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

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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 supergravity 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 formulation 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. The dimension-2 Bianchi identities a a δσE ¼ −σE ; ð2:14aÞ imply that 4 γ i α 1 α i αβ DαCβγδ ¼ D αCβγδ − iεα βDγδ S ⇒ D Cαβγ ¼ − DαβS: δ − σ − b γ D σ ð Þ ð Þ 3 σE ¼ 2 E 2 E ð bÞ β : ð2:14bÞ ð2:7Þ The gauge group of conformal supergravity is generated Throughout this section we make use of the definition by the local transformations (2.10) and (2.12). Due to the 2 α D ≔ D Dα. super-Weyl invariance, the above geometry describes the The definition of the torsion and curvature tensors, Weyl multiplet of N ¼ 1 conformal supergravity [38], a Eq. (2.5), can be recast in the language of superforms, which consists of the vielbein em ðxÞ and the gravitino α 4 which will be used in Sec. IV. Starting from the Lorentz ψ m ðxÞ (no auxiliary fields). connection ΩA given by (2.3), we introduce the connection A tensor superfield T is said to be (super-Weyl) primary one-form of weight w if its super-Weyl transformation law is

Ω ¼ ECΩ ; ΩV ¼ Ω BV ¼ ECΩ BV ; C A A B CA B δσT ¼ wσT: ð2:15Þ

VA ¼ðVa; ΨαÞ: ð2:8Þ Such superfields will be of primary importance in what Then the torsion and curvature two-forms are follows. 1 The action for conformal supergravity was constructed C ≔ B ∧ A C − C B ∧ Ω C T 2 E E TAB ¼ dE þ E B ; ð2:9aÞ for the first time by van Nieuwenhuizen [38] using the N ¼ 1 superconformal tensor calculus. More recently, it 1 was reformulated in superspace [39], as well as within the D ≔ B ∧ A D Ω D − Ω E ∧ Ω D RC 2 E E RABC ¼ d C C E : ð2:9bÞ superform approach [39,40]. The interested reader is referred to these publications for the technical details. The gauge group of conformal supergravity includes local transformations of the form B. Supersymmetric action 1 C cd To construct a locally supersymmetric and super-Weyl δKDA ¼½K; DA; K ¼ ξ EC þ K Mcd; ð2:10Þ 2 invariant action [37], one needs a real scalar Lagrangian L that is super-Weyl primary of weight þ2, with the gauge parameters ξCðzÞ and KbcðzÞ obeying natural reality conditions but otherwise arbitrary. Here C δσL ¼ 2σL: ð2:16Þ the supervector field ξ ¼ ξ EC describes a general coor- dinate transformation, and Kcd a local Lorentz transformation. The transformation (2.10) acts on a tensor superfield T The action is as follows: Z 3 2 A S ¼ i d xd θEL;E¼ BerðEM Þ: ð2:17Þ δKT ¼ KT: ð2:11Þ

The algebra of covariant derivatives is invariant under The action is super-Weyl invariant, since the super-Weyl super-Weyl transformations transformation of E proves to be δσE ¼ −2σE. 1 Instead of defining the action using the superspace δ D σD Dβσ σ α ¼ 2 α þ Mαβ; ð2:12aÞ integration, an alternative approach is to construct a

i γδ b c 4 S δσD σD γ DγσDδ ε D σM ; 2:12b The super-Weyl transformation of implies that its lowest a ¼ a þ 2 ð aÞ þ abc ð Þ S component jθ¼0 is a pure gauge.

126015-3 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) dimensionless super three-form Ξ3½L which is given For completeness we also give the equation of motion for in terms of L and possesses the following properties: the gravitational superfield (which is the N ¼ 1 super- (i) Ξ3½L is closed, dΞ3½L¼0; and (ii) Ξ3½L is supersymmetric analog of the gravitational field) 5 Weyl invariant, δσΞ3½L¼0. Modulo an overall numerical factor, these conditions prove to completely determine 1 −1 −2 Cαβγ ¼ 0; Cαβγ ≔− φ ðD αβDγ − 2CαβγÞφ : Ξ3½L to be 2 ð Þ

i ð2:22bÞ Ξ L γ ∧ β ∧ a γ L 3½ ¼2 E E E ð aÞβγ 1 See [46] for the technical details. The specific feature of γ ∧ b ∧ aε γc δD L S C þ 4 E E E abcð Þγ δ and αβγ is that they are super-Weyl invariant. Note that it is possible to choose a super-Weyl gauge in which φ ¼ 1 1 2 − c ∧ b ∧ aε D 8S L and, therefore, S and Cαβγ coincide with S and Cαβγ, 24 E E E abcði þ Þ : ð2:18Þ respectively. In this gauge, Eqs. (2.22) describe, locally, the This super three-form was originally constructed in N ¼ 1 AdS superspace [47]. [43,45]; however its super-Weyl invariance was first The action (2.21) can readily be reduced to components. φ 1 described in [39]. The action (2.17) is recast via Ξ3½L In the super-Weyl gauge ¼ we obtain as follows: Z 1 1 Z 3 R − 4 2 8 λ SSG ¼ κ d xe 2 S þ S þ fermions; S ¼ Ξ3½L; ð2:19Þ 3 M a e ¼ detðem Þ; ð2:23Þ where the integration is carried out over a spacetime M3 a ≔ a ≔ S being homotopic to the bosonic body of the curved where em ðxÞ Em jθ¼0 and SðxÞ jθ¼0. Integrating superspace M3j2 obtained by switching off the Grassmann out the auxiliary field S turns the action into variables. Z 1 1 3 R − Λ SSG ¼ d xe AdS þ fermions; C. AdS supergravity κ 2 Λ −4λ2 Both AdS and Poincar´e supergravity theories can be AdS ¼ : ð2:24Þ realized as super-Weyl invariant systems describing the coupling of conformal supergravity to a compensating D. Two-form supergravity multiplet. The standard choice for compensator is a nowhere vanishing scalar superfield φ, such that φ−1 exists, In this section we introduce a variant formulation for with the super-Weyl transformation N ¼ 1 AdS supergravity which is obtained by replacing the conformal compensator φ4 with a two-form multiplet.6 1 Let us first consider a massless two-form multiplet δσφ ¼ σφ: ð2:20Þ 2 coupled to conformal supergravity. It is described by a The action for N ¼ 1 AdS supergravity is given by real scalar superfield defined by Z 4 α − 3 2θ DαφD φ − 2Sφ2 λφ4 L ¼ D Λα; ð2:25Þ SSG ¼ κ i d xd Efi α þ g;

ð2:21Þ where the prepotential Λα is a primary real spinor superfield of dimension 3=2, where κ is the gravitational coupling constant, and the parameter λ determines the cosmological constant. Setting 3 λ 0 N 1 δ Λ σΛ ¼ in (2.21) gives the action for ¼ Poincar´e σ α ¼ 2 α: ð2:26Þ supergravity. The equation of motion for the compensator is This super-Weyl transformation implies that L is primary of dimension 2, S λ S ≔ φ−3 i D2 S φ ¼ ; 2 þ : ð2:22aÞ δσL ¼ 2σL: ð2:27Þ

5See [41–44] for the construction of locally supersymmetric invariants in D spacetime dimensions by using closed super 6In the case of Minkowski superspace, the two-form multiplet D-forms. was described in [36].

126015-4 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) The superfield L defined by (2.25) is a gauge-invariant field for the gravitational superfield, which corresponds to 1 strength with respect to gauge transformations of the form (2.31), is obtained from (2.22b) by replacing φ → L4.

δ Λ i DβD ζ 2Sζ Dαδ Λ 0 ζ α ¼ 2 α β þ α; ζ α ¼ ; ð2:28Þ E. Superform formulation for the two-form multiplet where the gauge parameter ζα is an arbitrary real spinor In this subsection we present a superform formulation for superfield. The gauge invariance of L follows from the the three-form multiplet coupled to conformal supergravity, identity as an extension of the flat-superspace construction given in [36]. Let us consider a gauge super two-form β 8i β βγ D DαDβ ¼ 4iSDα − ðD SÞMαβ − 2iCαβγM : ð2:29Þ 3 1 1 N ∧ M B ∧ A B2 ¼ 2 dz dz BMN ¼ 2 E E BAB; ð2:34Þ The gauge parameter in (2.28) is defined modulo arbitrary shifts of the form which is defined modulo gauge transformations of the form ζ → ζ0 ζ D ξ ξ¯ ξ α α ¼ α þ i α ; ¼ ; ð2:30Þ N B B2 → B2 þ dA1;A1 ¼ dz AN ¼ E AB; ð2:35Þ in the sense that δζ0 Λα ¼ δζΛα. This property means that the two-form multiplet is a gauge theory with linearly where the gauge parameter A1 is an arbitrary super one- dependent generators, in accordance with the terminology form. Associated with the potential B2 is the gauge- of the Batalin-Vilkovisky quantization [48]. invariant field strength We now assume L to be nowhere vanishing, such that 1 −1 P N M L exists. Then L can be used as a conformal compensator H3 ≔ dB2 ¼ dz ∧ dz ∧ dz ∂MBNP corresponding to a variant formulation of AdS supergravity. 2 1 4 1 Upon replacement φ → L = , the supergravity action (2.21) C B A D ¼ E ∧ E ∧ E fDABBC − TAB BDCg: ð2:36Þ turns into 2 Z 4 pffiffiffiffi i By construction, H3 is an exact super three-form, and hence − 3 2θ Dα D − 2S 0 SSG ¼ κ i d xd E L 16 ln L α ln L : it is closed, dH3 ¼ . We are interested in a closed super three-form H3 such ð2:31Þ that (i) its components are descendants of a scalar primary superfield L, and (ii) its lowest nonzero component is The supersymmetric cosmological term in (2.21) does not constrained to be Haβγ ¼ iðγaÞβγL. It turns out that the φ4 DαΛ contribute, since turns into L ¼ α, which is a total closure condition, dH3 ¼ 0, completely determine the derivative. Hence, the N ¼ 1 two-form supergravity does entire super three-form to be not allow for a supersymmetric cosmological term. This is analogous to the new minimal formulation for N ¼ 1 i γ ∧ β ∧ a γ supergravity in four dimensions [49–51]. However, the H3½L¼2 E E E ð aÞβγL difference from the new minimal supergravity is that a 1 γ ∧ b ∧ aε γc δD cosmological term is now generated dynamically. þ 4 E E E abcð Þγ δL For the theory with action (2.31), the equation of motion 1 for the compensator is − c ∧ b ∧ aε D2 8S 24 E E E abcði þ ÞL; ð2:37Þ −3 i 2 1 D S 0 S ≔ 4 D S 4 L → α ¼ ; L 2 þ L ; ð2:32Þ which is obtained from (2.18) by replacing L. In general, if L is an arbitrary scalar superfield, the superform H3 given by (2.37) is closed but not exact. and therefore α However, if we choose L ≔ D Λα in (2.37) then H3 turns out to be exact. In fact, the following super two-form S ¼ λ ¼ const: ð2:33Þ 1 If a solution with λ ≠ 0 is chosen, it describes an AdS Λ − β ∧ a γ γΛ − b ∧ aε γc ρτD Λ B2½ α¼ iE E ð aÞβ γ 4 E E abcð Þ ρ τ; background. Unlike the supergravity formulation (2.21), the action (2.31) does not contain a free parameter. ð2:38Þ The negative cosmological constant is generated dynamically. It should be pointed out that the equation of motion is such that

126015-5 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) α dB2½Λα¼H3½D Λα: ð2:39Þ A. Conformal supergravity We consider a curved N ¼ 2 superspace, M3j4, para- This proves that, if we consider the two- and three-forms m μ ¯ metrized by local bosonic (x ) and fermionic (θ , θμ) M m μ ¯ 1 coordinates z ¼ðx ; θ ; θμÞ, where m ¼ 0, 1, 2 and c ρτ B ¼ − ε ðγ Þ DρΛτ; ð2:40aÞ μ ¯ ab 2 abc μ ¼ 1, 2. The Grassmann variables θ and θμ are related to each other by complex conjugation: θμ ¼ θ¯μ. The 1 2 δ A a α ¯ H ¼ − ε ðiD þ 8SÞD Λδ; ð2:40bÞ supervielbein E ¼ðE ;E ; EαÞ and its inverse EA ¼ abc 4 abc ¯ α ðEa;Eα; E Þ are defined similarly to (2.1). Within the superspace formulation for N 2 conformal the latter is the field strength of the former, ¼ supergravity proposed in [53] and fully developed in [37], α the structure group is SLð2; RÞ ×Uð1Þ. The covariant H ¼ 3D B þ 2ε ðD SÞΛα: ð2:41Þ abc ½a bc abc derivatives have the form Using the super-Weyl transformation laws (2.14) and ¯ α (2.26), one can show that the superform (2.38) is super- DA ¼ðDa; Dα; D Þ¼EA þ ΩA; ΩA ≔ ΩA þ iΦAJ : Weyl invariant, ð3:1Þ α δσB2½Λα¼0 ⇒ δσH3½D Λα¼0: ð2:42Þ We recall that the Lorentz connection ΩA can be written in This result will be important for our analysis in Sec. IVA. several equivalent forms (2.3). The U(1) generator acts on Choosing B2 in the form (2.38) corresponds to a partial the covariant derivatives as follows: fixing of the gauge freedom (2.35). The residual gauge ¯ α ¯ α freedom is given by ½J ; Dα¼Dα; ½J ; D ¼−D : ð3:2Þ

δ Λ δ Λ ⇒ δ Λ 0 ζB2½ α¼B2½ ζ α d ζB2½ α¼ ; ð2:43Þ In general, the covariant derivatives have graded commutation relations of the form where δζΛα is defined by (2.28). D D CD III. TWO-FORM MULTIPLETS IN N =2 ½ A; Bg¼TAB C þ RAB; 1 SUPERGRAVITY ≔ cd J RAB 2 RAB Mcd þ iRAB : ð3:3Þ It is well known that the 3D AdS group is reducible, In order to describe the multiplet of conformal supergravity, SOð2; 2Þ ≅ ðSLð2; RÞ ×SLð2; RÞÞ=Z2; certain constraints should be imposed on the torsion tensor and so are its supersymmetric extensions, OSpðpj2; RÞ × [53]. Solving these constraints leads to the following OSpðqj2; RÞ. This implies that N -extended AdS super- algebra of covariant derivatives: gravity exists in several versions [52]. These are known as ¯ the ðp; qÞ AdS supergravity theories where the non- fDα; Dβg¼−4RMαβ; ð3:4aÞ negative integers p ≥ q are such that N ¼ p þ q.7 In this N 2 section we choose ¼ and describe four off-shell ¯ c Dα; Dβ −2i γ D − 2CαβJ − 4iεαβSJ formulations for (1,1) AdS supergravity and one for f g¼ ð Þαβ c γδ (2,0) AdS supergravity. Only one of these five off-shell þ 4iSMαβ − 2εαβC Mγδ; ð3:4bÞ supergravity theories is new, the so-called complex two- form supergravity; the others were presented in [15].

b γ c γ ¯ ¯ γ γ δρ ½Da; Dβ¼iεabcðγ Þβ C Dγ þðγaÞβ SDγ − iðγaÞβγRD − ðγaÞβ CγδρM 1 2 − 2D S D¯ ¯ − ε γb α 2D S D¯ ¯ c 3 ð β þ i βRÞMa 3 abcð Þβ ð α þ i αRÞM 1 1 − γ αγ γ γ 8D S D¯ ¯ J 2 ð aÞ Cαβγ þ 3 ð aÞβ ð γ þ i γRÞ ; ð3:4cÞ

7For any values of p and q allowed, the pure ðp; qÞ AdS supergravity was constructed in [52] as a Chern-Simons theory with the gauge group OSpðpj2; RÞ × OSpðqj2; RÞ.

