Generalised superconformal higher-spin multiplets

Published for SISSA by Springer Received: January 22, 2021 Accepted: February 10, 2021 Published: March 19, 2021 Sergei M. Kuzenko, Michael Ponds and Emmanouil S.N. Raptakis Department of Physics M013, The University of Western Australia, 35 Stirling Highway, Perth W.A. 6009, Australia E-mail: , , Abstract: We propose generalised N = 1 superconformal higher-spin (SCHS) gauge (t) multiplets of depth t, Υα(n)α̇(m) , 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 (t) gauge fields. Super-Weyl and gauge invariant actions describing the dynamics of Υα(n)α̇(m) 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 (t) Υα(n)α̇(m) , 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. Keywords: Higher Spin Symmetry, Supergravity Models, Superspaces ArXiv ePrint: 2011.11300 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP03(2021)183 JHEP03(2021)183 Generalised superconformal higher-spin multiplets Contents 1 2 Generalised superconformal models 2.1 Generalised superconformal prepotentials and field strengths 2.2 Generalised superconformal actions 2.3 Wess-Zumino gauge for minimal depth supermultiplets 3 4 5 6 3 Maximal-depth conformal graviton supermultiplet 3.1 Gauge invariant action in Bach-flat background 3.2 The component action 8 9 9 4 Superconformal non-gauge models 4.1 Chiral supermultiplets 4.2 Longitudinal linear supermultiplets 11 12 13 5 Discussion 14 A N = 1 conformal superspace in four dimensions 17 B Superspace realisations of the conformal hook model 19 1 Introduction Ever since its inception in 1985 [1], conformal higher-spin (CHS) theory has been the recipient of sustained interest from the physics community. 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 [6–8] 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 [9–16]. As is well known, consistent models for conformal spin-3/2 and spin-2 fields may be formulated at most on Bach-flat backgrounds. 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. An efficient way to describe the dynamics of CHS fields in curved space is within the setting of conformal gravity, whose geometry is encoded by the conformally covariant derivative ∇a , see eq. (A.9). In the two-component spinor formalism, an ordinary conformal gauge field hα(n)α̇(m) = h(α1 ...αn )(α̇1 ...α̇m ) , with n ≥ m ≥ 1, is characterised by the gauge transformation law δ` hα(n)α̇(m) = ∇(α1 (α̇1 `α2 ...αn )α̇2 ...α̇m ) , –1– (1.1) JHEP03(2021)183 1 Introduction and carries conformal weight1 2 − 12 (n + m) . Due to their low conformal weights and consequently high derivative Lagrangians, CHS fields are notoriously difficult to work with. This is why no closed form models for spin s > 2 have been constructed on Bach-flat backgrounds. However, there exists a broader class of conformal gauge fields, the so-called generalised
(t) ones hα(n)α̇(m) . The latter carry the conformal weight t+1− 12 (n+m) and are characterised by gauge transformations with depth t
(t) δ` hα(n)α̇(m) = ∇(α1 (α̇1 · · · ∇αt α̇t `αt+1 ...αn )α̇t+1 ...α̇m ) , 1 ≤ t ≤ min(n, m) . (1.2) 1 (2) δ` hab = ∇(a ∇b) ` − ηab ∇c ∇c ` . 4 (1.3) This is in contrast to the ordinary conformal graviton, whose transformation law is instead 1 (1) δ` hab = ∇(a `b) − ηab ∇c `c . 4 (1) (1.4) (2) The conformal weights of hab and hab are 0 and 1 respectively, which means that their corresponding Weyl and gauge invariant actions are fourth and second order in derivatives. Historically, the first generalised conformal field appeared in the seminal work by Deser and Nepomechie [17, 18], where they discussed the maximal-depth conformal graviton. This concept was later extended to tensors of a generic symmetry type by Vasiliev in [19], where the corresponding conformal and gauge invariant actions in d-dimensional Minkowski space Md were also given (see [20–24] for more recent related studies). In the cases d = 3, 4, these actions were lifted to conformally-flat backgrounds in [14]. Superconformal higher-spin (SCHS) gauge multiplets contain CHS fields at the component level. 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. It is therefore of interest to elaborate on generalised SCHS multiplets and their gauge invariant actions on curved backgrounds, which is the main subject of this paper. 1 A (super)conformal field Φ (with its indices suppressed) is said to carry conformal weight ∆ if it satisfies DΦ = ∆Φ, with D being the dilatation generator; see appendix A. 2 In terms of the torsion-free Lorentz covariant derivative Da , there are additional curvature dependent terms present in (1.3). Its specific form is given by (3.13). –2– JHEP03(2021)183 By virtue of their relatively high conformal weights and consequently lower derivative Lagrangians, they can provide a much friendlier environment in which to study CHS theory. Indeed, gauge invariant actions on Bach-flat backgrounds have been explicitly derived for conformal spin s = 5/2 and s = 3 gauge fields with maximal depth [15]. As an example, the maximal-depth conformal graviton corresponds to the case n = m = t = 2, and in the more familiar vector notation its gauge transformation law is2 2 Generalised superconformal models To begin with, we recall the structure of N = 1 superconformal higher-spin multiplets [11, 14, 42]. For n ≥ m > 0, such a multiplet is formulated in terms of a prepotential Ψα(n)α̇(m) , and its conjugate for n 6= m, defined modulo the gauge transformation ¯ (α̇ ηα(n)α̇ ...α̇ ) , δξ,η Ψα(n)α̇(m) = ∇(α1 ξα2 ...αn )α̇(m) + ∇ m 1 2 (2.1) ¯ α̇ ) with unconstrained gauge parameters ξα(n−1)α̇(m) and ηα(n)α̇(m−1) . Here ∇A = (∇a ,∇α ,∇ are the covariant derivatives (...truncated)