matter.

Using carefully chosen projections, we consider different Carroll limits of relativistic Dirac fermions in any spacetime dimensions. These limits define Carroll fermions of two types: electric and magnetic. The latter type transforms as a reducible but indecomposable representation of the Carroll group. We also build action principles for all Carro...

A bstract We provide holographic realisations in Minkowski spacetime of a free conformal Carrollian scalar field living at null infinity. To this end, we first show that the electric and magnetic limits of a relativistic conformal scalar are equivalent and we study the representation of the Carroll, Poincaré and BMS algebras that is realised on the...

A bstract Motivated by the generation of action principles from off-shell dualisation, we present a general class of free, topological theories in three dimensional Minkowski spacetime that exhibit higher-spin gauge invariance. In the spin-two case, we recover a dual reformulation of the triplet system already known, while the higher-spin systems t...

A bstract Ricci-flat solutions to Einstein’s equations in four dimensions are obtained as the flat limit of Einstein spacetimes with negative cosmological constant. In the limiting process, the anti-de Sitter energy-momentum tensor is expanded in Laurent series in powers of the cosmological constant, endowing the system with the infinite number of...

A bstract We study the renormalized action and the renormalized presymplectic potential for Maxwell fields on Anti de Sitter backgrounds of any dimensions. We then use these results to explicitly derive finite boundary charges for angle-dependent asymptotic symmetries. We consider both Poincaré and Bondi coordinates, the former allowing us to contr...

It was shown that the Lie algebra underlying higher-spin holography admits a contraction including a Poincaré subalgebra in any space-time dimensions. The associated curvatures, however, do not reproduce upon linearization those that are usually employed to formulate the equations of motion of free massless particles in Minkowski space. We show tha...

It was shown that the Lie algebra underlying higher-spin holography admits a contraction including a Poincar\'e subalgebra in any space-time dimensions. The associated curvatures, however, do not reproduce upon linearisation those that are usually employed to formulate the equations of motion of free massless particles in Minkowski space. We show t...

A bstract We perform a nontrivial check of Witten’s recently proposed admissibility criterion for complex metrics. We consider the ‘quasi-Euclidean’ metrics obtained from continuing the BTZ class of metrics to imaginary time. Of special interest are the overspinning metrics, which are smooth in this three-dimensional context. Their inclusion as sad...

A bstract We investigate the swampland distance conjecture in higher-spin gravity. To this end, we study multicritical generalizations of large- N vector models, bosonic and fermionic, and we compute the quantum information distance along selected renormalization-group trajectories toward the higher-spin limit. In contrast to the expected exponenti...

We show that, in any space-time dimension, the on-shell (electric) conformal Carrollian scalar can be interpreted as the flat-space limit of the singleton representation of the conformal algebra. In fact, a recently proposed higher-spin algebra for Minkowski spacetime amounts to the Poincaré enveloping algebra on the corresponding module. This high...

A bstract Relaxing the Bondi gauge, the solution space of three-dimensional gravity in the metric formulation has been shown to contain an additional free function that promotes the boundary metric to a Lorentz or Carroll frame, in asymptotically AdS or flat spacetimes. We pursue this analysis and show that the solution space also admits a finite s...

A bstract We explicitly establish the equivalence between the magnetic Carrollian limit of Einstein gravity defined through the Hamiltonian formalism and the Carrollian theory of gravity defined through a gauging of the Carroll algebra along the lines of standard Poincaré (or (A)dS) gaugings.

Higher Spin Gravity refers to extensions of gravity including at least one field of spin greater than two. These extensions are expected to provide manageable models of quantum gravity thanks to the infinite-dimensional (higher spin) gauge symmetry constraining them. One of the key aspects of Higher Spin Gravity/Symmetry is the range and diversity...

A bstract We present higher-spin algebras containing a Poincaré subalgebra and with the same set of generators as the Lie algebras that are relevant to Vasiliev’s equations in any space-time dimension D ≥ 3. Given these properties, they can be considered either as candidate rigid symmetries for higher-spin gauge theories in Minkowski space or as Ca...

Considering the Poincaré group ISO(d−1,1) in any dimension d>3, we characterise the coadjoint orbits that are associated with massive and massless particles of discrete spin. We also comment on how our analysis extends to the case of continuous spin.

A bstract We investigate asymptotic symmetries in flat backgrounds of dimension higher than or equal to four. For spin two we provide the counterpart of the extended BMS transformations found by Campiglia and Laddha in four-dimensional Minkowski space. We then identify higher-spin supertranslations and generalised superrotations in any dimension. T...