126015-6 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) 1 4i 2 D D ε γc αβεγδ − ¯ ε D¯ S ε D D ½ a; b¼2 abcð Þ iCαβδ þ 3 δðα βÞ þ 3 δðα βÞR γ 1 4i 2 ε γc αβεγδ − ε D S − ε D¯ ¯ D¯ þ 2 abcð Þ iCαβδ þ 3 δðα βÞ 3 δðα βÞR γ 1 1 c αβ τδ ¯ ¯ c 2 ¯ 2 ¯ − ε γ γ D τCδαβ D τCδαβ δ D R D R abc 4 ð Þ ð dÞ ði ð Þ þ i ð ÞÞþ6 dð þ Þ 2 c α ¯ c c ¯ 2 d δ D DαS − 4C C − 4δ RR S M þ 3 i d d dð þ Þ 1 ε γc αβ D D¯ S − εcefD C − 4SCc J þ i abc 2 ð Þ ½ α; β e f : ð3:4dÞ

The algebra involves four dimension-one torsion super- The important property of the algebra (3.4) is that its fields: a real scalar S, a complex scalar R, and its conjugate form is preserved under super-Weyl transformations of the ¯ R, and a real vector Ca. The U(1) charge of R is −2. These covariant derivatives [15,37] torsion superfields obey differential constraints implied by 1 the Bianchi identities, which are δ D σD Dγσ − D σJ σ α ¼ 2 α þ Mγα α ; ð3:10aÞ ¯ DαR ¼ 0; ð3:5aÞ 1 ¯ ¯ ¯ γ ¯ δσDα ¼ σDα þ D σMγα þ DασJ ; ð3:10bÞ ðD¯ 2 − 4RÞS ¼ 0; ð3:5bÞ 2

1 i γδ ¯ i γδ ¯ D C − ε D¯ ¯ 4 D S δσD ¼ σD − ðγ Þ D γσDδ − ðγ Þ D γσDδ α βγ ¼ iCαβγ 3 αðβð γÞR þ i γÞ Þ: ð3:5cÞ a a 2 a ð Þ 2 a ð Þ i ε Dbσ c − γ γδ D D¯ σJ In this paper we make use of the definitions þ abc M 8 ð aÞ ½ γ; δ ð3:10cÞ

2 α ¯ 2 ¯ ¯ α D ≔ D Dα; D ≔ DαD : ð3:6Þ and the torsion tensors

As follows from (3.5c), the complex dimension-3=2 sym- i α ¯ δσS ¼ σS þ D Dασ; ð3:10dÞ metric spinor Cαβγ, which appears in (3.4), is a descendant 4 of the torsion three-vector C , Cαβγ ¼ −iD αCβγ . a ð Þ 1 The definition of the torsion and curvature tensors, γδ ¯ δσCa ¼ σCa þ ðγaÞ ½Dγ; Dδσ; ð3:10eÞ Eq. (3.3), can be recast in the superform notation, which 8 will be used in Sec. IV. Associated with the connection Ω , A 1 Ω CΩ ¯ 2 Eq. (3.1), is the connection one-form ¼ E C. Its action δσR ¼ σR þ D σ: ð3:10fÞ on a real super-vector 4

¯ α Here the super-Weyl parameter σ is an unconstrained real V ¼ðV ; Ψα; Ψ Þ; J Ψα ¼ Ψα ð3:7Þ A a scalar superfield. It follows from (3.10) that the super-Weyl is given by transformation law of the supervielbein is

a a B B B δσE ¼ −σE ; ð3:11aÞ ΩVA ¼ ΩA VB ¼ ΩA VB þ iΦA VB; ð3:8Þ

B B 1 i with ΩA and ΦA being the Lorentz and U(1) connections, α α b αγ ¯ δσE ¼ − σE þ E ðγbÞ Dγσ; respectively. Using the definitions given, the torsion and 2 2 1 i curvature two-forms are δ ¯ − σ ¯ b γ Dγσ σEα ¼ 2 Eα þ 2 E ð bÞαγ : ð3:11bÞ 1 C ≔ B ∧ A C − C B ∧ Ω C T 2 E E TAB ¼ dE þ E B ; ð3:9aÞ The group of super-Weyl transformations must be a subgroup of the supergravity gauge group in order for 1 the superspace geometry under consideration to describe R D ≔ EB ∧ EAR D ¼ dΩ D − Ω E ∧ Ω D: ð3:9bÞ C 2 ABC C C E the multiplet of N ¼ 2 conformal supergravity.

126015-7 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) A tensor superfield T of a U(1) charge q, J T ¼ qT,is can be read off using the general formalism of integrating out said to be super-Weyl primary if its super-Weyl trans- fermionic dimensions, which was developed in [56]. formation law is The two actions, (3.14) and (3.17), are related to each other as follows: δσT ¼ wσT; ð3:12Þ Z Z d3xd2θd2θ¯EL ¼ d3xd2θEL ; for some constant parameter w which will be referred to as c the super-Weyl weight of T. 1 L ≔− D¯ 2 − 4 L The action for N ¼ 2 conformal supergravity was c 4 ð RÞ : ð3:19Þ constructed for the first time by Roček and van Nieuwenhuizen [54] using the N ¼ 2 superconformal This relation shows that the chiral action, or its conjugate tensor calculus. More recently, it was reformulated within antichiral action, is more fundamental than (3.14). the superform approach [40]. The interested reader is The chiral projection operator in (3.19) defined by referred to these publications for the technical details. 1 Δ¯ ≔− ðD¯ 2 − 4RÞð3:20Þ B. Supersymmetric actions 4 As in the 4D N ¼ 1 case, there are two (closely related) plays a fundamental role in N ¼ 2 supergravity. Among its locally supersymmetric and super-Weyl invariant actions in most important properties is the following: given a primary 3D N ¼ 2 supergravity [37]. ψ ¯ complex scalar satisfying Given a real scalar Lagrangian L ¼ L with the super- Weyl transformation law J ψ ¼ð2 − wÞψ; δσψ ¼ðw − 1Þσψ; ð3:21Þ

δσL ¼ σL; ð3:13Þ for some constant super-Weyl weight w, its descendant the action ϕ ¼ Δ¯ ψ ð3:22Þ Z 3 2 2 ¯ A is a primary chiral superfield of super-Weyl weight w, S ¼ d xd θd θEL;E¼ BerðEM Þð3:14Þ ¯ Dαϕ ¼ 0; J ϕ ¼ −wϕ; δσϕ ¼ wσϕ: ð3:23Þ is invariant under the supergravity gauge group. It is also super-Weyl invariant due to the transformation law For every primary chiral scalar superfield, its super-Weyl weight w and U(1) charge q are related to each other as δ −σ σE ¼ E: ð3:15Þ w þ q ¼ 0, in accordance with [37]. Any superfield ϕ with the properties (3.23) will be referred to as a weight-w chiral L Given a covariantly chiral scalar Lagrangian c of super- scalar. Weyl weight two, The chiral action, Eq. (3.17), can be represented as an integral over the full superspace, D¯ L 0 JL −2L δ L 2σL α c ¼ ; c ¼ c; σ c ¼ c; Z ð3:16Þ 3 2θ 2θ¯ CL Sc ¼ d xd d E c; ð3:24Þ the following chiral action if we make use of an improved complex linear superfield C Z defined by the two properties: 3 2θEL (i) C obeys the constraint Sc ¼ d xd c ð3:17Þ Δ¯ C ¼ 1; ð3:25aÞ is locally supersymmetric and super-Weyl invariant. Action (3.17) involves integration over the chiral subspace of the (ii) the transformation properties of C are full superspace, with E the chiral density possessing the properties δσC ¼ −σC; J C ¼ 2C: ð3:25bÞ C JE ¼ 2E; δσE ¼ −2σE: ð3:18Þ A possible choice for is η¯ 1 The explicit expression for E in terms of the supergravity ¯ C ¼ ¯ ; Dαη ¼ 0; δση ¼ ση; ð3:26Þ prepotentials is given in [55]. Alternatively, the chiral density Δ η¯ 2

126015-8 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) for some covariantly chiral superfield η such that Δη is component results of [59], for the cosmological constant nowhere vanishing. In case C is not required to be super- one obtains Weyl primary, it can be identified with R−1, Z Λ ¼ −4jμj2: ð3:32Þ L AdS 3 2θ 2θ¯ c Sc ¼ d xd d E ; ð3:27Þ R The above minimal formulation for (1,1) AdS supergravity (which was called type I minimal supergravity in provided R is nowhere vanishing. This representation is [15]) is the 3D analog of the old minimal formulation for analogous to that discovered by Siegel [57] and Zumino 4D N ¼ 1 supergravity [60–62]. [58] in 4D N ¼ 1 supergravity. For the supergravity theory with action (3.31), the The chiral action can also be described using the super equation of motion for the chiral compensator is three-form constructed in [59] R ¼ μ; R ≔ Φ−3Δ¯ Φ¯ : ð3:33aÞ Ξ L −2 ¯ ∧ ¯ ∧ a γ βγL 3½ c¼ Eγ Eβ E ð aÞ c

i ¯ b a d γδ We also reproduce the equation of motion for the N ¼ 2 − Eγ ∧ E ∧ E ε ðγ Þ DδL 2 abd c gravitational superfield8 1 c ∧ b ∧ aε D2 − 16 ¯ L þ E E E abcð RÞ c: ð3:28Þ 1 24 C 0 C ≔− D D¯ − 4C ΦΦ¯ −1 αβ ¼ ; αβ 4 ð½ ðα; βÞ αβÞð Þ ; This superform is closed and super-Weyl invariant, ð3:33bÞ

dΞ3½L ¼0; δσΞ3½L ¼0: ð3:29Þ c c see [46] for the technical details. The specific feature of R and Cαβ is that they are super-Weyl invariant. The super- The chiral action is equivalently represented as Weyl and local U(1) transformations can be used to choose Z the gauge Φ ¼ 1, which implies that S ¼ 0 and R and Cαβ Ξ L Sc ¼ 3½ c; ð3:30Þ R Cαβ M3 coincide with the torsion superfields and , respectively. In this gauge, every solution to the equations (3.33) where the integration is carried out over a spacetime M3 is locally diffeomorphic to the (1,1) AdS superspace [47]. being homotopic to the bosonic body of the curved Within the nonminimal formulation for (1,1) AdS super- M3j4 gravity [15], the conformal compensators are an improved superspace obtained by switching off the Grassmann ¯ variables. complex linear scalar Γ and its conjugate Γ. The former has the transformation properties C. AdS supergravity δσΓ −σΓ; J Γ 2Γ : There are two off-shell formulations for (1,1) AdS ¼ ¼ ð3 34aÞ supergravity developed in [15], minimal and nonminimal ones, which do not have gauge two-forms in the sector of and obeys the improved linear constraint auxiliary fields. ΔΓ¯ ¼ μ ¼ const; ð3:34bÞ 1. (1,1) AdS supergravity In the minimal case, the conformal compensators are a compare with (3.25). The supergravity action is ¯ ¯ weight-1=2 chiral scalar Φ, DαΦ ¼ 0, and its conjugate Φ. Z −1 2 Of course, Φ has to be nowhere vanishing, such that Φ nonminimal 3 2 2 ¯ ¯ −1=2 S 1 1 ¼ − d xd θd θEðΓΓÞ : ð3:35Þ exists, in order to serve as a conformal compensator. ð ; ÞSG κ The supergravity action is Z As demonstrated in [15], this theory is dual to the minimal 4 3 2 2 AdS supergravity, Eq. (3.31). The theory under consid- Sminimal ¼ − d xd θd θ¯EΦΦ¯ ð1;1ÞSG κ eration is the 3D analog of the nonminimal N ¼ 1 AdS Z μ supergravity in four dimensions [64]. Both formulations 3 2θEΦ4 þ κ d xd þ c:c: ; ð3:31Þ lead to the (1,1) AdS superspace [15,47] as the maximally supersymmetric solution. where μ is a complex parameter. The second terms in the action is the supersymmetric cosmological term. Using the 8The N ¼ 2 gravitational superfield was introduced in [55,63].

126015-9 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) 2. (2,0) AdS supergravity is locally diffeomorphic to the (2,0) AdS super- The conformal compensator for (2,0) AdS supergravity space [15,47]. is a linear multiplet [15,37,53] describing the field strength The above supergravity theory (called type II minimal of an Abelian vector multiplet. It is realized in terms of a supergravity in [15]) is the 3D analog of the new minimal N 1 – real scalar superfield L ¼ L¯ subject to the constraint for ¼ supergravity in four dimensions [49 51]. The latter theory is known to allow no supersymmetric cos- Δ¯ L ¼ 0 ⇔ ΔL ¼ 0; ð3:36Þ mological term. Such a supersymmetric cosmological term does exist in the 3D case, and it is given by the Chern- ξ ξ ≠ 0 which is consistent with the super-Weyl transformation Simons -term in (3.41).For the theory possesses a law maximally supersymmetric solution, which is the (2,0) AdS superspace [15,47] corresponding to the (2,0) AdS supersymmetry [52]. δσL ¼ σL: ð3:37Þ

The constraint (3.36) is solved in terms of a real uncon- D. Two-form supergravity strained prepotential V, There is one more variant off-shell formulation for (1,1) AdS supergravity proposed in [15]. Its conformal compen- α ¯ ¯ L ¼ iD DαV; V ¼ V; ð3:38Þ sator is the so-called two-form multiplet, which is the 3D cousin of the well-known three-form multiplet in 4D which is defined modulo gauge transformations of the N ¼ 1 supersymmetry, which was proposed by Gates form [65] and reviewed in [36,66]. ¯ ¯ In curved superspace, the two-form multiplet is δλV ¼ λ þ λ; Dαλ ¼ 0: ð3:39Þ described by a real unconstrained scalar prepotential P ¼ P¯ which enters any action functional, S ¼ S½Π; Π¯ , To reproduce the super-Weyl transformation (3.37),it only via the covariantly chiral descendant suffices to choose Π ¼ Δ¯ P ð3:43Þ δσV ¼ 0: ð3:40Þ and its conjugate Π¯ . In order for Π to be a primary In order to be used as a conformal compensator, L has to P −1 superfield, the prepotential should possess the super- be nowhere vanishing, such that L exists. The action for Weyl transformation law (2,0) AdS supergravity was constructed in [15].Itis

