Conformal Interactions Between Matter and Higher‐Spin (Super)Fields

Abstract

In even spacetime dimensions, the interacting bosonic conformal higher‐spin (CHS) theory can be realised as an induced action. The main ingredient in this definition is the model scriptSfalse[φ,hfalse]$\mathcal {S}[\varphi ,h]$ describing a complex scalar field φ coupled to an infinite set of background CHS fields h, with scriptSfalse[φ,hfalse]$\mathcal {S}[\varphi ,h]$ possessing a non‐abelian gauge symmetry. Two characteristic features of the perturbative constructions of scriptSfalse[φ,hfalse]$\mathcal {S}[\varphi , h]$ given in the literature are: (i) the background spacetime is flat; and (ii) conformal invariance is not manifest. In the present paper we provide a new derivation of this action in four dimensions such that (i) scriptSfalse[φ,hfalse]$\mathcal {S}[\varphi , h]$ is defined on an arbitrary conformally‐flat background; and (ii) the background conformal symmetry is manifestly realised. Next, our results are extended to the scriptN=1$\mathcal {N}=1$ supersymmetric case. Specifically, we construct, for the first time, a model scriptSfalse[normalΦ,Hfalse]$\mathcal {S}[\Phi , H]$ for a conformal scalar/chiral multiplet Φ coupled to an infinite set of background higher‐spin superfields H. Our action possesses a non‐abelian gauge symmetry which naturally generalises the linearised gauge transformations of conformal half‐integer superspin multiplets. The other fundamental features of this model are: (i) scriptSfalse[normalΦ,Hfalse]$\mathcal {S}[\Phi , H]$ is defined on an arbitrary conformally‐flat superspace background; and (ii) the background scriptN=1$\mathcal {N}=1$ superconformal symmetry is manifest. Making use of scriptSfalse[normalΦ,Hfalse]$\mathcal {S}[\Phi , H]$, an interacting superconformal higher‐spin theory can be defined as an induced action.

Covariant action for conformal higher spin gravity

“…□ k ϕ = 0, k > 1 for the scalar matter corresponds to F (0) = (p 2 ) k ; (b) one can choose different types of matter, e.g. fermion ψ, or, more generally, a mixed-symmetry (spin-)tensor field, see [114] for the discussion of the CHS gravity based on the free fermion (called Type-B) and [115,116] for further extensions; (c) supersymmetric extensions should also be possible and be based on the Clifford-Weyl algebra, see the recent [45,61] for N = 1.…”

“…We expect that the approach of this paper provides an efficient way to attack some of the problems of CHS fields that have been around for a while: whether conformal gravity is a consistent truncation of CHS gravity [39][40][41]? ; what are the gravitational backgrounds that admit free CHS fields [39][40][41][42][43][44][45]? ; the structure of (HS) Weyl anomaly and, hence, the problem of quantum consistency of CHS gravity.…”

“…The induced action approach should be generalizable to mixed-symmetry and supersymmetric cases. In particular, it has recently been advanced to N = 1 supersymmetric CHS case [45,61] and the quadratic part of the action has been obtained via the induced action technique [61].…”

“…Having such a formulation is important for any extension of gravity and should facilitate the study of these theories in the future, e.g. propagation of CHS fields on gravitational backgrounds [39][40][41][42][43][44][45].…”

Three‐Point Functions of Higher‐Spin Supercurrents in 4D N=1${{\cal N}}=1$ Superconformal Field Theory

“…For instance, explicit realisations in terms of free scalar and vector multiplets are known for $$J_{\alpha (r) {\dot{\alpha }}(r)}$$, in Minkowski, [ 63–69 ] AdS [ 70 ] and conformally‐flat backgrounds. [ 71 ] Higher‐spin supercurrent $$J_{\alpha (r) {\dot{\alpha }}(r)}$$ may also be realised in terms of the on‐shell, gauge‐invariant, chiral field strengths $$W_{\alpha (r)}$$ obeying $$ \bar{D}_{\dot{\beta }}W_{\alpha (r)} =0$$ and $$D^\beta W_{\beta \alpha (r-1)} =0$$ . Its realisation is given by [70, 72, 73]: $$$$\begin{eqnarray} J_{\alpha (r) {\dot{\alpha }}(r) } =W_{\alpha (r)} \bar{W}_{{\dot{\alpha }}(r)} .$$…”