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Two-Form Supergravity, Superstring Couplings, and Goldstino Superfields in Three Dimensions

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Two-Form Supergravity, Superstring Couplings, and Goldstino Superfields in Three Dimensions PHYSICAL REVIEW D 96, 126015 (2017) Two-form supergravity, superstring couplings, and Goldstino superfields in three dimensions † ‡ Evgeny I. Buchbinder,1,* Jessica Hutomo,1, Sergei M. Kuzenko,1, and Gabriele Tartaglino-Mazzucchelli2,§ 1School of Physics and Astrophysics M013, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Australia 2Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium (Received 12 October 2017; published 27 December 2017) We develop off-shell formulations for N ¼ 1 and N ¼ 2 anti-de Sitter supergravity theories in three spacetime dimensions that contain gauge two-forms in the auxiliary field sector. These formulations are shown to allow consistent couplings of supergravity to the Green-Schwarz superstring with N ¼ 1 or N ¼ 2 spacetime supersymmetry. In addition to being κ-symmetric, the Green-Schwarz superstring actions constructed are also invariant under super-Weyl transformations of the target space. We also present a detailed study of models for spontaneously broken local supersymmetry in three dimensions obtained by coupling the known off-shell N ¼ 1 and N ¼ 2 supergravity theories to nilpotent Goldstino superfields. DOI: 10.1103/PhysRevD.96.126015 I. INTRODUCTION by the requirement of classical κ-symmetry of the Green- Schwarz superstring. In regard to the 3D case, it should be The Green-Schwarz superstring action with N ¼ 1 or kept in mind that at the time when Refs. [12,13] were N ¼ 2 supersymmetry [1] exists for spacetime dimensions written, those off-shell formulations for N ¼ 1 and N ¼ 2 D ¼ 3, 4, 6 and 10. However, its light-cone quantization supergravity theories, which are suitable to describe con- breaks Lorentz invariance unless either D ¼ 10 (see, e.g., sistent superstring propagation, had not been described in [2]), which corresponds to critical superstring theory, or the literature. One such theory, the so-called N 2 D ¼ 3 [3,4]. Due to the exceptional status of the D ¼ 3 ¼ two-form supergravity, was formulated six years ago case, it is of interest to study in more detail three- N 2 dimensional (3D) superstring actions in supergravity back- [15]. A new ¼ supergravity theory will be given in grounds. In order for such a coupling to supergravity to be the present paper. N 1 consistent, the superstring action must possess a local The present work aims at developing (i) ¼ and N 2 fermionic invariance (known as the κ-symmetry) which ¼ anti-de Sitter (AdS) supergravity theories that was first discovered in the cases of massive [5,6] and contain gauge two-forms in the auxiliary field sector; massless [7] superparticles.1 The κ-symmetry, in its turn, (ii) consistent couplings of these supergravity theories to N 1 N 2 requires the superstring action to include a Wess-Zumino the Green-Schwarz superstring with ¼ or ¼ term associated with a closed super three-form in curved supersymmetry; and (iii) models for spontaneously broken superspace such that (i) it is the field strength of a gauge 3D supergravity obtained by coupling the off-shell N ¼ 1 super two-form, and (ii) it reduces to a nonvanishing or N ¼ 2 supergravity theories to Goldstino superfields. invariant super three-form in the flat superspace limit. The first two goals are related to the above discussion. As to The latter requirement means that only certain supergravity point (iii), it requires additional comments. formulations are suitable to describe string propagation in In the last 3 years, there has been considerable interest in curved superspace. The constraints on the geometry of models for spontaneously broken N ¼ 1 local supersym- curved D ¼ 3, 4, 6, 10 superspace, which are required metry in four dimensions [16–27], including the models for for the coupling of supergravity to the Green-Schwarz off-shell supergravity coupled to nilpotent Goldstino super- superstring, were studied about 30 years ago [10–13]. fields. One of the reasons for this interest is that a positive Nevertheless, there still remain some open questions and contribution to the cosmological constant is generated once unexplored cases, as can be seen from the recent work by the local supersymmetry becomes spontaneously broken. Tseytlin and Wulff [14] that determined the precise con- For instance, if the supergravity multiplet is coupled to an straints imposed on the 10D target superspace geometry irreducible Goldstino superfield [20,25,28–30] (with the Volkov-Akulov Goldstino [31,32] being the only indepen- * universal [email protected] dent component field of the superfield), a positive † 2 [email protected] contribution to the cosmological constant is generated, ‡ [email protected] §[email protected] 1For reviews of various aspects of the κ-symmetry, see, 2The gravitino becomes massive in accordance with the super- e.g., [8,9]. Higgs effect [33–35]. 