Z δσP σP; 3:44 4 ¼ ð Þ 3 2θ 2θ¯ − 4 S 4ξ Sð2;0ÞSG ¼ d xd d EfL ln L V þ VLg; κ which implies ð3:41Þ δσΠ ¼ 2σΠ; J Π ¼ −2Π: ð3:45Þ where the parameter ξ determines the cosmological constant. The equations of motion for this theory can be written The chiral scalar (3.43) is a gauge-invariant field strength in the form [46] for gauge transformations of the form

δ Δ¯ 0 ¯ S ξ S ≔−i −1 DγD¯ 4 S LP ¼ L; L ¼ ; L ¼ L: ð3:46Þ ¼ ; 4 L ð γ ln L þ i Þ; ð3:42aÞ Here the linear gauge parameter can be expressed via an 1 ¯ −1 unconstrained superfield V as in (3.38). Since V is defined Cαβ ¼ 0; Cαβ ≔− ð½D α; Dβ − 4CαβÞL ; ð3:42bÞ 4 ð Þ modulo gauge transformations (3.39), we conclude that S S Π; Π¯ S C 9 any system with action ¼ ½ , which describes the with and αβ being super-Weyl invariant. The super- dynamics of the two-form multiplet, is a gauge theory with 1 Weyl gauge freedom can be used to set L ¼ , which linearly dependent generators. 0 S C implies R ¼ , and then and αβ turn into the torsion Lagrangian quantization of the two-form multiplet can superfields S and Cαβ, respectively. Under the gauge be carried out similarly to that of the 4D N ¼ 1 three-form condition chosen, every solution to the equations (3.42) multiplet coupled to supergravity [67] (see [68] for a review). 9 Φ4 → Π The vector superfield Cαβ should not be confused with Upon replacement in (3.31) the supergravity (3.33b). action turns into

126015-10 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) Z βγ γ β 4 1 1 Π − ¯ ∧ ¯ ∧ a γ Π− ∧ ∧ a γ Π¯ − 3 2 2 ¯ ¯ H3½ ¼ iEγ Eβ E ð aÞ iE E E ð aÞβγ Stwo form ¼ − d xd θd θE ðΠΠÞ4 − mP ð1;1ÞSG κ 2 1 Z ¯ b a d γδ þ Eγ ∧ E ∧ E εabdðγ Þ DδΠ 4 3 2 2 1 4 − θ θ¯ ΠΠ¯ 4 ¼ κ d xd d Eð Þ 1 Z − γ ∧ b ∧ aε γd D¯ δΠ¯ 4E E E abdð Þγδ m 3 2θEΠ þ d xd þ c:c: ; ð3:47Þ i κ c ∧ b ∧ aε þ 48E E E abc 2 ¯ ¯ 2 ¯ where m is a real parameter. In the second form, the action × ððD − 16RÞΠ −ðD − 16RÞΠÞ; ð3:51Þ is manifestly invariant under gauge transformations (3.46). The equation of motion for the compensator is and, hence, it is constructed solely in terms of the compensator Π and its conjugate Π¯ , with Π being related −3 4 1 4 to P as in (3.43). R þ R¯ ¼ 2m; R ≔ Π = Δ¯ Π¯ = ; ð3:48Þ The relation H3½Π¼dB2½P implies that the top components of B2½P and H3½Π, and therefore 1 c ρτ ¯ c Bab ¼ − εabcððγ Þ ½Dρ; Dτ − 8C ÞP; ð3:52aÞ R ¼ μ ¼ const: ð3:49Þ 8

− i ε D¯ 2 − 16 Π − D2 − 16 ¯ Π¯ The action for type I minimal supergravity (3.31) Habc ¼ 8 abcðð RÞ ð RÞ Þ; ð3:52bÞ involves two real parameters, Reμ and Imμ, which appear in the supersymmetric cosmological term. The action for are connected to each other as two-form supergravity (3.47) contains only one real 3D ε Dα − 2D¯ αS D parameter, m, which determines the corresponding super- Habc ¼ ½aBbc þ abcði R Þ αP symmetric cosmological term. As is seen from (3.49), the ¯ α ¯ α ¯ þ ε ðiD R þ 2D SÞDαP: ð3:53Þ second parameter Imμ is generated dynamically. At the abc component level, the cosmological constant in the theory Equations (3.52) and (3.53) tell us that the imaginary part (3.49) is given by (3.32). of the top component field of the chiral superfield Π, The two-form supergravity theory described above is the defined by F ¼ − 1 D2Πj, is the field strength of a gauge 3D analog of the variant formulation for 4D N ¼ 1 4 supergravity known as three-form supergravity. The latter two-form. was proposed for the first time by Gates and Siegel [66] The gauge transformation (3.46) of the prepotential P is and fully developed at the component level in [69,70]. The equivalent to the following transformation of the super super-Weyl invariant formulation for the three-form super- two-form (3.50): gravity was given in [71]. Our formulation of the 3D two- δ ⇒ δ Π 0 form supergravity is similar to [71]. LB2½P¼B2½L LH3½ ¼ : ð3:54Þ

This allows us to interpret B2½P as a gauge two-form E. Superform formulation for the two-form and H3½Π as its gauge-invariant field strength. The closed multiplet super two-form B2½L in (3.54) is actually exact, We now present a geometric formulation for the two- B2½L¼dA1, where A1 is the gauge potential of a vector form multiplet used in the previous section. Let us multiplet. introduce a super two-form B2 defined by Using the super-Weyl transformation laws (3.11) and (3.44), one can check that the superform (3.50) is invariant i under arbitrary super-Weyl transformations, − ¯ ∧ α β ∧ a γ Dγ B2½P¼ Eα E P þ 2 E E ð aÞβγ P δσB2½P¼0 ⇒ δσH3½Π¼0: ð3:55Þ i ¯ a βγ ¯ þ Eβ ∧ E ðγaÞ DγP 2 This property will be important for our analysis in 1 b a c ρτ ¯ c Sec. IV B. − ε E ∧ E γ Dρ; Dτ − 8C P 3:50 16 abc ðð Þ ½ Þ ð Þ Ξ L Let us recall the closed super three-form 3½ c, defined by Eq. (3.28), which generates the supersymmetric invari- ≔ L Π Π and consider its exterior derivative H3 dB2. It is not ant (3.30). If we choose c ¼ , with given by (3.43), difficult to check that H3 is given by the following then the exact super three-form H3½Π proves to be the expression: imaginary part of Ξ3½Π,

126015-11 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) i i ¯ ¯ We wish the chiral scalar Υ to be super-Weyl primary, H3½Π¼ Ξ3½Π − Ξ3½Π: ð3:56Þ 2 2 which means

The real part of Ξ3½Π, on the other hand, is not exact and δσΥ ¼ wΥσΥ; J Υ ¼ −wΥΥ; ð3:61Þ generates a nontrivial supersymmetric invariant, which may be realized as the full superspace integral (3.14), with P in accordance with (3.23). The transformation properties playing the role of the Lagrangian L. (3.60) and (3.61) are consistent with (3.58) only if wΣ 1, The local U(1) and super-Weyl transformations may be ¼ and therefore used to choose the gauge Π ¼ 1. This condition implies that S ¼ 0 and the algebra of covariant derivatives reduces δ Υ 2σΥ J Υ −2Υ to that of type I minimal supergravity [15,37] with one extra σ ¼ ; ¼ : ð3:62Þ constraint: the imaginary part of R is now the divergence of a vector (related, by Poincar´e duality, to a two-form The chiral scalar Υ defined by (3.58) is a gauge-invariant potential). To see this it suffices to write the super three- field strength under gauge transformations of the form form H3½Π in the gauge Π ¼ 1 ¯ ¯ ¯ δLΣ ¼ L1 þ iL2; ΔLi ¼ 0; Li ¼ Li ð3:63Þ γ β a ¯ ¯ a βγ H3 ¼ −iE ∧ E ∧ E ðγaÞβγ − iEγ ∧ Eβ ∧ E ðγaÞ 1 For many purposes such as Lagrangian quantisation, it is c ∧ b ∧ aε − ¯ þ 3 E E E abciðR RÞ; ð3:57Þ advantageous to work with unconstrained superfields. The antilinear superfield Σ¯ can always be represented as keeping in mind that H3 ¼ dB2½P. Note that a similar N 1 ¯ α constraint appears in the case of the 4D ¼ three-form Σ ¼ D Ψα; ð3:64Þ supergravity where iðR − R¯ Þ is also the divergence of a vector [69,70,72]. for some unconstrained complex spinor prepotential Ψα. The chiral scalar Υ defined by (3.58) is a gauge-invariant F. Complex two-form supergravity field strength under gauge transformations of the form In the framework of 4D N ¼ 1 Poincar´e supersymmetry, ¯ β the complex three-form multiplet was introduced by Gates δΨα ¼ DαZ þ D Λαβ; Λαβ ¼ Λβα; ð3:65Þ and Siegel [66] as a conformal compensator for the Stelle- West formulation for 4D N ¼ 1 supergravity [61],in Λ with unconstrained complex gauge parameters Z and ðαβÞ. which the complex auxiliary field F was realized as the Here the gauge transformation generated by Λαβ leaves the field strength of a complex gauge three-form. The name superfield (3.64) invariant. The gauge transformation gen- “complex three-form multiplet” was coined in [36]. This erated by Z is equivalent to (3.63) when acting on Σ¯ .Any multiplet was recently used in [73] (under the name of dynamical system with action S Υ; Υ¯ , which is realized in “double three-form multiplet”) to construct a super-Weyl ½ Ψ Ψ¯ invariant formulation for the complex three-form super- terms of the unconstrained prepotentials α and α,isa gravity of [61], in the spirit of the super-Weyl invariant gauge theory with linearly dependent generators of an formulation [71] for three-form supergravity [66,70]. Here infinite stage of reducibility, following the terminology of we propose a 3D N ¼ 2 cousin of the complex three-form the Batalin-Vilkovisky quantization [48]. Φ4 → Υ multiplet. Upon replacement in (3.31) the supergravity A complex two-form multiplet coupled to conformal action turns into supergravity is described in terms of a covariantly chiral Z 4 Υ Υ¯ Υ complex two-form 3 2 2 ¯ ¯ 1 scalar and its conjugate , with being defined by S ¼ − d xd θd θEðΥΥÞ4: ð3:66Þ ð1;1ÞSG κ Υ ¼ Δ¯ Σ¯ ; ð3:58Þ This complex two-form supergravity allows no supersym- where Σ is a complex linear superfield constrained by metric cosmological term, and the action involves no free parameter, unlike the actions for type I supergravity (3.31) ΔΣ¯ ¼ 0: ð3:59Þ and two-form supergravity (3.47). However, the equation of motion for Ψα is In general, if Σ is chosen to transform homogeneously −3=4 ¯ ¯ 1=4 under the super-Weyl transformations, its U(1) charge is DαR ¼ 0; R ≔ Υ ΔΥ ; ð3:67Þ determined by the super-Weyl weight [37] and it implies that R ¼ μ ¼ const. Thus the complex δ Σ σΣ ⇒ J Σ 1 − Σ σ ¼ wΣ ¼ð wΣÞ : ð3:60Þ cosmological parameter μ is generated dynamically.

126015-12 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) G. Superform formulation for the complex The local U(1) and super-Weyl transformations may be two-form multiplet used to choose the gauge Υ ¼ 1. In this gauge, S ¼ 0 and Similarly to the real two-form multiplet, the complex the algebra of covariant derivatives reduces to that of type I two-form multiplet has a geometric superform origin. Let minimal supergravity [15,37] with one extra constraint: the us consider the following complex super two-form: torsion R is the divergence of a vector (related, by Poincar´e duality, to a complex two-form potential). This follows ¯ ¯ α ¯ β a γ ¯ from the fact that Ξ3½Υ in the gauge Υ ¼ 1 is given by C2½Σ¼2iEα ∧ E Σ þ E ∧ E ðγaÞβγD Σ ¯ a βγ ¯ ¯ β ∧ γ DγΣ ¯ ¯ a βγ þ E E ð aÞ Ξ3 ¼ dC2 ¼ −2Eγ ∧ Eβ ∧ E ðγaÞ

i b a c ρτ ¯ c ¯ 2 þ ε E ∧ E ððγ Þ ½Dρ; Dτ − 8C ÞΣ: ð3:68Þ − c ∧ b ∧ aε ¯ 8 abc 3 E E E abcR: ð3:74Þ

¯ 1 B A All coefficients CAB of C2½Σ¼2 E ∧ E CAB are descend- ¯ ¯ IV. GREEN-SCHWARZ SUPERSTRINGS ants of Σ. For the exterior derivative of C2½Σ we get COUPLED TO TWO-FORM ¯ ¯ ¯ a βγ SUPERGRAVITY dC2½Σ¼−2Eγ ∧ Eβ ∧ E ðγaÞ Υ i In this section we will show that the N ¼ 1 and N ¼ 2 − ¯ ∧ b ∧ aε γd γδD Υ 2 Eγ E E abdð Þ δ two-form supergravity theories provide consistent back- 1 grounds for the Green-Schwarz superstring. c ∧ b ∧ aε D2 − 16 ¯ Υ ≡ Ξ Υ þ 24 E E E abcð RÞ 3½ : A. 3D N = 1 Green-Schwarz superstring in ð3:69Þ curved superspace ¯ In the case of 3D N ¼ 1 Green-Schwarz superstring, we Thus, all coefficients of dC2½Σ are descendants of Υ. draw on the results obtained by Bergshoeff et al. [13]. The expression for Ξ3½Υ is obtained from (3.28) by To describe the dynamics of a superstring propagating in a replacement L → Υ. Since both L and Υ are chiral primary c c two-form supergravity background, we propose the follow- superfields of the same weight, we conclude that Ξ3½Υ is ing action: super-Weyl invariant, δσΞ3½Υ¼0. A stronger result is that Z the superform (3.68) is also super-Weyl invariant 1 ffiffiffiffiffiffi 2 p ij a b ij B A S ¼ T2 d ξ −γγ LE E η − ϵ E E B : ¯ 2 i j ab i j AB δσC2½Σ¼0: ð3:70Þ ð4:1Þ ¯ Our result Ξ3½Υ¼dC2½Σ implies that the top compo- Σ¯ Ξ Υ i nents of the superforms C2½ and 3½ , Here ξ ¼ðτ; σÞ are the world-sheet coordinates, γij is the 1 ij kl 12 world-sheet metric, γ ¼ det γij ¼ 2 ϵ ϵ γikγjl with ϵ ¼ i c ρτ ¯ c ¯ ϵ21 1. Both the kinetic and Wess-Zumino terms in (4.1) Cab ¼ εabcððγ Þ ½Dρ; Dτ − 8C ÞΣ; ð3:71aÞ ¼ 4 involve certain target space fields associated with two-form A 1 supergravity, which are the supervielbein EM entering the Ξ ε D2 − 16 ¯ Υ abc ¼ 4 abcð RÞ ; ð3:71bÞ action via the pull-back supervielbein A ∂ M A are related to each other as Ei ¼ iz EM ; ð4:2Þ

