Off-shell superconformal higher spin multiplets in four dimensions

Published for SISSA by Springer Received: February 20, 2017 Accepted: June 26, 2017 Published: July 7, 2017 Sergei M. Kuzenko,a Ruben Manvelyanb and Stefan Theisenc a School of Physics and Astrophysics M013, The University of Western Australia, 35 Stirling Highway, Crawley, W.A. 6009, Australia b Yerevan Physics Institute, Alikhanian Br. St. 2, 0036 Yerevan, Armenia c Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany E-mail: , , Abstract: We formulate off-shell N = 1 superconformal higher spin multiplets in four spacetime dimensions and briefly discuss their coupling to conformal supergravity. As an example, we explicitly work out the coupling of the superconformal gravitino multiplet to conformal supergravity. The corresponding action is super-Weyl invariant for arbitrary supergravity backgrounds. However, it is gauge invariant only if the supersymmetric Bach tensor vanishes. This is similar to linearised conformal supergravity in curved background. Keywords: Higher Spin Symmetry, Conformal Field Theory, Superspaces ArXiv ePrint: 1701.00682 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP07(2017)034 JHEP07(2017)034 Off-shell superconformal higher spin multiplets in four dimensions Contents 1 2 Superconformal transformations 2 3 Off-shell superconformal multiplets in flat space 3.1 Half-integer superspin 3.2 Integer superspin 3.3 Superconformal gravitino multiplet 3.4 Generalisations 4 4 6 8 8 4 Off-shell superconformal multiplets in supergravity 4.1 General considerations 4.2 Superconformal gravitino multiplet 4.3 Linearised conformal supergravity 8 9 11 13 5 Concluding comments 13 A The Grimm-Wess-Zumino superspace geometry 16 B Super-Weyl transformations 17 1 Introduction The role of conformal field theories as cornerstones for the exploration of more general quantum field theories, which are connected to them via renormalization group flows, has been appreciated since a long time ago. Higher spin gauge theories [1–7] have an even longer history and have attracted considerable interest recently. It is quite natural to combine the two symmetry principles and to study conformal higher spin theories [8]. A further symmetry which is compatible with conformal and higher spin symmetry is supersymmetry. This leads to superconformal higher spin theories, first advocated in [9], which are the main focus of this note. More specifically, we introduce off-shell N = 1 superconformal higher spin multiplets in four dimensions and analyse in some detail the problem of lifting such supermultiplets to curved backgrounds. Our main technical tool, as far as the supersymmetry and supergravity aspects are concerned, is superspace and we refer to [10] for a thorough introduction to this formalism. We first study superconformal higher spin theories in flat superspace. In section 2 we review superconformal transformations and the important notion of superconformal primaries. In section 3 we construct off-shell superconformal higher spin multiplets: starting from prepotentials and their transformation laws under higher spin gauge transformations –1– JHEP07(2017)034 1 Introduction There are different ways to describe N = 1 conformal supergravity in superspace.1 The simplest option is to make use of the superspace geometry of [11, 12], which underlies the Wess-Zumino approach [14] to the old minimal formulation for N = 1 supergravity developed independently in [15, 16]. Another option is to work with the U(1) superspace proposed by Howe [17, 18]. Finally, one can make use of the so-called conformal superspace [19]. The three superspace approaches to N = 1 conformal supergravity are equivalent, although each of them has certain advantages and disadvantages (see [19] for a detailed discussion of the relationship between these formulations). In this paper we make use of the oldest and simplest superspace setting [11, 12]. 2 Superconformal transformations In this section we briefly recall the structure of N = 1 superconformal transformations in Minkowski superspace M4|4 , see [10] for more details. We denote by z A = (xa , θα , θ̄α̇ ) the Cartesian coordinates for M4|4 , and use the notation DA = (∂a , Dα , D̄α̇ ) for the superspace covariant derivatives. Let ξ = ξ B DB = ξ b ∂b + ξ β Dβ + ξ¯ D̄β̇ be a real supervector field on M4|4 . It is called β̇ conformal Killing if it obeys the equation   1 bc ξ + K [ξ]Mbc , DA + δσ[ξ] DA = 0 , 2 (2.1) for some local Lorentz (K bc [ξ]) and super-Weyl (σ[ξ]) parameters. The super-Weyl transformation of the covariant derivatives is defined in (B.1). Choosing A = α and A = α̇ in (2.1) implies that the spinor components of ξ A as well as the parameters K bc [ξ] and σ[ξ] 1 See [8] for a nice review of N = 1 conformal supergravity and the complete list of references. –2– JHEP07(2017)034 and under superconformal transformations, we construct field strengths and invariant actions. The two cases of half-integer and integer superspin as well as the superconformal gravitino multiplet have to be treated separately. The component fields of these multiplets are totally symmetric traceless tensor and tensor-spinor fields. More general fields will be briefly discussed in the last part of section 3. In section 4 we couple the superconformal higher spin multiplets to conformal supergravity where the notion of superconformal transformations is replaced by that of super-Weyl transformations. While super-Weyl invariance is easy to achieve, gauge invariance requires non-minimal couplings. We explicitly discuss the gravitino multiplet, but defer the general case to the future. Section 5 contains concluding comments, including the explicit expressions for conserved higher spin current multiplets that correspond to the superconformal higher spin prepotentials. The main body of the paper is accompanied by two technical appendices. Appendix A contains those results concerning the Grimm-Wess-Zumino superspace geometry [11, 12], which are important for understanding the supergravity part of this paper. Appendix B contains the essential information about the super-Weyl transformations [13]. are expressed in terms of the vector components of ξ A : i ξ α = − D̄α̇ ξ α̇α , 8 Kαβ [ξ] = D(α ξβ) , 1 σ[ξ] = (Dα ξ α + 2D̄α̇ ξ¯α̇ ) , 3 D̄γ̇ ξ α = 0 , (2.2a) D̄γ̇ Kαβ [ξ] = 0 , (2.2b) D̄γ̇ σ[ξ] = 0 . (2.2c) The vector components of ξ A obey the equations ⇐⇒ D̄(α̇ ξβ̇)β = 0 , (2.3) D2 ξβ β̇ = 0 ⇐⇒ D̄2 ξβ̇β = 0 , (2.4) which imply as well as the ordinary conformal Killing equation 1 ∂a ξb + ∂b ξa = ηab ∂c ξ c . 2 (2.5) A useful corollary of (2.1) with A = α is Dγ K αβ [ξ] = δγ(α Dβ) σ[ξ] =⇒ D2 σ[ξ] = 0 . (2.6) Another consequence of (2.1) is ∂a σ[ξ] = ∂a σ̄[ξ] =⇒ ∂a Dβ σ[ξ] = 0 . (2.7) The most general conformal Killing supervector field proves to be 1 α̇β β̇α β̇α α̇β α̇α α̇ α ξ+ = aα̇α + (σ + σ̄) xα̇α + + K̄ β̇ x+ + x+ Kβ + x+ bβ β̇ x+ (...truncated)