Symmetries of curved superspace in five dimensions

Sergei M. Kuzenko 0 1 2 Joseph Novak 0 1 2 Gabriele Tartaglino-Mazzucchelli 0 1 2 0 Open Access , c The Authors 1 35 Stirling Highway , Crawley W.A. 6009 , Australia 2 School of Physics M013, The University of Western Australia We develop a formalism to construct supersymmetric backgrounds within the superspace formulation for five-dimensional (5D) conformal supergravity given in arXiv:0802.3953. Our approach is applicable to any off-shell formulation for 5D minimal Poincare and anti-de Sitter supergravity theories realized as the Weyl multiplet coupled with two compensators. For those superspace backgrounds which obey the equations of motion for (gauged) supergravity, we naturally reproduce the supersymmetric solutions constructed a decade ago by Gauntlett et al. For certain supersymmetric backgrounds with eight supercharges, we construct a large family of off-shell supersymmetric sigma models such that the superfield Lagrangian is given in terms of the Kahler potential of a real analytic Kahler manifold. ArXiv ePrint: 1406.0727 1 Introduction 2 The Weyl multiplet in superspace 3 (Conformal) isometries Conformal isometries Conformally related superspaces 3.3 Isometries 3.1 3.2 4.1 4 Supersymmetric backgrounds: general formalism 5 Supersymmetric backgrounds: eight supercharges Conformal Killing spinors Killing spinors The case s 6= 0 The case s = 0 and caij 6= 0 5.3 The case s = 0 and caij = 0 6 Vector multiplet compensator Supersymmetric backgrounds 6.2 The dilaton Weyl multiplet 7 O(2) multiplet compensator Supersymmetric backgrounds 7.2 Supersymmetric backgrounds with eight supercharges 8 Off-shell supergravity Supersymmetric backgrounds 8.2 Supersymmetric backgrounds with eight supercharges The case s 6= 0 The case s = 0 9 Supersymmetric solutions in Poincare and anti-de Sitter supergravities 26 10 Concluding comments A (Conformal) isometries in curved space B Conformal Killing spinors and bilinears matter couplings in five dimensions (5D) [13].1 As concerns the Weyl multiplet of 5D conformal supergravity, its formulation given in [3] may be simply thought of as an alternative realization of the one discovered a few years earlier within the component superconformal tensor calculus [712].2 However, the real power of the superspace approach of [13] is that it offers a generating formalism to realize the most general locally supersymmetric This is achieved by making use of the concept of covariant projective supermultiplets [13]. These supermultiplets are a curved-superspace extension of the so-called superconformal shell projective multiplets pioneered by Lindstrom and Rocek [16, 17]. Among the most interesting covariant projective supermultiplets are polar ones that have infinitely many auxiliary fields. Such off-shell supermultiplets are practically impossible to engineer or to deal with in the framework of superconformal tensor calculus. This is why they had never appeared within the component settings of [412], which deal only with hypermultiplets either with a gauged central charge [49] or that are on-shell [1012]. The superspace formulation developed in [13] provides a universal setting to generate off-shell supersymmetric field theories on curved backgrounds. For instance, the general 5D read off from the supergravity-matter systems proposed in [13] by properly freezing the supergravity fields. Of course, the problem of constructing supersymmetric field theories on a given spacetime is well formulated only if this manifold is a supersymmetric background, i.e. it admits rigid supersymmetries. Thus one is naturally led to the more general problem of looking for those curved superspaces that possess (conformal) isometries. In the case of years ago. The approach presented in [19] is universal, for in principle it may be generalized to supersymmetric backgrounds associated with any supergravity theory formulated in superspace. In particular, it has already been used to construct rigid supersymmetric Recently, a number of publications have appeared devoted to the construction of supersymmetric backgrounds associated with off-shell supergravity theories in diverse dimensions, see [2744] and references therein. Inspired by [27], these works used component is used, e.g., in [412]. The reason for this choice is that dimensional reduction of five-dimensional theories by Howe in 1981 [13] (using the supercurrent multiplet constructed in [14]) and fully elaborated in [1, 2]. It was re-discoverd by Zucker [46] who elaborated on the component implications of [13]. the key component results of, e.g., [27, 34] from the more general superspace construction of [19]. Recently, the formalism of [45] was extended to construct supersymmetric gravity [24, 47]. The results obtained are in agreement with the component considerations of [38, 39, 43]. In the present paper, we apply the ideas and techniques developed in [45, 46] This paper is organized as follows. Section 2 contains a brief review of the superspace formulation for 5D conformal supergravity [3]. In section 3 we study (conformal) isometries of a background superspace. In section 4 we study bosonic backgrounds that possess at least one (conformal) Killing spinor. Maximally supersymmetric backgrounds are described in section 5. Sections 6 and 7 are concerned with additional restrictions on the background geometry, which arise when a single conformal compensator, a vector multiplet or an O(2) multiplet, is turned on. Section 8 is devoted to supersymmetric backgrounds in off-shell supergravity. Supersymmetric solutions in Poincare and anti-de Sitter supergravity theories are studied in section 9. Finally, concluding comments are given in section 10. The main body of the paper is accompanied by two technical appendices. In appendix A we recall how the problem of computing the (conformal) isometries of a curved spacetime is addressed within the Weyl-invariant formulation for gravity. In appendix B we discuss the properties of bilinears constructed from a conformal Killing spinor. The Weyl multiplet in superspace In this section we briefly review the superspace description [3] of the Weyl multiplet of 5D conformal supergravity. Our notation and conventions follow those introduced in [48] (see also the appendix of [2]). condition (i) = i = ij . The tangent-space group is chosen to be SO(4, 1)SU(2) j Here EA = E M (z) M is the (inverse) supervielbein, with M = /zM , A bc Mbc = M , A A Mab = Mba , is the Lorentz connection, and Jkl = Jlk is the SU(2) connection. The Lorentz generators with vector indices (Mab) and spinor of SO(4, 1) SU(2) act on the covariant derivatives as follows:3 [J kl, Di] = i(kDl) , [M, Dk] = (Dk) , [Mab, Dc] = 2c[aDb] , The supergravity gauge group is generated by local transformations of the form Kcd(z)Mcd + Kkl(z)Jkl , with all the gauge parameters obeying natural reality conditions but are otherwise arbitrary. Given a tensor superfield U (z) (with its indices supp (...truncated)