α Ξ 3D 2ε Dα 2 D¯ αS D Σ the super two-form B and the compensator L ¼ D Lα abc ¼ ½aCbc þ abcð R þ i Þ α AB (the dilaton superfield). 2ε D¯ α ¯ − 2 DαS D¯ Σ þ abcð R i Þ α : ð3:72Þ The classical consistency of the Green-Schwarz superstring action requires that it be invariant under gauge Υ This confirms that the F component of is the field fermionic transformations (κ-symmetry) of the form strength of a complex two-form. ¯ The gauge transformation of Σ, Eq. (3.63), can be viewed a α αβ 1 a i δE ¼ 0; δE ¼ 2ðγ Þ L4E κβ; ð4:3Þ as the following superform transformation: a i δ A ≔ δ M A δ Σ¯ ⇒ δ Ξ Υ 0 where we have defined E z EM . The gauge param- LC2½ ¼C2½L1 þ iL2 L 3½ ¼ : ð3:73Þ i eter κα is a real 3D spinor and also a 2D vector satisfying 1 ¯ ij −2 ij This allows us to interpret C2½Σ as a gauge complex two- the self-duality condition ðγ − ð−γÞ ϵ Þκαj ¼ 0. form and Ξ3½Υ as the corresponding gauge-invariant field It can be shown that the action (4.1) is invariant under the strength. gauge transformation (4.3) provided the super three-form

126015-13 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) A M A ¯ H3 ¼ dB2 is given by Eq. (2.37) and the world-sheet metric where δE ≔ δz EM , δEα is given by the complex transforms as δ α κα ≡ κ¯α conjugate of E and i i . We point out the relation ffiffiffiffiffiffi ffiffiffiffiffiffi p ij p 1 α −1 αβ c δ −γγ −2 −γL4 4iE − L γ E DβL δ A ∂ δ A − 2δ C BΩ A δ C B A ð Þ¼ ð k ð cÞ k Þ Ei ¼ i E E Ei ½BCÞ þ E Ei TBC ; ð4:7Þ kði jÞl ij kl × ð2γ γ − γ γ Þκ α: ð4:4Þ l where we have used the definitions (3.8) and (3.9a). Then it 1=4 i Let us point out that one can absorb the factor of L into κα. is not difficult to show that the variation of the action is After this redefinition, the action (4.1) and the κ trans- given by the following expression (compare with [12]) formations (4.3) and (4.4) become similar to those in [13]. Z 1 pffiffiffiffiffiffi The action (4.1) is invariant under arbitrary super-Weyl δ 2ξ δ −γγij ΦΦ¯ 2 a bη S ¼ T2 d 2 ð Þð Þ Ei Ej ab transformations of the target space, as a consequence of the ffiffiffiffiffiffi p ij ¯ 2 B A c d relations (2.14a), (2.26) and (2.42). The super-Weyl gauge − −γγ ðΦΦÞ Ei δE TAB Ej ηcd 1 ffiffiffiffiffiffi freedom may be fixed by setting L ¼ . p ij a b ¯ 2 α 2 ¯ ¯ ¯ α ¯ þ −γγ E E η ðΦΦ δE DαΦ þ Φ ΦδEαD ΦÞ i j ab B. 3D N = 2 Green-Schwarz superstring þ ϵijE CE BδEAH ; ð4:8Þ in curved superspace i j ABC

Now we turn to constructing the covariant action for the 1 C B A where H3 ≔ E ∧ E ∧ E H ¼ dB2. 3D N ¼ 2 superstring in a two-form supergravity back- 6 ABC ground, and make use of the results by Grisaru et al. [12] Let us first consider the case of two-form supergravity, with Φ4 Π. To show that the variation (4.8) vanishes, we concerning the 10D N ¼ 2 superstring. We propose the ¼ following superstring action have to make use of the geometrical data specific for the Z two-form supergravity. The only nonvanishing torsion 1 ffiffiffiffiffiffi 2 p ij ¯ 2 a b appearing in the variation (4.8) is the dimension-zero S T2 d ξ −γγ ΦΦ E E η ¼ 2 ð Þ i j ab torsion which is 1 − ϵij B A βc −2 γc β 2 Ei Ej BAB ; ð4:5Þ Tα ¼ ið Þα : ð4:9Þ

A where the pull-back supervielbein Ei is defined similarly The nontrivial components of the super three-form H3 to (4.2). The dilaton ðΦΦ¯ Þ2 is constructed in terms of the given by Eq. (3.51), which enter the variation (4.8), are 1 2 conformal compensator described by a weight- = chiral 1 scalar superfield Φ and its conjugate Φ¯ . The concrete −2 γ Φ¯ 4 − ε γd D¯ δΦ¯ 4 Hαβc ¼ ið cÞαβ ;Habγ ¼ 2 abdð Þγδ structure of Φ depends on the supergravity formulation chosen. In the case of three-form supergravity, the con- ð4:10Þ formal compensator is the three-from multiplet, and then Φ4 ¼ Π ¼ Δ¯ P. On the other hand, the choice Φ4 ¼ Υ ¼ together with their complex conjugates. Substituting the Δ¯ Σ¯ corresponds to complex three-form supergravity. expressions (4.9) and (4.10) into the variation (4.8) and Both the real and complex two-form supergravities using the identities possess a real super two-form B2 which can be used as 1 γ β β c β i½j kl jk il −1 ðγ Þα ðγ Þγ ¼ η δα þ ε ðγ Þα ; γ γ ¼ ϵ ϵ γ ; the Kalb-Ramond field BAB in the action (4.5). For two- a b ab abc 2 form supergravity, the choice of B2 is unique, modulo an ð4:11Þ overall numerical factor, and is given by B2½P, Eq. (3.50). In the case of complex two-form supergravity, there is a one can show that the Green-Schwarz action is indeed whole family of possible super two-forms that can be put in invariant provided the κ-transformation law of the world- a one-to-one correspondence with a circle U(1). However, sheet metric is postulated to be all these choices are equivalent. For concreteness, we ffiffiffiffiffiffi ffiffiffiffiffiffi p p 3 ¯ −1 choose B2 to be the real or imaginary part of the super δð −γγijÞ¼2 −γð2γkðiγjÞl − γijγklÞΦ2Φ 2 ¯ two-form C2½Σ given by Eq. (3.68). 2 E¯ Φ−1 γ E cDβΦ κα Let us show that the action (4.5) is κ symmetric once × ð i kα þ ð cÞαβ k Þ l −1 ¯ 3 we consider a background of real or complex two-form − 2ðϵkðiγjÞl þ ϵlðiγjÞkÞΦ 2Φ2 supergravity. We postulate the following κ-symmetry 2 α Φ¯ −1 γ αβ cD¯ Φ¯ κ transformation × ð iEk þ ð cÞ Ek β Þ lα þ c:c: ð4:12Þ α 3 −1 αβ −1 δ a 0 δ Φ2Φ¯ 2 a γ γijκ − −γ 2ϵijκ¯ E ¼ ; E ¼ Ei ð aÞ ð jβ ð Þ jβÞ; The superstring action constructed is invariant under ð4:6Þ arbitrary super-Weyl transformations of the background

126015-14 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) fields, as follows from the transformation laws (3.11), The Goldstino superfield action is (3.45) and (3.55). Z It is clear that the analysis in the case of the complex two- 3 2 i α 0 3 S i d xd θE D XDαX 2f φ X ; 5:4 form supergravity is identical to the one presented above X ¼ 2 þ ð Þ with the only difference that Φ4 is replaced with Υ instead of Π. In fact, in proving the κ-invariance of the action only for some nonzero parameter f0 which characterizes the the real closed super three-form H3 enters the computations scale of supersymmetry breaking. The second term in rather than its potential B2. Therefore we have proven that the action involves the compensator, φ,ofN ¼ 1 AdS both the real and complex two-form supergravities are supergravity; see Sec. II C. The action is super-Weyl consistent backgrounds for the 3D N ¼ 2 Green-Schwarz invariant. superstring. The nilpotency constraint (5.1) is invariant under local arbitrary rescalings of X, V. GOLDSTINO SUPERFIELDS COUPLED ~ ρ TO SUPERGRAVITY X → X ¼ e X; ð5:5Þ In this section we present various models for sponta- ρ N 1 N 2 for any real scalar . Such a rescaling (5.5) acts on the neously broken local ¼ and ¼ supersymmetry component fields of X as that are obtained by coupling the off-shell supergravity theories, which have been described in the previous ψ 2 ψ → ψ~ ρj ψ D ρ sections, to nilpotent Goldstino superfields. It should be α α ¼ e α þ 4 ð α Þj ; ð5:6aÞ pointed out that the first model for spontaneously broken F N 1 local ¼ supersymmetry was constructed in 1977 [74] 1 ψ 2 N 1 ~ ρj α 2 by coupling on-shell ¼ supergravity to the Volkov- F → F ¼ e F − ψ ðDαρÞj − ðD ρÞj : ð5:6bÞ Akulov action. 2 16F We often make use of the notion of reducible and irreducible Goldstino superfields introduced in [25]. Each of these transformations is a local rescaling accom- By definition, an irreducible Goldstino superfield contains panied by a nilpotent shift of the field under consideration, 1 ~ −1 Goldstone spin-2 fermion(s) as the only independent and therefore F is well defined. Requiring the action component field(s). Every reducible Goldstino superfield (5.4) to be stationary under (5.5) (following the 4D works also contains some auxiliary field(s) along with the [26,27]) gives the constraint Goldstone spin-1 fermion(s). 2 i i 0φ3 D2 Δ Δ ≔ D2 S f X ¼ 2 X X ¼ X X; 2 þ : ð5:7Þ A. N = 1 Goldstino superfields A reducible Goldstino multiplet is described by a real Here XΔX is manifestly a super-Weyl primary. As follows scalar superfield X subject to the nilpotency constraint from (5.6a) and (5.6b), the F component of the nonlinear constraint (5.7) is equivalent to a sum of the equation of X2 ¼ 0: ð5:1Þ motion for F and a linear combination of the equations of motion for ψ α. We also require D2X to be nowhere vanishing so that Consider an irreducible Goldstino superfield X con- ðD2XÞ−1 is well defined, and therefore (5.1) implies strained by

α 2 3 D XDαX X ¼ 0;f0φ X ¼ XΔX; ð5:8Þ X ¼ − : ð5:2Þ D2X with ΔX being nowhere vanishing. This superfield is As a result, X has two independent component fields, a irreducible because the Goldstino ψ α ¼ −iDαXj is the only spinor ψ αðxÞ and a real auxiliary scalar FðxÞ, that may be independent component field of X. Indeed, the second ψ D − 1 D2 −1 defined as i α ¼ αXj and iF ¼ 4 Xj, where F is constraint in (5.8) proves to express the auxiliary field F in well defined. The lowest component of the Goldstino terms of the Goldstino; see Appendix A. The dynamics of superfield, Xj, is a composite field as a consequence X is described by the action of (5.2). Z 1 2 We postulate X to be super-Weyl primary of weight = , 0 3 2 3 SX if d xd θEφ X; 5:9 which means the super-Weyl transformation law of X is ¼ ð Þ

1 which is obtained from (5.4) by making use of the non- δσX ¼ σX: ð5:3Þ 2 linear constraint obeyed by X. The Goldstino theories (5.4)

126015-15 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) and (5.9) prove to be equivalent, which may be shown Making use of (5.14) in order to express X in terms of the by extending the 4D analyses given in [75–77].10 This supergravity fields, the action (5.12) can be recast as a issue is discussed in more detail in Appendix A. The higher-derivative supergravity theory flat-superspace limit of our Goldstino theory defined by Z 8 λ Eqs. (5.8) and (5.9) is analogous to the 2D N ¼ 1 i 3 2 4 S ¼ d xd θEφ S þ Goldstino model pioneered by Roček [79]. 3κ 2 Z It is not difficult to check that the constraints (5.8) are 32 − 3 2θ φ2DαSD S satisfied if 2 d xd E α : ð5:16Þ ð3f0κÞ X X 0φ3 In four dimensions, various approaches to nilpotent N ¼ 1 ¼ f Δ ; ð5:10Þ X supergravity were developed, e.g., in [16,17,20,23,26, where X is only subject to the nilpotency constraint (5.1). 80,81]. Our presentation here is similar to [20]. The important property of X defined by (5.10) is that it is To conclude this subsection, we note that the nilpotent Goldstino superfield X can also be coupled to the two-form invariant under arbitrary local rescalings (5.5), supergravity constructed in Sec. II D. For this we should simply replace the action (5.4) with δρX ¼ ρX ⇒ δρX ¼ 0; ð5:11Þ Z i ρ ~ 3 2θ Dα D 2 0 3=4 for arbitrary real superfield , compare with the 4D analysis SX ¼ i d xd E 2 X αX þ f L X : ð5:17Þ in [27]. This remarkable property actually can be explained by recalling at the component transformation law (5.6b) Then, the equation of motion (5.14) turns into implied by (5.5). The point is the superfield transformation (5.5) implies an arbitrary local rescaling of the auxiliary 3 X ρ D S 0κ 0 X F → jF X α þ f 1 4 ¼ ; ð5:18Þ field of , e . Since does not contain an 8 L = independent auxiliary field, it should remain invariant under (5.5). where S is now defined as in (2.32). Let us consider the model for spontaneously broken local supersymmetry which is obtained by coupling the super- B. Reducible N = 2 Goldstino superfields gravity theory (2.21) to the Goldstino superfield X. The N 2 dynamics of this system is described by the action The family of nilpotent ¼ Goldstino superfields, both reducible and irreducible, is more populous than in the N ¼ 1 case.11 However practically all 3D N ¼ 2 Goldstino S ¼ S þ S : ð5:12Þ SG X superfields can be obtained from the known 4D N ¼ 1 Goldstino supermultiplets by dimensional reduction, at least The component structure of this theory will be discussed in the flat superspace case. This is why our discussion of elsewhere. Here we only present the corresponding cos- nilpotent N ¼ 2 Goldstino superfields will be reasonably mological constant. It is obtained upon eliminating all the concise. We will try to emphasize only conceptual con- auxiliary fields, and is given by structions and those results that are truly new or have not 1 1 received much discussion in the 4D case. Λ 02κ Λ 02κ − 4λ2 ¼ 2 f þ AdS ¼ 2 f : ð5:13Þ 1. Nilpotent chiral scalar superfield The supergravity-matter system (5.12) may be reformu- To begin with, we consider a 3D N ¼ 2 locally super- lated as a model for nilpotent supergravity. Varying (5.12) symmetric counterpart of the reducible Goldstino superfield with respect to the compensator φ gives the equation introduced by Casalbuoni et al. [84] and independently by Komargodski and Seiberg [75]. We choose it to be a 3 X 1 2 S − λ − 0κ covariantly chiral scalar X of super-Weyl weight þ = , ¼ 8 f φ ; ð5:14Þ ¯ 1 1 DαX ¼ 0; δσX ¼ σX ⇒ J X ¼ − X; ð5:19Þ where S is defined by (2.22a). Since X is nilpotent, the 2 2 equation of motion implies which is subject to the nilpotency constraint ðS − λÞ2 ¼ 0: ð5:15Þ 11One can also introduce spinor Goldstino superfields, by analogy with the 4D N ¼ 1 constructions given in [29,82,83]. 10Reference [76] is a considerably generalized and extended However such superfields are not particularly useful in the version of [78]. supergravity framework.

