We develop a manifestly conformal approach to describe linearised (super)conformal higher-spin gauge theories in arbitrary conformally flat backgrounds in three and four spacetime dimensions. Closed-form expressions in terms of gauge prepotentials are given for gauge-invariant higher-spin (super) Cotton and (super) Weyl tensors in three and four dimensions, respectively. The higher-spin (super) Weyl tensors are shown to be conformal primary (super)fields in arbitrary conformal (super)gravity backgrounds, however they are gauge invariant only if the background (super) Weyl tensor vanishes. The proposed higherspin actions are (super) Weyl-invariant on arbitrary curved backgrounds, however the appropriate higher-spin gauge invariance holds only in the conformally flat case. We also describe conformal models for generalised gauge fields that are used to describe partially massless dynamics in three and four dimensions. In particular, generalised higher-spin Cotton and Weyl tensors are introduced.

We elaborate on conformal higher-spin gauge theory in three-dimensional (3D) curved space. For any integer n > 2 we introduce a conformal spin-n 2 gauge field h (n) = h α 1 ...αn (with n spinor indices) of dimension (2 − n/2) and argue that it possesses a Weyl primary descendant C (n) of dimension (1 + n/2). The latter proves to be divergenceless and gauge invariant in any conformally flat space. Primary fields C (3) and C (4) coincide with the linearised Cottino and Cotton tensors, respectively. Associated with C (n) is a Chern-Simons-type action that is both Weyl and gauge invariant in any conformally flat space. These actions, which for n = 3 and n = 4 coincide with the linearised actions for conformal gravitino and conformal gravity, respectively, are used to construct gauge-invariant models for massive higher-spin fields in Minkowski and anti-de Sitter space. In the former case, the higher-derivative equations of motion are shown to be equivalent to those first-order equations which describe the irreducible unitary massive spin-n 2 representations of the 3D Poincaré group. Finally, we develop N = 1 supersymmetric extensions of the above results.

For every conformal gauge field h α(n)α(m) in four dimensions, with n ≥ m > 0, a gauge-invariant action is known to exist in arbitrary conformally flat backgrounds. If the Weyl tensor is non-vanishing, however, gauge invariance holds for a pure conformal field in the following cases: (i) n = m = 1 (Maxwell's field) on arbitrary gravitational backgrounds; and (ii) n = m + 1 = 2 (conformal gravitino) and n = m = 2 (conformal graviton) on Bach-flat backgrounds. It is believed that in other cases certain lower-spin fields must be introduced to ensure gauge invariance in Bach-flat backgrounds, although no closedform model has yet been constructed (except for conformal maximal depth fields with spin s = 5/2 and s = 3). In this paper we derive such a gauge-invariant model describing the dynamics of a conformal gauge field h α(3)α coupled to a self-dual two-form. Similar to other conformal higher-spin theories, it can be embedded in an off-shell superconformal gaugeinvariant action. To this end, we introduce a new family of N = 1 superconformal gauge multiplets described by unconstrained prepotentials Υ α(n) , with n > 0, and propose the corresponding gauge-invariant actions on conformally-flat backgrounds. We demonstrate that the n = 2 model, which contains h α(3)α at the component level, can be lifted to a Bach-flat background provided Υ α(2) is coupled to a chiral spinor Ω α. We also propose families of (super)conformal higher-derivative non-gauge actions and new superconformal operators in any curved space. Finally, through considerations based on supersymmetry, we argue that the conformal spin-3 field should always be accompanied by a conformal spin-2 field in order to ensure gauge invariance in a Bach-flat background.

The problem of constructing gauge-invariant actions for conformal higher-spin fields in curved backgrounds is known to be notoriously difficult. In this paper we present gauge-invariant models for conformal maximal depth fields with spin s = 5/2 and s = 3 in four-dimensional Bach-flat backgrounds. We find that certain lower-spin fields must be introduced to ensure gauge invariance when s > 2, which is analogous to a conjecture made earlier in the literature for conformal higher-spin fields of minimal depth.

We elaborate on the traceless and transverse spin projectors in four-dimensional de Sitter and anti-de Sitter spaces. The poles of these projectors are shown to correspond to partially massless fields. We also obtain a factorisation of the conformal operators associated with gauge fields of arbitrary Lorentz type (m/2, n/2), with m and n positive integers.1 In vector notation this reads D a , D b = −2S 2 M ab .