New locally (super)conformal gauge models in Bach-flat backgrounds

For every conformal gauge field $$ {h}_{\alpha (n)\overset{\cdot }{\alpha }(m)} $$ 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 closed-form 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 $$ {h}_{\alpha (3)\overset{\cdot }{\alpha }} $$ 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 gauge-invariant action. To this end, we introduce a new family of $$ \mathcal{N} $$ N = 1 superconformal gauge multiplets described by unconstrained prepotentials ϒα(n), with n > 0, and propose the corresponding gauge-invariant actions on conformally-flat backgrounds. We demonstrate that the n = 2 model, which contains $$ {h}_{\alpha (3)\overset{\cdot }{\alpha }} $$ h α 3 α ⋅ at the component level, can be lifted to a Bach-flat background provided ϒα(2) is coupled to a chiral spinor Ωα. 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.