126015-16 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) Z Z 2 0 4 μ X ¼ ; ð5:20Þ − 3 2θ 2θ¯ ΦΦ¯ − 3 2θEΦ4 S ¼ 3κ d xd d E 3κ d xd þ c:c: Z 4 2 in conjunction with the requirement that the descendant 3 2 2 ¯ ¯ 2 2 þ d xd θd θEΦΦjR − μj : ð5:26Þ D X is nowhere vanishing. The nilpotency condition implies 3fκ that X has two independent component fields, a complex Goldstino ψ αðxÞ and a complex auxiliary field FðxÞ, which Here the expression in the first line differs from the super- 1ﬃﬃ 1 2 we define as ψ α ¼ p DαXj and F ¼ − 4 D Xj, respectively. gravity action (3.31) only by new values for the parameters 2 κ → 3κ μ → −μ The constraints on X do not make use of any super- involved, and . The functional form of the N 1 gravity compensator, which means that X is defined in any action (5.26) differs from its 4D ¼ counterpart derived conformal supergravity background. However, a compen- in [20] (see also [26]) in the sense that the supersymmetric sator is required in order to define an action functional Einstein-Hilbert term completely canceled out in the latter for the Goldstino superfield. Here we choose the chiral case. compensator Φ corresponding to the minimal (1,1) AdS The nilpotency condition (5.20) is preserved if X is supergravity described in Sec. III C 1. The dynamics of this locally rescaled, supermultiplet is described by the action τ ¯ X → e X; Dατ ¼ 0; ð5:27Þ Z Z τ 3 2θ 2θ¯ ¯ − 3 2θEΦ3 where the parameter is neutral under U(1). Requiring the SX ¼ d xd d EXX f d xd X þ c:c: ; action (5.21) to be stationary under such rescalings of X (compare with [26]) gives the nonlinear equation ð5:21Þ XΔ¯ X¯ ¼ fΦ3X: ð5:28Þ in which the supersymmetry breaking parameter, f, may be chosen to be real. This nonlinear constraint proves to express the auxiliary We now consider a model for spontaneously broken field F in terms of the Goldstini ψ α and ψ¯ α and their N ¼ 2 local supersymmetry which is obtained by coupling derivatives, see Appendix B. the Goldstino superfield X to the minimal (1,1) AdS The constraints (5.19), (5.20) and (5.28) define an supergravity reviewed in Sec. III C 1. The complete action is irreducible Goldstino superfield X, 1 minimal ¯ 2 S S S ; 5:22 DαX ¼ 0; δσX ¼ σX; X ¼ 0; ¼ ð1;1ÞSG þ X ð Þ 2 XΔ¯ X¯ ¼ fΦ3X: ð5:29Þ minimal where the supergravity action Sð1;1ÞSG is given by Eq. (3.31). This theory proves to generate the following cosmological It is the 3D N ¼ 2 analog of the 4D N ¼ 1 Goldstino constant superfield used by Lindström and Roček [28] in their off- shell model for spontaneously broken N ¼ 1 local super- 12 2 2 2 symmetry. The corresponding action can be given in two Λ ¼ f κ þ Λ ¼ f κ − 4jμj : ð5:23Þ AdS different but equivalent forms: Z Z Varying the action (5.22) with respect to the chiral com- − 3 2θ 2θ¯ XX¯ − 3 2θEΦ3X pensator gives the equation of motion SX ¼ d xd d E ¼ f d xd : ð5:30Þ 3 X R − μ ¼ − fκ ; ð5:24Þ 4 Φ So far we have considered the coupling of the nilpotent Goldstino superfield X to the minimal (1,1) AdS where the super-Weyl neutral chiral scalar R is defined by (3.33a).SinceX is nilpotent, the above equation implies 12Reference [28] is the first work on off-shell de Sitter supergravity in four dimensions. Terminology “de Sitter supergravity” R − μ 2 0; 5:25 was introduced by Bergshoeff et al. [18]. The only difference ð Þ ¼ ð Þ between the supergravity models put forward in [18,28] is that they made use of different Goldstino superfields—the 4D N ¼ 1 X and thus the torsion superfield ðR − μÞ becomes nilpotent. analogs of X and , respectively. The two supergravity models ¯ are equivalent on shell [25]. However, the power of the approach Equation (5.24) can be used to eliminate X and X from the advocated in [18] is that the nilpotency condition X2 ¼ 0 is model action (5.22), resulting with the following geometric higher- independent, which implies that the Goldstino superfield can be derivative supergravity action: readily coupled to matter multiplets.

126015-17 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) supergravity. Its coupling to the two-form (or complex two- The dynamics of the nilpotent superfield V is described form) supergravity is obtained simply by replacing the by the action chiral compensator Φ in (5.21) with Π1=4 (Υ1=4 in the case Z 2 of complex three-form supergravity). However, there is a 3 2 2 jΔVj S ¼ d xd θd θ¯E − 2fV ; ð5:35Þ different universal approach to couple a nilpotent chiral V ðΦΦ¯ Þ3 supermultiplet to any off-shell supergravity. It consists in replacing X, defined by (5.19) and (5.20), with a super- with f the supersymmetry breaking parameter.13 Weyl primary scalar X with the properties The constraints (5.33) are preserved if V is locally rescaled, ¯ 2 DαX ¼ 0; δσX ¼ 2σX; X ¼ 0: ð5:31Þ V → eρV; ð5:36Þ The action (5.21) has to be replaced with for any real scalar ρ. Requiring the action (5.35) to be Z Z stationary under such rescalings of V gives the nonlinear XX¯ 3 2 2 ¯ 3 2 SX ¼ d xd θd θE − f d xd θEX þ c:c: ; equation W 1 Φ−3Δ¯ Φ¯ −3Δ ð5:32Þ 2 Vf ; gV ¼ fV: ð5:37aÞ where W is a real scalar primary superfield of weight þ3 Due to the constraints (5.33), this may equivalently be such that (i) it is nowhere vanishing, and (ii) it is a rewritten as composite of the supergravity compensators. In particular, 3 Φ−3Δ¯ Φ¯ −3Δ Φ¯ −3Δ Φ−3Δ¯ W ¼ðΦΦ¯ Þ in the case of minimal (1,1) AdS supergravity, V ð VÞ¼V ð VÞ¼fV: ð5:37bÞ W ¼ L3 for (2,0) AdS supergravity, W ¼ðΠΠ¯ Þ3=4 for the This nonlinear constraint proves to express the auxiliary two-form supergravity, and so on. field D in terms of the Goldstini. The constraints (5.33) and (5.37) define an irreducible 2. Nilpotent real scalar superfield Goldstino superfield V.Itisa3DN ¼ 2 counterpart of the We now introduce a 3D N ¼ 2 analog of the reducible Goldstino superfield introduced in [25]. The corresponding Goldstino superfield proposed in [26]. It is a real scalar action can be written in two equivalent forms superfield subject to the nilpotency conditions: Z Z ΔV 2 3 2 2 ¯ j j 3 2 2 ¯ 2 SV ¼ − d xd θd θE ¼ −f d xd θd θEV: V ¼ 0; ð5:33aÞ ðΦΦ¯ Þ3 ð5:38Þ VDADBV ¼ 0; ð5:33bÞ The Goldstino models (5.35) and (5.38) are equivalent on D D D 0 V A B CV ¼ : ð5:33cÞ the mass shell. The irreducible Goldstino superfields X and V are The super-Weyl transformation of V is postulated to be related to each other as follows δ σ σV ¼ V: ð5:34Þ fV ¼ XX¯ ; ð5:39aÞ 1 Δ Δ¯ We also require that the descendant 2 f ; gV is nowhere X ¼ Φ−3Δ¯ V: ð5:39bÞ vanishing. The nilpotency conditions (5.33) imply that V has three independent component fields (see Appendix B These relations are analogous to those given in [28] in the for more details) that may be chosen as follows: the 4D case. 1 χ pﬃﬃ D Δ¯ χ¯ complex Goldstino αðxÞ¼ 2 α Vj, its conjugate αðxÞ 1 ¯ −1 and a real auxiliary field DðxÞ¼2 fΔ; ΔgVj, with D 3. Relating X and V being well defined. Starting from the nilpotent chiral superfield X described The constraints (5.33) imposed on V do not make use of in Sec. VB1, we define any supergravity compensator, which means that V is defined ¯ in any conformal supergravity background. However, a fV ≔ XX; ð5:40Þ compensator is required in order to formulate an action functional for the Goldstino superfield. As in the previous 13 Φ Had we chosen V to be an unconstrained real scalar section, here we again choose the chiral compensator superfield, the action (5.35) would have described the dynamics corresponding to the minimal (1,1) AdS supergravity of a two-form multiplet (with a linear superpotential) coupled to (minimal type I supergravity) described in Sec. III C 1. the minimal type I supergravity.

126015-18 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) as a generalization of (5.39). The superfield V introduced The component structure of this model will be discussed in satisfies all the requirements imposed on the nilpotent Appendix B.4. Here we would like just to point out that Goldstino superfield V in Sec. VB2. One of the two the Goldstino superfield Y contains two independent auxiliary fields of X does not contribute to the right-hand auxiliary fields, F ¼ H þ iG, of which H is a scalar and side of (5.40). G is the divergence of a vector. In supergravity, both H and Implementing the field redefinition (5.40) in the G produce positive contributions to the cosmological Goldstino superfield action (5.35) leads to the following constant. While the contribution from H is universal and higher-derivative action: uniquely determined by the parameter of the supersym- Z metry breaking f in (5.47), the contribution from G is 1 jXΔXj2 dynamical. We believe that the latter may be used to S ½X; X¯ ¼ d3xd2θd2θ¯E − 2XX¯ : HD f2 ðΦΦ¯ Þ3 neutralize the negative contribution from the supersymmetric cosmological term. ð5:41Þ

Its important property is C. Irreducible N = 2 Goldstino superfields Using the nilpotent chiral superfield X described in X X¯ SHD½ ; ¼SX ; ð5:42Þ Sec. VB1, we introduce a composite superfield ¯ with SX given by (5.30). Unlike the Goldstino action X Σ ¼ f : ð5:48Þ (5.21), (5.41) is invariant under the discrete transformation Δ¯ X¯ X → −X. The model (5.41) will be studied in more detail in Appendix C. It has the following transformation properties:

δσΣ ¼ −σΣ; J Σ ¼ 2Σ; ð5:49Þ 4. Nilpotent two-form Goldstino superfield As a generalization of the 4D N ¼ 1 models proposed in as well as it identically satisfies the improved linear [24,27], we introduce a nilpotent two-form Goldstino constraint multiplet. It is described by a chiral scalar superfield ΔΣ¯ ¼ f; ð5:50aÞ 1 − D¯ 2 − 4 ¯ Y ¼ 4 ð RÞU; U ¼ U; ð5:43aÞ compare with (3.25). By construction, it is nilpotent,

2 which is constrained to be nilpotent, Σ ¼ 0: ð5:50bÞ

Y2 ¼ 0: ð5:43bÞ It also obeys the nonlinear constraint 1 The prepotential U in (5.43) is defined modulo gauge D Σ − Σ D¯ 2 − 4 D Σ f α ¼ 4 ð RÞ α ; ð5:50cÞ transformations of the form ¯ ¯ which is equivalent to δLU ¼ L; ΔL ¼ 0; L ¼ L; ð5:44Þ ¯ β fDαΣ ¼ −iΣDαβD Σ: ð5:51Þ and Y and Y¯ are gauge-invariant field strengths. The super-Weyl transformation of the prepotential U is Thus Σ is a 3D N ¼ 2 counterpart of the irreducible Goldstino superfield introduced in [30]. Unlike other δσU ¼ σU; ð5:45Þ irreducible Goldstino superfields, such as X and V, the constraints obeyed by Σ, Eq. (5.50), do not make use of any which implies supergravity compensator. In other words, Σ couples to conformal supergravity. δσY ¼ 2σY: ð5:46Þ The remarkable feature of Σ and its conjugate is that these superfields are invariant under local rescalings of X To describe dynamics of the nilpotent two-form multiplet, and its conjugate, Eq. (5.27), we propose the action δ τ ⇒ δ Σ 0 D¯ τ 0; Z Z τX ¼ X τ ¼ ; α ¼ ð5:52Þ YY¯ S ¼ d3xd2θd2θ¯E − f d3xd2θEY þ c:c: : Y ðΦΦ¯ Þ3 compare with [27]. In complete analogy with the 4D N ¼ 1 case [30], every irreducible Goldstino superfield ð5:47Þ is a descendant of Σ and Σ¯ , for instance

126015-19 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) fV ¼ðΦΦ¯ Þ3ΣΣ¯ : ð5:53Þ by the Interuniversity Attraction Poles Program initiated by the Belgian Science Policy (P7/37) and the C16/16/005 Therefore we conclude that all irreducible Goldstino super- grant of the KULeuven. fields are invariant under local rescalings (5.27). As pointed out above, the Goldstino superfields Σ and Σ¯ APPENDIX A: COMPONENT STRUCTURE couple to conformal supergravity. Relation (5.53) clearly N shows that the conformal compensators have to be used in OF = 1 GOLDSTINO MODELS order to define V as a composite superfield constructed In this appendix we will discuss the component actions ¯ 14 from Σ and Σ. for the N ¼ 1 Goldstino models introduced in Sec. VA.For simplicity we will perform our analysis in flat superspace. 3 2 VI. CONCLUDING COMMENTS Here we specialize the superspace M j of Sec. II A to be the standard N ¼ 1 Minkowski superspace M3j2 para- The results obtained in this work may lead to several α metrized by Cartesian real coordinates zA ¼ðxa; θ Þ. The interesting developments including the following: D D D M3j2 (i) The work by Ovrut and Waldram [70] provided covariant derivatives A ¼ð a; αÞ on , defined by membrane solutions in the 4D N ¼ 1 three-form Eq. (2.2), become the flat-superspace ones supergravity. In a similar way, the two-form super- ∂ ∂ γa θβ∂ ∂ θβ∂ gravity theories described in the present paper DA ¼ð a;DαÞ;Dα ¼ α þ ið Þαβ a ¼ α þ i αβ: should possess string solutions. It is of interest to ðA1Þ derive such solutions explicitly. (ii) In three dimensions, consistent models for massive Making use of the anticommutation relation supergravity can be constructed by adding certain higher-derivative terms to the standard supergravity fDα;Dβg¼2i∂αβ ðA2Þ action. These include N ¼ 1 and N ¼ 2 topologically massive [59,85,86] and new massive [46,87–89] allows us to obtain a number of useful properties including supergravity theories. Coupling these theories to the the following: Goldstino superfields described in Sec. V should give consistent models for spontaneously broken massive 1 2 α DαDβ ¼ i∂αβ þ εαβD ;DDβDα ¼ 0; supergravity. 2 N 3 N 4 2 2 (iii) It is of interest to construct ¼ and ¼ D D ¼ −4□: ðA3Þ Goldstino superfields, as an extension of the 4D results given in [26,90]. The N 3 case is espe- 2 α ¼ We recall that D ¼ D Dα. Given a supersymmetric action cially interesting since it has no 4D analog. Z (iv) Since we formulated the 3D Green-Schwarz super- 3 2θL L¯ L string action, with N ¼ 1 and N ¼ 2 spacetime S ¼ i d xd ; ¼ ; ðA4Þ supersymmetry, in off-shell supergravity backgrounds, the quantum superstring analysis given with some superfield Lagrangian L, the component action in [3,4] may be extended from the Minkowski is computed by the rule superspace to other maximally supersymmetric Z backgrounds including the AdS one. − i 3 2L S ¼ 4 d xD j: ðA5Þ ACKNOWLEDGMENTS ≔ As usual, the bar projection is defined by Uj Ujθ¼0, for E. I. B. would like to thank the Center for Theoretical any superfield Uðx; θÞ. Physics at Tomsk State Pedagogical University, where Let us now consider a real scalar superfield X. We define some of this work was done, for warm hospitality. The its real component fields ϕðxÞ, ψ αðxÞ and FðxÞ as work of E. I. B. was supported by the ARC Future Fellowship FT120100466. The work of J. H. is supported 1 ϕ ψ − 2 by an Australian Government Research Training Program ¼ Xj; i α ¼ DαXj; iF ¼ 4 D Xj: ðA6Þ (RTP) Scholarship. The work of S. M. K. is supported in part by the Australian Research Council, Project Introducing a free supersymmetric model with action No. DP160103633. The work of G. T.-M. was supported Z 3 2 i α 0 SX ¼ i d xd θ D XDαX þ 2f X ; ðA7Þ 14 −3 2 However, if V is replaced with V ≔ VðΦΦ¯ Þ , then the constraints obeyed by V are independent of any compensator, and therefore V couples to conformal supergravity. at the component level we obtain

