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.

In four spacetime dimensions, all N = 1 supergravity-matter systems can be formulated in the so-called U(1) superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background U(1) superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields ℓ (α 1 ...αm)(α 1 ...αn) , with m and n non-negative integers, m + n > 0, and elaborate on their significance in the following cases: (i) m = n = 1; (ii) m − 1 = n = 0; and (iii) m = n > 1. The (conformal) Killing vector superfields ℓ αα generate the (conformal) isometries of curved superspace, which are symmetries of every (conformal) supersymmetric field theory. The (conformal) Killing spinor superfields ℓ α generate extended (conformal) supersymmetry transformations. The (conformal) Killing tensor superfields with m = n > 1 prove to generate all higher symmetries of the (massless) massive Wess-Zumino operator.

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General $$ \mathcal{N} $$ N = (1, 0) supergravity-matter systems in six dimensions may be described using one of the two fully fledged superspace formulations for conformal supergravity: (i) SU(2) superspace; and (ii) conformal superspace. With motivation to develop rigid supersymmetric field theories in curved space, this paper is devoted to the study of the geometric symmetries of supergravity backgrounds. In particular, we introduce the notion of a conformal Killing spinor superfield ϵα, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector ξa and tensor ζa(n) superfields. The former parametrise conformal isometries of supergravity backgrounds, which in turn yield symmetries of every superconformal field theory. Meanwhile, the conformal Killing tensors of a given background are associated with higher symmetries of the hypermultiplet. By studying the higher symmetries of a non-conformal vector multiplet we introduce the concept of a Killing tensor superfield. We also analyse the problem of computing higher symmetries for the conformal d’Alembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on a limited class of backgrounds, including all conformally flat ones.

We propose generalised $$ \mathcal{N} $$ N = 1 superconformal higher-spin (SCHS) gauge multiplets of depth t, $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t , with n ≥ m ≥ 1. At the component level, for t > 2 they contain generalised conformal higher-spin (CHS) gauge fields with depths t − 1, t and t + 1. The supermultiplets with t = 1 and t = 2 include both ordinary and generalised CHS gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t on conformally-flat superspace backgrounds are then derived. For the case n = m = t = 1, corresponding to the maximal-depth conformal graviton supermultiplet, we extend this action to Bach-flat backgrounds. Models for superconformal non-gauge multiplets, which are expected to play an important role in the Bach-flat completions of the models for $$ {\Upsilon}_{\alpha (n)\overset{\cdot }{\alpha }(m)}^{(t)} $$ ϒ α n α ⋅ m t , are also provided. Finally we show that, on Bach-flat backgrounds, requiring gauge and Weyl invariance does not always determine a model for a CHS field uniquely.