We explore memory effects associated with both Abelian and non-Abelian radiation getting to null infinity, in arbitrary even spacetime dimensions. Together with classical memories, linear and nonlinear, amounting to permanent kicks in the velocity of the probes, we also discuss the higher-dimensional counterparts of quantum memory effects, manifest...

We describe the dynamics of two-dimensional relativistic and Carrollian fluids. These are mapped holographically to three-dimensional locally anti-de Sitter and locally Minkowski spacetimes, respectively. To this end, we use Eddington-Finkelstein coordinates, and grant general curved two-dimensional geometries as hosts for hydrodynamics. This requi...

Starting from the dual Lagrangians recently obtained for (partially) massless spin-2 fields in the Stueckelberg formulation, we write the equations of motion for (partially) massless gravitons in (A)dS in the form of twisted-duality relations. In both cases, the latter admit a smooth flat limit. In the massless case, this limit reproduces the gravi...

Weinberg's celebrated factorisation theorem holds for soft quanta of arbitrary integer spin. The same result, for spin one and two, has been rederived assuming that the infinite-dimensional asymptotic symmetry group of Maxwell's equations and of asymptotically flat spaces leave the S-matrix invariant. For higher spins, on the other hand, no such in...

We provide a unified treatment of electric-magnetic duality, at the action level and with manifest Lorentz invariance, for massive, massless as well as partially-massless gravitons propagating in maximally symmetric spacetimes of any dimension n>3. For massive and massless fields, we complete previous analyses that use parent-action techniques by g...

We provide a unified treatment of electric-magnetic duality, at the action level and with manifest Lorentz invariance, for massive, massless as well as partially-massless gravitons propagating in maximally symmetric spacetimes of any dimension n>3. For massive and massless fields, we complete previous analyses that use parent-action techniques by g...

We analyse the conservation laws associated with large gauge transformations of massless fields in Minkowski space. Our aim is to highlight the interplay between boundary conditions and finiteness of the asymptotically conserved charges in any space-time dimension, both even and odd, greater than or equal to three. After discussing non-linear Yang-...

We study the formulation of massless higher-spin gravity on AdS3 in a gauge in which the fundamental variables satisfy free field Poisson brackets. This gauge choice leaves a small portion of the gauge freedom unfixed, which should be further quotiented out. We show that doing so leads to a bulk version of the Coulomb gas formalism for W N CFT’s: t...

We analyse the conservation laws associated with large gauge transformations of massless fields in Minkowski space. Our aim is to highlight the interplay between boundary conditions and finiteness of the asymptotically conserved charges in any space-time dimension, both even and odd, greater than or equal to three. After discussing non-linear Yang-...

We study the formulation of massless higher-spin gravity on AdS$_3$ in a gauge in which the fundamental variables satisfy free field Poisson brackets. This gauge choice leaves a small portion of the gauge freedom unfixed, which should be further quotiented out. We show that doing so leads to a bulk version of the Coulomb gas formalism for $W_N$ CFT...

We study the large gauge transformations of massless higher-spin fields in four-dimensional Minkowski space. Upon imposing suitable fall-off conditions, providing higher-spin counterparts of the Bondi gauge, we observe the existence of an infinite-dimensional asymptotic symmetry algebra. The corresponding Ward identities can be held responsible for...

We study the large gauge transformations of massless higher-spin fields in four-dimensional Minkowski space. Upon imposing suitable fall-off conditions, providing higher-spin counterparts of the Bondi gauge, we observe the existence of an infinite-dimensional asymptotic symmetry algebra. The corresponding Ward identities can be held responsible for...

We build the asymptotic higher-spin charges associated with "improper" gauge transformations for fermionic higher-spin gauge fields on Anti de Sitter backgrounds of arbitrary dimension. This is achieved within the canonical formalism. We consider massless fields of spin s+1/2, described by a symmetric spinor-tensor of rank s in the Fang-Fronsdal ap...

We build the asymptotic higher-spin charges associated with "improper" gauge transformations for fermionic higher-spin gauge fields on Anti de Sitter backgrounds of arbitrary dimension. This is achieved within the canonical formalism. We consider massless fields of spin s+1/2, described by a symmetric spinor-tensor of rank s in the Fang-Fronsdal ap...

We study asymptotic charges for symmetric massless higher-spin fields on Anti de Sitter backgrounds of arbitrary dimension within the canonical formalism. We first analyse in detail the spin-3 example: we cast Fronsdal's action in Hamiltonian form, we derive the charges and we propose boundary conditions on the canonical variables that secure their...