126015-20 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) Z 1 i i − 3 ∂aϕ∂ ϕ ψ α∂ ψ β − 2 2 2 0 XD2X ¼ f0X; ðA15Þ SX ¼ d x 2 a þ 2 αβ F þ f F : 2

ðA8Þ which allows one to solve for the auxiliary field F in terms of ψ. Evaluating the top component of (A15) gives Let us turn to the component analysis of the N ¼ 1 0 2 2 Goldstino model (5.4) in flat superspace. To describe f i α β 1 ψ ψ F − − ψ ∂αβψ − □ ¼ 0: ðA16Þ a reducible Goldstino multiplet, we subject X to the 2 4F 64 F2 F nilpotency condition This equation can be solved by repeated substitution to X2 ¼ 0 ðA9Þ result with

−1 0 1 and assume that F is well defined. The nilpotency f i α β 2 β αγ F ¼ þ ψ ∂αβψ − ψ ∂αβψ ∂ ψ γ constraint allows us to solve for ϕ in terms of the 2 2f0 4f03 Goldstino ψ and the auxiliary field F: 1 þ ψ 2□ψ 2: ðA17Þ 8f03 iψ 2 ϕ ¼ : ðA10Þ 4F Plugging this into (A11) leads to the component action Z With the constraint (A9) imposed, the supersymmetric action 02 3 f i α β 1 2 β αγ SX ¼ − d x þ ψ ∂αβψ − ψ ∂αβψ ∂ ψ γ (A7) defines a nonlinear interacting theory. Making use of 2 2 4f02 (A8) and (A10) leads to the following action: 1 Z þ ψ 2□ψ 2 : ðA18Þ 1 ψ 2 ψ 2 8f02 3 i α β 2 0 S ¼ − d x ψ ∂αβψ þ □ − 2F þ 2f F : X 2 32 F F Comparing the two expressions for F, which are given ðA11Þ by Eqs. (A13) and (A17) and which correspond to the models SX and SX , respectively, we see that they are The equation of motion for F is different. The final Goldstino actions (A14) and (A18) also have different quartic terms. Nevertheless, the two models δ 1 ψ 2 ψ 2 SX are equivalent. Indeed, it was pointed out in Sec. VA that ¼ 4F þ □ − 2f0 ¼ 0: ðA12Þ δF 16 F2 F the top component of (A15) is equivalent to a sum of the equation of motion for F and a linear combination of the This equation can be solved by repeated substitution which equations of motion for ψ α, both equations of motion gives corresponding to the action (A11). One can readily check that the left-hand side of (A16) can be represented as f0 1 F ¼ − ψ 2□ψ 2: ðA13Þ 2 8 03 0 2 2 α f f i α β 1 ψ ψ 1 δSX ψ δSX F − − ψ ∂αβψ − □ ¼ þ ; 2 4F 64 F2 F 4 δF F δψ α Substituting it back into (A11) gives the following action for the Goldstino: ðA19Þ Z 02 1 and therefore the two expressions for F coincide on the ~ 3 f i α β 2 2 SX ¼ − d x þ ψ ∂αβψ þ ψ □ψ : ðA14Þ mass shell. Moreover, it may be shown that every solution 2 2 8f02 to the equation of motion for the Goldstino action (A14) is a Since the auxiliary field possesses a nonvanishing expect- solution to the equation of motion for (A18) and vice versa. 1 0 This follows from the identity ation value, hFi¼2 f , the supersymmetry is spontaneously Z broken. The constant term in the integrand (A14) generates a 1 δ~ δ~ ~ 3 2 αβ SX SX positive contribution to the cosmological constant in SX ¼ S þ d xψ ε : ðA20Þ X 4 02 δψ α δψ β supergravity. f Our next goal is to study the component structure of the irreducible Goldstino model SX , Eq. (5.9), in Minkowski APPENDIX B: COMPONENT STRUCTURE superspace. We recall that it is obtained from the reducible OF N = 2 GOLDSTINO MODELS Goldstino theory defined by Eqs. (A7) and (A9) by requiring the action (A7) to be stationary under local In this appendix we will discuss the component actions for rescalings X → eρX. This gives the constraint N ¼ 2 Goldstino models in flat superspace. We specialize

126015-21 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) the superspace M3j4 of Sec. III A to be the standard N ¼ 2 The action for X follows from (5.21) Minkowski superspace M3j4 parametrized by Cartesian Z Z A a α ¯ ¯α 3 2 2 ¯ 3 2 coordinates z ¼ðx ; θ ; θαÞ,withθ being the complex S ¼ d xd θd θ XX¯ − f d xd θX þ c:c: : ðB9Þ α X conjugate of θ . The covariant derivatives DA ¼ D D D¯ α M3j4 ð a; α; Þ on , defined by Eq. (3.1), become the ¯ ¯ α The integral over θ and θ can be performed using (B7), (B8) flat-superspace ones D ¼ð∂ ;Dα; D Þ. Here the spinor A a to give the following component action covariant derivatives have the form Z 1 ψ¯ 2 ψ 2 ¯β a ¯β 3 ¯ ¯ Dα ¼ ∂α þ iθ ðγ Þαβ∂ ¼ ∂α þ iθ ∂αβ; S ¼ d x − ðhuiþhu¯iÞ þ □ þ FF − fðF þ FÞ ; a 2 2F¯ 2F ¯ ¯ β Dα ¼ −∂α − iθ ∂αβ ðB1Þ ðB10Þ and obey the anticommutation relations where we have defined ¯ ¯ fDα;Dβg¼0; fDα; Dβg¼0; α β β α hui¼iψ ∂αβψ¯ ; hu¯i¼−i∂αβψ ψ¯ : ðB11Þ ¯ fDα; Dβg¼−2i∂αβ: ðB2Þ The superfield X defined by (B5) describes a reducible Given a supersymmetric action multiplet containing the Goldstino ψ α and an auxiliary Z Z field F. 3 2θ 2θ¯L 3 2θL As was explained in the previous appendix there are S ¼ d xd d þ d xd c þ c:c: ; two approaches to define an irreducible Goldstino multiplet. ¯ L¯ L ¯ L 0 We can eliminate F and F from the action (B10) using the ¼ ; Dα c ¼ ; ðB3Þ equations of motion L L with some real and chiral c superfield Lagrangians, the ψ¯ 2 ψ 2 ψ 2 ψ¯ 2 component action is computed using the formula F ¼ f þ □ ; F¯ ¼ f þ □ : ðB12Þ 2F¯ 2 2F 2F2 2F¯ Z Z 1 1 3 2 ¯ 2L − 3 2L Solving equations (B12) by repeated substitution yields S ¼ 16 d xD D j 4 d xD cjþc:c: : ðB4Þ 1 −3 2 2 3 −7 2 2 2 2 2 2 F f f ψ¯ □ψ − f ψ ψ¯ □ψ □ψ¯ : The contractions D and D¯ are defined as in (3.6). ¼ þ 4 16 ð Þð Þ ðB13Þ 1. The N = 2 chiral scalar Goldstino superfield Then the Goldstino action becomes Let us consider a model of a chiral scalar superfield X Z satisfying 1 1 − 3 2 ¯ ∂aψ¯ 2 ∂ ψ 2 S ¼ d x f þ ðhuiþhuiÞ þ 2 ð Þð a Þ ¯ 2 2 4f DαX ¼ 0;X¼ 0: ðB5Þ 1 þ ψ 2ψ¯ 2ð□ψ 2Þð□ψ¯ 2Þ : ðB14Þ This model defines a reducible Goldstino superfield model 16f6 analogous to the 4D N ¼ 1 chiral model studied in [75,84]. Hence, our analysis will be similar to those in [75–77]. Alternatively, we can require that the action be stationary τ ¯ A general chiral superfield can be written as under rescalings X → e X, Dατ ¼ 0 which gives the ffiffiffi constraint p α 2 X ¼ ϕ þ 2θ ψ α þ θ F; ðB6Þ 1 − XD¯ 2X¯ ¼ fX: ðB15Þ so that the components can be defined as 4 Equations (B5), (B15) define a Goldstino model (5.28) as 1 1 2 ϕ ¼ Xj; ψ α ¼ pffiffiffi DαXj;F¼ − D Xj: ðB7Þ was discussed in Sec. V. From (B15) we find the following 2 4 equation for the auxiliary field: 2 The nilpotency condition X ¼ 0 gives 1 ¯ −1ψ¯ α∂ ψ β − ¯ −2ψ¯ 2□ −1ψ 2 F ¼ f þ iF αβ 4 F ðF Þ: ðB16Þ ψ 2 ψ¯ 2 ϕ ¼ ; ϕ¯ ¼ : ðB8Þ 2F 2F¯ Solving it by repeated substitution we obtain

126015-22 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) 1 1 −1 −3 1 2 2 α F f f u¯ − f u u¯ ψ¯ □ψ ψ α ¼ pffiffiffi Wαj;D¼ − D Wαj: ðB23Þ ¼ þ h i h ih iþ4 2 4 þ f−5ðhui2hu¯iþhu¯i2huiÞ Since Wα satisfies 1 f−5 u¯ ψ 2□ψ¯ 2 2 u ψ¯ 2□ψ 2 ψ¯ 2□ ψ 2 u¯ þ 4 ðh i þ h i þ ð h iÞÞ α ¯ ¯ α D Wα ¼ DαW ðB24Þ 1 − 3f−7 hui2hu¯i2 þ ψ 2ψ¯ 2□ðhui2 − huihu¯iþhu¯i2Þ 4 we see that D is real. Using Eqs. (B19), (B21), (B22), (B23) 1 we obtain ψ 2ψ¯ 2□ψ 2□ψ¯ 2 þ 16 ; ðB17Þ ϱ¯ βϱ2 i ¯β i ψ α ϱα − ∂αβλ ϱα − ∂αβ ; ¼ 2 ¼ 4 D2 where hui is given in Eq. (B11). Comparing Eqs. (B13) and (B17) we see that the solution for F is different in our two 1 1 ϱ2ϱ¯ 2 D D − □v D − □ : B25 approaches but the difference is related to the equation of ¼ 4 ¼ 16 D3 ð Þ motion for the Goldstino as was explained at the end of the previous appendix. From here we can derive the following useful relations

2 α 2 α 2 α 2 α 2 2 2 2 2. The N = 2 real scalar Goldstino superfield ϱ ϱ¯ ¼ ψ ψ¯ ; ϱ¯ ϱ ¼ ψ¯ ψ ; ϱ ϱ¯ ¼ ψ ψ¯ ; The real scalar Goldstino superfield is defined to obey ðB26Þ the constraints which, in turn, allow us to invert (B25) to get 2 V ¼ 0;VDADBV ¼ 0;VDADBDCV ¼ 0: i ψ¯ βψ 2 1 ψ 2ψ¯ 2 ðB18Þ ϱα ¼ ψ α þ ∂αβ ; D ¼ D þ □ : 4 D2 16 D3 We will start with a general N ¼ 2 real scalar superfield ðB27Þ pffiffiffi pffiffiffi α ¯ ¯α 2 ¯2 ¯ α ¯β V ¼ v þ 2θ λα þ 2θαλ þ θ F þ θ F þ θ θ Aαβ Substituting (B27) into (B21) we obtain the components of ffiffiffi ffiffiffi V in terms of ðψ α;DÞ p ¯2 α p 2 ¯ α 2 ¯2 þ 2θ θ ϱα þ 2θ θαϱ¯ þ θ θ D: ðB19Þ ψ 2ψ¯ 2 ψ ψ¯ 2 ψ¯ ψ 2 λ α λ¯ α ~ a v ¼ 3 ; α ¼ 2 ; α ¼ 2 ; Here Aαβ describes both a vector A and a scalar φ: 4D 2D 2D ψ¯ 2 ψ¯ 2 ψ 2 ψ 2 ¯ ¯ γ ~ a ϵ φ F ¼ þ 3 hui; F ¼ þ 3 hui; Aαβ ¼ð aÞαβA þ i αβ : ðB20Þ 2D 4D 2D 4D 2ψ ψ¯ α β − i ψ 2ψ¯ γ ∂ ψ¯ i ψ¯ 2 ∂ ψ ψ γ λ λ¯ ¯ Aαβ ¼ 3 ð αγ βÞþ 3 ð βγ αÞ Imposing conditions (B18) we find that v; α; α;Aαβ;F;F D 2D 2D can be solved in terms of ϱα, ϱ¯ α; D as follows: 1 1 2 2 δ γ 2 2 a − ψ ψ¯ ∂αγ∂βδ ψ ψ¯ − ψ ψ¯ ∂ ψ α∂ ψ¯ β: 8 5 ð Þ 4 5 a 2 2 2 2 D D ϱ ϱ¯ ϱαϱ¯ ¯ ϱ¯ αϱ v ; λα ; λα ; ðB28Þ ¼ 4D3 ¼ 2D2 ¼ 2D2 ϱ¯ 2 ϱ2 2ϱ ϱ¯ ¯ α β Either ðϱα; DÞ or ðψ α;DÞ can be used to describe a F ¼ ; F ¼ ;Aαβ ¼ : ðB21Þ 2D 2D D reducible Goldstino multiplet in this model. Relations (B25) and (B27) allow one to quickly transform from Hence, we have explicitly shown that the model (B18) one description to another. Since the components of V are ϱ D describes a reducible Goldstino multiplet ð α; Þ consisting simpler when written in terms of ðϱα; DÞ below we will of the Goldstino ϱα and an auxiliary field D. use this pair of fields. Alternatively, we can define the Goldstino as follows. The action for the Goldstino superfield can be taken as Let the flat superspace limit of (5.35) 1 Z Z ¯ 2 1 Wα ¼ − D DαV: ðB22Þ 3 2θ 2θ¯ 2 ¯ 2 − 2 3 2θ 2θ¯ 4 S ¼ 16 d xd d D VD V f d xd d V:

Let us define ðB29Þ

126015-23 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) Using the nilpotency conditions (B18) the first term of the Goldstino. Examining Eqs. (B31), (B34) one can show action (B29) can also be written as that the last three terms in (B34) do not contribute to the Z Z 1 1 action and, hence, can be ignored. Keeping this in mind, we − 3 2θ 2θ¯ α 3 2θ α obtain 4 d xd d VD Wα ¼ 4 d xd W Wα: ðB30Þ 1 1 However, we find that Eq. (B29) is more convenient to use. D − ϱαϱ¯ β∂ ∂ ϱγϱ¯ δ − □ ϱ2ϱ¯ 2 ¼ f 4 3 αβ γδð Þ 16 3 ð Þ In terms of ðϱα; DÞ the action (B29) reads f f 1 1 Z ϱ2□ ϱ¯ 2 ϱ¯ 2□ ϱ2 þ 3 ð Þþ 3 ð Þþ; ðB35Þ 3 D2 − 2 D − i ϱα ∂ ϱ¯ β 8f 8f S ¼ d x f 2 ð αβ Þ ϱαϱ¯ β ϱγϱ¯ δ where the ellipsis stands for the terms which do not i ∂ ϱβ ϱ¯ α − ∂ ∂ þ 2 ð αβ Þ 4D αβ γδ D contribute to the action. Substituting Eq. (B35) into (B31) we find the following action for the Goldstino: 1 ϱ2ϱ¯ 2 1 ϱ ϱ¯ 2 1 ϱ¯ αϱ2 □D − ϱα□ α − ϱ¯ □ Z þ 8 ð Þ D3 4 D2 4 α D2 1 1 − 3 2 ¯ ϱα□ ϱ ϱ¯ 2 S ¼ d x f þ ðhwiþhwiÞ þ 2 ð α Þ 1 ϱ¯ 2 ϱ¯ βϱ2 ϱαϱ¯ 2 2 4f ϱ2□ i ∂ □ þ þ αβ 2 2 1 1 4D D 16 D D ϱ¯ □ ϱ¯ αϱ2 − ϱ2□ ϱ¯ 2 þ 2 α ð Þ 2 ð Þ 1 4f 4f þ ϱ2ϱ¯ 2□2ðϱ2ϱ¯ 2Þ : ðB31Þ 1 256D6 2 i α 2 β 2 þ ðhwi − hw¯ iÞ þ ϱ ϱ¯ ∂αβ□ðϱ¯ ϱ Þ 4f2 16f4

Again, there are two approaches to define an irreducible 1 2 2 2 D þ ϱ ϱ¯ □ðhwi − hw¯ iÞ Goldstino multiplet. We can eliminate using its equation 64f6 of motion: 1 α β γ δ 2 2 þ ϱ ϱ¯ ∂αβ∂γδðϱ ϱ¯ Þ□ðϱ ϱ¯ Þ ; ðB36Þ 1 ϱαϱ¯ β ϱγϱ¯ δ 1 ϱ2ϱ¯ 2 32f6 D f − ∂αβ∂γδ − □ ¼ 4 D2 D 16 D3 α β where hwi¼iϱ ∂αβϱ¯ . 3 ϱ2ϱ¯ 2 1 ϱαϱ¯ 2 1 ϱ¯ ϱ2 □D − □ϱ − α □ϱ¯ α þ 16 ð Þ D4 4 D3 ð αÞ 4 D3 ð Þ 1 ϱ¯ 2 1 ϱ2 3. From V to equivalent two-form multiplet þ ϱ2□ þ ϱ¯ 2□ 8D2 D 8D2 D There is another possibility to study the model from the ϱαϱ¯ 2 ϱ¯ βϱ2 ϱ¯ αϱ2 ϱβϱ¯ 2 previous subsection. For this we will introduce − i ∂ □ − i ∂ □ 3 αβ 2 3 αβ 2 16 D D 16 D D 1 1 2 ¯ 2 3 Ψ ¼ − D¯ V; Ψ ¼ − D V: ðB37Þ ϱ2ϱ¯ 2□2 ϱ2ϱ¯ 2 4 4 þ 256D7 ð Þ; ðB32Þ The action in Eq. (B29) can be equivalently written as which can be solved by repeated substitutions. The second approach is to require that the action (B29) be stationary Z Z Z ρ 3 2 2 ¯ 3 2 3 2 ¯ under local rescalings V → e V which yields the constraint S ¼ d xd θd θ¯ ΨΨ− f d xd θΨ − f d xd θ¯ Ψ : 1 2 ¯ 2 ðB38Þ 32 VfD ; D gV ¼ fV; ðB33Þ Since Ψ is chiral we can define its components as as was discussed in Sec. V. Here for simplicity we will follow the first approach and eliminate D using the equation 1 1 2 of motion (B32). From Eq. (B32) we see that, the solution ϕ ¼ Ψj; χα ¼ pffiffiffi DαΨj;F1 þ iF2 ¼ − D Ψj: 2 4 for D has to be of the following form: ðB39Þ α β 2 2 ¯ 2 α D ¼ f þ ϱ ϱ¯ Aαβ þ ϱ B þ ϱ¯ B þ ϱ¯ ϱ Cα 2 2 ¯ α 2 2 From Eq. (B18) it follows that Ψ ¼ 0, hence, þ ϱ ϱ¯ αC þ ϱ ϱ¯ F; ðB34Þ χ2 where A, B, C, F depend on ϱ only through derivatives. ϕ ¼ : ðB40Þ Note that in Eq. (B34) there are no terms linear in 2ðF1 þ iF2Þ

126015-24 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) Z 2 2 Note that in addition to the Goldstino χα, Ψ contains two 1 χ¯ χ S ¼ d3x − ðhviþhv¯iÞ þ □ auxiliary fields F1 and F2. However, as we will see below 2 2ðF1 − iF2Þ 2ðF1 þ iF2Þ F2 is a function of the Goldstino and F1. Therefore, it is the χ 2 2 pair ð α;F1Þ which describes a reducible Goldstino multi- þ F1 þ F2 − 2fF1 ; ðB48Þ plet which, of course, is equivalent to the ones studied in the previous subsection up to a nonlinear transformation χα∂ χ¯β which we will derive below. To express F2 in terms of χα where hvi¼i αβ and F2 is given by (B47). We will χ and F1 we note that not present the final action in terms of α since it is substantially more complicated than the one in Eq. (B36). 2 ¯ 2 ¯ αβ ¯ Out of the three possible Goldstino fields ϱ, ψ and χ it is D Ψ − D Ψ ¼ i∂ ½Dα; DβV: ðB41Þ ϱ which has the simplest action. Using the fact that 4. Nilpotent two-form Goldstino superfield 1 ¯ Aαβ ¼ ½Dα; DβVjðB42Þ Here we will discuss the component structure of the 2 model introduced in Sec. VB4. As before, we will take the flat space limit. The two-form Goldstino multiplet is which follows from (B19) we find that described by a chiral scalar superfield Y satisfying the following conditions: ϱ ϱ¯ 1 αβ 1 αβ α β F2 − ∂ Aαβ − ∂ : B43 ¼ 4 ¼ 2 D ð Þ 1 − ¯ 2 2 0 Y ¼ 4 D U; Y ¼ ; ðB49Þ Hence, we see that F2 is expressed in terms of the Goldstino and the remaining auxiliary field. The relation where U is an unconstrained real superfield. Since Y is between ðϱα; DÞ and ðχα;F1Þ can be obtained using the chiral we can define its components in the usual way defining equation (B37) as well as the definition of the component (B19), (B39). We get 1 1 2 ϕ ¼ Yj; ξα ¼ pffiffiffi DαYj;F¼ − D Yj: ðB50Þ 2 4 ϱ¯ βϱ2 1 ϱ2ϱ¯ 2 χ ϱ i ∂ D □ α ¼ α þ 4 αβ D2 ;F1 ¼ þ 16 D3 : From (B49) it follows that

ðB44Þ 2 ¯ 2 ¯ αβ ¯ D Y − D Y ¼ i∂ ½Dα; DβU: ðB51Þ

Using the identities This means that the imaginary part of the auxiliary field F

2 α 2 α 2 α 2 α 2 2 2 2 is the divergence of a vector. Let us denote F ¼ H þ iG. ϱ ϱ¯ ¼ χ χ¯ ; ϱ¯ ϱ ¼ χ¯ χ ; ϱ ϱ¯ ¼ χ χ¯ ; ðB45Þ a a Then we have G ¼ ∂aC , where C is an auxiliary vector field. The action for the superfield Y is given by the flat we can invert (B44): space limit of Eq. (5.47): Z Z χ¯βχ2 1 χ2χ¯2 ϱ χ − i ∂ D − □ 3 2 2 ¯ ¯ 3 2 α ¼ α αβ 2 ; ¼ F1 3 : SY ¼ d xd θd θ YY− f d xd θY þ c:c: : ðB52Þ 4 F1 16 F1 ðB46Þ Just like in the theory of three-form multiplet in four dimensions this action has to be supplemented with the Using the relations (B43) and (B46), we can express F2 in boundary term [91–93] terms of the fields χα and F1. The result is Z 1 γ 2 γ 2 3 2 2 ¯ α 1 1 χ χ¯ χ¯ χ B d xd θd θD YDαU − UDαY c:c − ∂αβ 4χ χ¯ χ ∂ χ¯ ∂ Y ¼ 4 ð Þþ F2 ¼ α β þi α βγ 2 þi α βγ 2 Z 8 F1 F1 F1 3 a 1 1 ¼ −2 d x∂aðC GÞþ; ðB53Þ χ2χ¯2 ∂ ∂ χ¯γχδ − ∂aχ ∂ χ¯ þ 5 αγ βδð Þ α a β : ðB47Þ F1 2 where the ellipsis stands for the boundary terms which do Since the action (B38) is the same as the action for a not play a role and can be set to zero. chiral superfield X in (B9) it has the identical component Since the action (B52) is the same as the action for a structure: chiral superfield it is given by

126015-25 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) Z ξ2 ξ¯2 3 i α ¯β i α ¯β the contribution from the boundary term yields the follow- S ¼ d x □ þ ð∂αβξ Þξ − ξ ð∂αβξ Þ Y 2F 2F¯ 2 2 ing Goldstino action: Z 2 2 þ H þ G − 2fH ; ðB54Þ − 3 2 − i ∂ ξα ξ¯β i ξα ∂ ξ¯β SY þ BY ¼ d x jhj 2 ð αβ Þ þ 2 ð αβ Þ 2 1 2 3 2 Y 0 f þ g 2 2 where we have used the fact that ¼ and, hence, þ ∂aξ ∂ ξ¯ ϕ ¼ ξ2=ð2FÞ. Now we will eliminate the auxiliary fields 4 jhj4 a using their equations of motion. Varying the action (B54) 1 2 7 2 f þ g 2 ¯2 2 ¯2 with respect to H and C gives the following equations: þ ξ ξ □ξ □ξ : ðB59Þ a 16 jhj8 ξ¯2 ξ2 ξ2 ξ¯2 H − f − □ − □ 0; 4 ¯ 2 2 4 2 2 ¯ ¼ F F F F APPENDIX C: GOLDSTINO MULTIPLET FROM ξ¯2 ξ2 ξ2 ξ¯2 ∂ □ − □ 0 A HIGHER-DERIVATIVE THEORY a G þ i 4 ¯ 2 2 i 4 2 2 ¯ ¼ : ðB55Þ F F F F In this appendix we will analyze the higher-derivative model (5.41) in Minkowski superspace. We first consider The second equation implies that the case when the dynamical variable X is an unconstrained ¯ 0 ξ¯2 ξ2 ξ2 ξ¯2 chiral superfield, DαX ¼ , which obeys no nilpotency G þ i □ − □ ¼ g; ðB56Þ condition. Then the model with action 4F¯ 2 2F 4F2 2F¯ Z 1 where g is an arbitrary constant. Hence, we find that S ¼ d3xd2θd2θ¯ XD2XX¯ D¯ 2X¯ − 2XX¯ ðC1Þ 16f2 ξ¯2 ξ2 F h □ ;hf g: has two phases, one with unbroken supersymmetry, and the ¼ þ 2 ¯ 2 2 ¼ þ i ðB57Þ F F other with spontaneously broken one. In the unbroken Solving this equation by repeated substitution yields phase, the equations of motion have free massless solutions 2 1 3 D X ¼ 0: ðC2Þ 1 −4ξ¯2□ξ2 − −8ξ2ξ¯2□ξ2□ξ¯2 F ¼ h þ 4 jhj 16 jhj ; However, the kinetic term in (C1) has a wrong sign and thus jhj2 ¼ f2 þ g2: ðB58Þ the theory is ill defined at the quantum level. We therefore turn to the phase with spontaneously broken supersymmetry TheR boundary term on the solution G ¼ g þ gives in which F develops a nonzero expectation value, hFi¼f. −2 d3xðg2 þ total derivativeÞ. Substituting Eq. (B58) Defining the components of X as in Eq. (B7) we obtain into the bulk action (B54) and combining the result with the component action: Z 3 a ¯ α β ¯ S ¼ 2 d x½∂ ϕ∂aϕ þ iψ ∂αβψ¯ − FF Z 1 þ d3x FF¯ ðϕ¯ □ϕ þ ϕ□ϕ¯ Þþϕϕ¯ ð□ϕÞð□ϕ¯ ÞþðFF¯ Þ2 − ∂ ðϕF¯ Þ∂aðϕ¯ FÞ f2 a 3 3 − i ¯ ψ α∂ ψ¯ β − i ¯ ψ¯ α∂ ψ β − i ¯ ∂ ψ αψ¯ β i ∂ ¯ ψ αψ¯ β 2 FF αβ 2 FF αβ 2 F αβF þ 2 F αβF α ¯ ¯ α αγ β ¯ αγ ¯ β − ϕFψ¯ α□ψ¯ − ϕ F ψ □ψ α þ F∂ ϕψ¯ γ∂αβψ¯ − F∂ ϕψ γ∂αβψ i ϕ¯ ∂αγϕ − ϕ∂αγϕ¯ ∂ ψ¯ β∂ ψ δ i ϕϕ¯ ∂ ψ¯ β□ψ α ∂ ψ β□ψ¯ α þ 2 ð Þ αβ γδ þ 2 ð αβ þ αβ Þ ¯ α β ¯ α β α β γ δ − iðϕ□ϕÞψ¯ ∂αβψ − iðϕ□ϕÞψ ∂αβψ¯ − ðψ ∂αβψ¯ Þðψ¯ ∂γδψ Þ : ðC3Þ

The equation of motion for F¯ is

2 2 ¯ ¯ ¯ α β α β ¯ α β ¯ α − 2Ff þ 2F F þ Fϕ□ϕ þ Fϕ□ϕ − 2iFψ ∂αβψ¯ − iFψ¯ ∂αβψ þ ϕ□ðϕFÞ − i∂αβFψ ψ¯ − ϕψ □ψ α αγ ¯ β − ∂ ϕψγ∂αβψ ¼ 0: ðC4Þ