2470-0010=2017=96(12)=126015(31) 126015-1 © 2017 American Physical Society EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) which is proportional to f2, with the parameter f setting the We follow the notation and make use of the results of scale of supersymmetry breaking. The same positive [37]. Every supergravity theory will be realized as a super- contribution is generated by the reducible Goldstino super- Weyl invariant coupling of conformal supergravity to a fields used in the models studied in [18,19,26].3 There is compensating supermultiplet. one special reducible Goldstino superfield, the nilpotent three-form multiplet introduced in [24,27], which yields A. Conformal supergravity a dynamical positive contribution to the cosmological N 1 M3j2 constant. Consider a curved ¼ superspace, , parame- M m θμ 0 Since our Universe is characterized by a positive trized by local real coordinates z ¼ðx ; Þ, with m ¼ , μ 1 m θμ cosmological constant, and a theoretical explanation for 1, 2 and ¼ , 2, of which x are bosonic and this positivity is required, 4D supergravity theories with fermionic. We introduce a preferred basis of one-forms A a α nilpotent Goldstino superfields deserve further studies. E ¼ðE ;E Þ and its dual basis EA ¼ðEa;EαÞ, In this respect, it is also of some interest to construct A M A M models for spontaneously broken local 3D N ¼ 1 and E ¼ dz EM ;EA ¼ EA ∂M; ð2:1Þ N ¼ 2 supersymmetry that are obtained by coupling off- shell 3D supergravity to nilpotent superfields. This is one of which will be referred to as the supervielbein and its the objectives of the present work. inverse, respectively. This paper is organized as follows. Sections II and III The superspace structure group is SLð2; RÞ, the double N 1 N 2 provide thorough discussions of the ¼ and ¼ off- cover of the connected Lorentz group SO0ð2; 1Þ. The shell supergravity theories, respectively. Section IV covariant derivatives have the form describes consistent couplings of the two-form supergrav- ity theories to the Green-Schwarz superstring with N ¼ 1 DA ¼ðDa; DαÞ¼EA þ ΩA; ð2:2Þ or N ¼ 2 spacetime supersymmetry. The nilpotent Goldstino superfields and their couplings to various off- where shell supergravity theories are presented in Sec. V. Here we introduce only those reducible Goldstino superfields that 1 1 are defined in the presence of conformal supergravity bc b βγ ΩA ¼ ΩA Mbc ¼ −ΩA Mb ¼ ΩA Mβγ ð2:3Þ without making use of any conformal compensator. 2 2 Section VI contains concluding comments and a brief discussion of the results obtained. The main body of the is the Lorentz connection. The Lorentz generators with two − paper is accompanied by three technical appendices which vector indices (Mab ¼ Mba), one vector index (Ma), and are devoted to the analysis of the component structure of two spinor indices (Mαβ ¼ Mβα) are related to each other 1 ε bc γa several Goldstino superfield models in the flat superspace by the rules Ma ¼ 2 abcM and Mαβ ¼ð ÞαβMa. These limit. generators act on a vector Vc and a spinor Ψγ as follows: 2η Ψ ε Ψ II. TWO-FORM MULTIPLET IN N =1 MabVc ¼ c½aVb;Mαβ γ ¼ γðα βÞ: ð2:4Þ SUPERGRAVITY The covariant derivatives are characterized by graded In this section we describe two off-shell formulations for commutation relations N ¼ 1 AdS supergravity, with 4 þ 4 off-shell degrees of freedom, which differ from each other by their auxiliary 1 C cd fields. One of them is known since the late 1970s (see [36] ½DA; DBg¼TAB DC þ RAB Mcd; ð2:5Þ for a review), and its auxiliary field is a scalar. The other 2 formulation is obtained by replacing the auxiliary scalar C cd field with the field strength of a gauge two-form, which where TAB and RAB are the torsion and curvature requires the use of a different compensating supermultiplet. tensors, respectively. To describe supergravity, the covar- As was pointed out in [12,13], the latter formulation is iant derivatives have to obey certain torsion constraints [36] required for consistent coupling to the Green-Schwarz such that the algebra (2.5) takes the form superstring. However, the technical details of this formu- lation have not been described in the literature, to the best fDα; Dβg¼2iDαβ − 4iSMαβ; ð2:6aÞ of our knowledge. γ δρ ½Da; Dβ¼ðγaÞβ ½SDγ − CγδρM 2 3 c c b γ The notion of irreducible and reducible Goldstino superfields − ½DβSδa − 2εab ðγ ÞβγD SMc; ð2:6bÞ was introduced in [25]. 3 126015-2 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) 1 2 with the parameter σ being a real unconstrained superfield, D D ε γc αβγ − γc βγD S D ½ a; b¼ abc i 2 ð ÞαβC 3 ð Þ β γ provided the torsion superfields transform as i γc αβ γd γδD 3 1 þ ð Þ ð Þ ðαCβγδÞ i 2 2 δσS ¼ σS − D σ; δσCαβγ ¼ σCαβγ − D αβDγ σ: 4 2 2 ð Þ 2i D2S 4S2 ηcd ð2:13Þ þ 3 þ Md : ð2:6cÞ The super-Weyl transformation of the vielbein is Here the scalar S is real, while the symmetric spinor Cαβγ ¼ CðαβγÞ is imaginary.
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