We study asymptotic charges for symmetric massless higher-spin fields on Anti de Sitter backgrounds of arbitrary dimension within the canonical formalism. We first analyse in detail the spin-3 example: we cast Fronsdal's action in Hamiltonian form, we derive the charges and we propose boundary conditions on the canonical variables that secure their...

We build unitary representations of the BMS algebra and its higher-spin extensions in three dimensions, using induced representations as a guide. Our prescription naturally emerges from an ultrarelativistic limit of highest-weight representations of Virasoro and 𝒲 algebras, which is to be contrasted with nonrelativistic limits that typically give n...

We evaluate one-loop partition functions of higher-spin fields in thermal flat space with angular potentials; this computation is performed in arbitrary space-time dimension, and the result is a simple combination of Poincar\'e characters. We then focus on dimension three, showing that suitable products of one-loop partition functions coincide with...

The asymptotic structure of three-dimensional higher-spin anti-de Sitter gravity is analyzed in the metric approach, in which the fields are described by completely symmetric tensors and the dynamics is determined by the standard Einstein-Fronsdal action improved by higher order terms that secure gauge invariance. Precise boundary conditions are gi...

The conical defect solutions in higher-spin gauge theories on 2+1 dimensional space-times with AdS-asymptotics are conjectured to correspond to certain primary fields in the dual conformal field theory on the boundary. In this note we prove that indeed all higher-spin charges match.

We give a brief overview of some three-dimensional toy models for higher-spin interactions. We first review the construction of pure higher-spin gauge theories in terms of Chern-Simons theories. We then discuss how this setup could be modified along the lines of the known topologically massive theories.

We consider the coupling of a symmetric spin-3 gauge field μνρ to three-dimensional gravity in a second-order metric-like formulation. The action that corresponds to an Chern–Simons theory in the frame-like formulation is identified to quadratic order in the spin-3 field. We apply our result to compute corrections to the area law for higher spin bl...

We construct a class of smooth solutions in three-dimensional Vasiliev higher spin theories based on the gauge algebra hs[\lambda]. These solutions naturally generalize the previously constructed conical defect solutions in higher spin theories with sl(N) gauge algebra, to which they reduce when \lambda is taken to be equal to N. We provide evidenc...

We show how implementing invariance under divergence-free gauge transformations leads to a remarkably simple Lagrangian description of massless bosons of any spin. Our construction covers both flat and (A)dS backgrounds and extends to tensors of arbitrary mixed-symmetry type. Irreducible and traceless fields produce single-particle actions, while w...

We discuss how to systematically compute the asymptotic symmetry algebras of generic three-dimensional bosonic higher-spin gauge theories in backgrounds that are asymptotically AdS. We apply these techniques to a one-parameter family of higher-spin gauge theories that can be considered as large N limits of SL(N) x SL(N) Chern-Simons theories, and w...

We discuss the emergence of W-algebras as asymptotic symmetries of higher-spin gauge theories coupled to three-dimensional Einstein gravity with a negative cosmological constant. We focus on models involving a finite number of bosonic higher-spin fields, and especially on the example provided by the coupling of a spin-3 field to gravity. It is desc...

We review the structure of local Lagrangians and field equations for free bosonic and fermionic gauge fields of mixed symmetry in flat space. These are first presented in a constrained setting extending the metric formulation of linearized gravity, and then the ($\gamma$-)trace constraints on fields and gauge parameters are eliminated via the intro...

This paper is a sequel of arXiv:0810.4350 [hep-th], and is also devoted to the local "metric-like" unconstrained Lagrangians and field equations for higher-spin fields of mixed symmetry in flat space. Here we complete the previous constrained on-shell formulation of Labastida for Fermi fields, deriving the corresponding constrained Lagrangians both...

We review the structure of local Lagrangians and field equations for free bosonic and fermionic gauge fields of mixed symmetry in flat space. These are first presented in a constrained setting extending the metric formulation of linearized gravity, and then the ($\gamma$-)trace constraints on fields and gauge parameters are eliminated via the intro...

This is the first of two papers devoted to the local "metric-like" unconstrained Lagrangians and field equations for higher-spin gauge fields of mixed symmetry in flat space. Here we complete the previous constrained formulation of Labastida for Bose fields. We thus recover his Lagrangians via the Bianchi identities, before extending them to their...

We review the structure of local Lagrangians and field equations for free bosonic and fermionic gauge fields of mixed symmetry in flat space. These are first presented in a constrained setting and then the ($\g$-)trace constraints on fields and gauge parameters are eliminated via the introduction of a number of auxiliary fields.