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

Abstract

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.

Symmetries of $$ \mathcal{N} $$ = (1, 0) supergravity backgrounds in six dimensions

“…There exists a class of combined local dilatations and special conformal transformations preserving the gauge B = 0. These exactly reproduce the super-Weyl transformations(2.11), see e.g [29,37,95]…”

Generalised superconformal higher-spin multiplets

“…Hence, an effective method of studying various CHS models is to study the corresponding SCHS models which induce them. In the case of minimal depth CHS fields, such studies have already been initiated [11,14,16]. However, the supersymmetric multiplets containing generalised CHS gauge fields have not yet appeared in the literature, neither at the superspace nor component level.…”

“…Superconformal gauge-invariant actions for the multiplets (2.1) were constructed in [11] in Minkowski superspace, while for arbitrary conformally flat backgrounds they were derived in [14]. Superconformal gauge-invariant actions for the multiplets (2.2) with n > 1 were derived in [16] for conformally flat backgrounds. The action for the superconformal gravitino multiplet, which corresponds to n = 1 in (2.2), was described earlier in Bach-flat backgrounds [11] (see also [16]).…”

“…There are a number of reasons for this, including a consistent Lagrangian formulation of bosonic CHS fields at the cubic [2,3] and full non-linear level [4,5] (see [6][7][8] for later developments). One of its central open problems is the construction of gauge invariant models for CHS fields on curved gravitational backgrounds, where much effort has been directed [9][10][11][12][13][14][15][16]. As is well known, consistent models for conformal spin-3/2 and spin-2 fields may be formulated at most on Bach-flat backgrounds.…”

“…As is well known, consistent models for conformal spin-3/2 and spin-2 fields may be formulated at most on Bach-flat backgrounds. While this is thought also to be true for spins greater than two, recent studies [10,12,15,16] indicate that in this case it is necessary to switch on non-minimal couplings to subsidiary conformal fields.…”