126015-26 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) It shows that F and F¯ are no longer auxiliary fields since above inconsistencies do not occur. In doing so, we will they cannot be expressed in terms of the off-shell physical obtain correctly normalized kinetic terms for the physical fields ϕ and ψ α and their conjugates. One could try to look fields. Indeed, since in the supersymmetry breaking phase for F as a series in powers of the fields ϕ, ψ α and their F ¼ f þ, for the relevant terms in (C3) we get derivatives, Z 3 a ¯ α β ¯ ¯ α β α β 2 d x½∂ ϕ∂ ϕ þ iψ ∂αβψ¯ F ¼ f þ a1ϕ□ϕ þ a2ϕ□ϕ þ a3ψ ∂αβψ¯ þ a4ψ¯ ∂αβψ a Z þ; ðC5Þ 1 þ d3x FF¯ ϕ¯ □ϕ þ ϕ□ϕ¯ f2 where a1;a2; … are some constants which have to be found 3i by substituting (C5) into (C4) and working order by order − ψ α∂ ψ¯ β ψ¯ α∂ ψ β ϕ ¯ □ ϕ¯ 2 ð αβ þ αβ Þ þ F ð FÞ in perturbation theory. Such a solution would correspond Z to a supersymmetry breaking phase (note that F ¼ f for 3 a ¯ α β ¼ − d x½∂ ϕ∂ ϕ þ iψ ∂αβψ¯ þ ðC7Þ ϕ ¼ 0; ψ α ¼ 0). However, it is not difficult to show that no a solution for F exists: substitution of (C5) into (C4) yields inconsistent equations where the ellipsis stands for cubic and higher order terms in the fields ϕ, ψ α and their conjugates. 2 2¯ −1 0 2 2¯ 2 −1 0 a2 þ a1 þ f ¼ ; a1 þ a2 þ f ¼ ; ðC6Þ We now restrict our study to the case of model (C1) with X chosen to be nilpotent, and similarly for a3, a4. This means that it is impossible to solve the equation of motion for the field F and substitute 2 0 the solution into Eq. (C3) to find the action for the off-shell X ¼ : ðC8Þ physical fields. The procedure of eliminating the auxiliary field F can be fulfilled only when the physical fields are Then ϕ can be expressed as in (B8) and we have a reducible also on shell. In other words, the equation (C4) and its Goldstino model. The component action of this model is conjugate have to be solved in conjunction with the given by (C3) with ϕ replaced according to Eq. (B8). equations of motion for the physical fields, and then the The equation for the auxiliary field now reads

f2 ψ¯ 2 ψ 2 1 ψ¯ 2 1 ψ¯ 2 − 2f2F þ □ þ ψ 2□ − □ðF¯ ψ 2Þ 2 F¯ 2 F 4 F¯ 4 F¯ 2 1 ψ 2ψ¯ 2 1 ψ 2 Fψ¯ 2 1 Fψ¯ 2 F¯ ψ 2 − □ψ 2□ψ¯ 2 þ 2F2F¯ þ □ − □ 8 F2F¯ 3 4 F F¯ 4 F¯ 2 F 3 3 − i ψ α∂ ψ¯ β − i ψ¯ α∂ ψ β − i ψ αψ¯ β ∂ − i ∂ ψ αψ¯ β 2 F αβ 2 F αβ 2 ð αβFÞ 2 αβðF Þ 1 ψ¯ 2 1 ψ¯ 2 ψ¯ 2 ψ 2 αγ β αγ ¯ β i αγ β δ − ∂ ψ γ∂αβψ − ∂ ðFψ γ∂αβψ Þ − ∂ ∂αβψ¯ ∂γδψ 2 F¯ 2 F¯ 2 8 F¯ 2 F 2 2 i ψ¯ αγ ψ β δ i 2 2 β α β α − ∂ ∂αβψ¯ ∂γδψ − ψ ψ¯ ½∂αβψ¯ □ψ þ ∂αβψ □ψ¯ 8 F¯ 2 F 8FF¯ 2 ψ 2 ψ 2 i ¯2 ¯ −2 α β i ¯2 ¯ −2 α β þ ψ F □ ψ¯ ∂αβψ þ ψ F ψ ∂αβψ¯ □ ¼ 0: ðC9Þ 4 F 4 F

One can show that just like in the case of Eq. (C4) it is not Since D¯ 2X¯ is nowhere vanishing, this condition is equiv- possible to solve this equation for F in terms of the physical alent to fields ψ and ψ¯ . The procedure of eliminating the field F can ¯ 2 ¯ 2 2 be performed only if the Goldstino is on shell. Therefore, XD ðXD XÞ¼16f X: ðC11Þ we will follow the other approach: instead of considering the equations of motion for F, we will require that the The problem of solving Eqs. (C10), (C11) can be reformu- action (C1) be stationary under rescaling X → eτX, which lated as follows. Let us define the superfield Y by the rule yields 1 ¯ 2 ¯ 2 ¯ 2 ¯ 2 ¯ − ¯ 2 ¯ D ðXXD XD X − 16f XXÞ¼0: ðC10Þ 4 XD X ¼ fY: ðC12Þ

126015-27 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) 1 It then follows from Eq. (C10) that Y has the properties − −3 − a4 ¼ 4 f a5;

¯ 1 −5 −2 DαY ¼ 0; − − 2 a7 ¼ 4 f f a5; 2 Y ¼ 0; 1 −4 − 2 1 a12 ¼ f a5 a5 ðC17Þ − ¯ 2 ¯ 2 4 YD Y ¼ fY: ðC13Þ and cannot be fixed uniquely. The solution (B17) corre- 1 −3 1 −5 3 −7 sponds to a4 ¼ 0, a5 ¼ − f , a7 ¼ f , a12 ¼ − f . That is Y defines an irreducible Goldstino multiplet whose 4 4 16 The ambiguity that we can have more than one solution to auxiliary field is uniquely solved in terms of the Goldstini (C14) is expected to be related to the fact that we can add to with the solution given in Eq. (B17). Therefore, the F and to the action terms proportional to the equations of problem can be stated as to find X using Eq. (C12) given motion as in (A19) and (A20), but we will not discuss this Y. Comparing Eqs. (C12) and (C13) we see that there is an issue in detail in this paper. obvious solution X ¼ Y.15 However, this solution is not Let us now clarify why Eq. (C11), or equivalently unique. To show it we will examine Eq. (C11) in compo- Eq. (C14), has a solution for F despite the fact that nents. Let us consider the equation for F followed from Eq. (C9) does not. For this we will consider the equation (C11). We obtain of motion for the superfield X. Since X is nilpotent to find it we have to add the term 2 2 ¯ α β α β Z − 2f F þ 2F F − 2iFψ ∂αβψ¯ − 2iFψ¯ ∂αβψ 3 2 2 1 ψ¯ 2 ψ 2 1 ψ 2 ψ¯ 2 d xd θλX þ c:c: ðC18Þ − 2 ∂ ψ αψ¯ β F □ □ F ið αβFÞ þ 2 ¯ þ 2 ¯ F F F F λ 2 to the action (C1), where is a Lagrange multiplier. Thus, α ψ¯ βγ þ ψ ∂αβ ∂ ψ γ ¼ 0: ðC14Þ we obtain the following equation of motion for X: F¯ D¯ 2½XD¯ 2XD¯ 2X¯ þD¯ 2D2½XX¯ D¯ 2X¯ − 32f2D¯ 2X¯ 2 Note that we cannot solve this equation by repeated − 128f λX ¼ 0: ðC19Þ substitution. However, we can solve it by expanding F Multiplying it by X we get the constraint (C10). However, in powers in the Goldstino and its derivatives the equation of motion for X contains not just the equation of motion for F (C9) but also the equation for the α β α β F ¼ f þ a1ðiψ ∂αβψ¯ Þþa2ðiψ¯ ∂αβψ Þþ: ðC15Þ Goldstino. Hence, in obtaining Eq. (C14) equations of motion for both F and ψ are taken into account and that is ψ why it has a solution. Since is nilpotent this expansion is finite. Substituting it With the nilpotency condition (C8) imposed, the action into (C14) we can fix the coefficients. From the analysis (C1) can be rewritten as presented above we know that there is a solution for F Z 1 given by (B17). Therefore, we will look for a solution in 3 2 2 ¯ α ¯ ¯ ¯ β ¯ ¯ S ¼ d xd θd θ D XDαXDβXD X − 2XX : the form of (B17): 16f2 ðC20Þ 2 2 F ¼ f þ a1huiþa2hu¯iþa3huihu¯iþa4ψ □ψ¯ Similar supersymmetric higher derivative models have been ψ¯ 2□ψ 2 2 ¯ ¯ 2 ¯ ψ 2□ψ¯ 2 þ a5 þ a6ðhui huiþhui huiÞ þ a7hui considered in the literature in the case when X is an 2 2 2 2 2 2 þ a8huiψ¯ □ψ þ a9ψ¯ □ðψ hu¯iÞ þ a10hui hu¯i unconstrained chiral superfield. In particular, an action of 2 2 2 2 the type (C20) was studied in [94]. In their case they could þ a11ψ ψ¯ □ðhui − huihu¯iþhu¯i Þ solve for the auxiliary field in terms of the off-shell physical 2 2 2 2 þ a12ψ ψ¯ □ψ □ψ¯ : ðC16Þ scalar field provided the fermions were ignored. However, if we take into account the fermions as well we can show that it is also impossible to solve for the auxiliary field unless the Substituting this ansatz into (C14) we find that the fermions are on-shell. Unlike in our case, Eq. (C7),inthe coefficients a1, a2, a3, a6, a8, a9, a10, a11 are fixed as model studied in [94] the kinetic term for scalars completely in (B17), whereas the remaining coefficients satisfy canceled in the supersymmetry breaking phase. Ref. [95] studied a model with canonically normalized kinetic term. −2 ¯ → ¯ 15Note that if X is a solution to (C12) then so is −X. Hence, we It is obtained from (C20) by replacement XX XX. have two supersymmetry breaking phases. For concreteness we It was shown in [95] that the resulting model cannot break select the phase in which hFi¼f. supersymmetry.

126015-28 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) [1] M. B. Green and J. H. Schwarz, Covariant description of [23] I. Bandos, L. Martucci, D. Sorokin, and M. Tonin, Brane superstrings, Phys. Lett. 136B, 367 (1984). induced supersymmetry breaking and de Sitter supergravity, [2] M. B. Green, J. H. Schwarz, and E. Witten, Superstring J. High Energy Phys. 02 (2016) 080. Theory, Cambridge Monographs on Mathematical Physics [24] F. Farakos, A. Kehagias, D. Racco, and A. Riotto, Scanning (Cambridge University Press, Cambridge, England, 1987), of the supersymmetry breaking scale and the gravitino mass Vols. 1 and 2. in supergravity, J. High Energy Phys. 06 (2016) 120. [3] L. Mezincescu and P. K. Townsend, Anyons from Strings, [25] I. Bandos, M. Heller, S. M. Kuzenko, L. Martucci, and D. Phys. Rev. Lett. 105, 191601 (2010). Sorokin, The Goldstino brane, the constrained superfields [4] L. Mezincescu and P. K. Townsend, Quantum 3D super- and matter in N ¼ 1 supergravity, J. High Energy Phys. 11 strings, Phys. Rev. D 84, 106006 (2011). (2016) 109. [5] J. A. de Azcarraga and J. Lukierski, Supersymmetric par- [26] S. M. Kuzenko, I. N. McArthur, and G. Tartaglino- ticles with internal symmetries and central charges, Phys. Mazzucchelli, Goldstino superfields in N ¼ 2 supergravity, Lett. 113B, 170 (1982). J. High Energy Phys. 05 (2017) 061. [6] J. A. de Azcarraga and J. Lukierski, Supersymmetric par- [27] E. I. Buchbinder and S. M. Kuzenko, Three-form multiplet 2 ticles in N ¼ superspace: Phase space variables and and supersymmetry breaking, J. High Energy Phys. 09 Hamiltonian dynamics, Phys. Rev. D 28, 1337 (1983). (2017) 089. [7] W. Siegel, Hidden local supersymmetry in the supersym- [28] U. Lindström and M. Roček, Constrained local superfields, metric particle action, Phys. Lett. 128B (1983) 397. Phys. Rev. D 19, 2300 (1979). [8] E. Sezgin, in Salamfestschrift, edited by A. Ali, J. Ellis, and [29] S. Samuel and J. Wess, A superfield formulation of the S. Randjbar-Daemi (World Scientific, Singapore, 1994), nonlinear realization of supersymmetry and its coupling to – pp. 478 498. supergravity, Nucl. Phys. B221, 153 (1983). [9] D. P. Sorokin, Superbranes and superembeddings, Phys. [30] S. M. Kuzenko and S. J. Tyler, Complex linear superfield as Rep. 329, 1 (2000). σ a model for Goldstino, J. High Energy Phys. 04 (2011) 057. [10] M. Henneaux and L. Mezincescu, A -model interpretation [31] D. V. Volkov and V. P. Akulov, Possible universal neutrino of Green-Schwarz covariant superstring action, Phys. Lett. interaction, Pis’ma Zh. Eksp. Teor. Fiz. 16, 621 (1972) 152B, 340 (1985). [JETP Lett. 16, 438 (1972]; Is the neutrino a Goldstone [11] E. Witten, Twistor-like transform in ten dimensions, Nucl. particle?, Phys. Lett. 46B, 109 (1973). Phys. B266, 245 (1986). [32] V. P. Akulov and D. V. Volkov, Goldstone fields with spin [12] M. T. Grisaru, P. S. Howe, L. Mezincescu, B. Nilsson, and 1=2, Teor. Mat. Fiz. 18, 39 (1974) [Theor. Math. Phys. 18, P. K. Townsend, N ¼ 2 superstrings in a supergravity back- 28 (1974]. ground, Phys. Lett. 162B, 116 (1985). [33] D. V. Volkov and V. A. Soroka, Higgs effect for Goldstone [13] E. Bergshoeff, E. Sezgin, and P. K. Townsend, Superstring particles with spin 1=2, Pis’ma Zh. Eksp. Teor. Fiz. 18, 529 actions in D ¼ 3, 4, 6, 10 curved superspace, Phys. Lett. 169B, 191 (1986). (1973) [JETP Lett. 18, 312 (1973]. [14] A. A. Tseytlin and L. Wulff, Kappa-symmetry of super- [34] D. V. Volkov and V. A. Soroka, Gauge fields for symmetry string sigma model and generalized 10d supergravity group with spinor parameters, Teor. Mat. Fiz. 20, 291 (1974) equations, J. High Energy Phys. 06 (2016) 174. [Theor. Math. Phys. 20, 829 (1974]. [15] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, Three- [35] S. Deser and B. Zumino, Broken Supersymmetry and dimensional N ¼ 2 (AdS) supergravity and associated Supergravity, Phys. Rev. Lett. 38, 1433 (1977). č supercurrents, J. High Energy Phys. 12 (2011) 052. [36] S. J. Gates Jr., M. T. Grisaru, M. Ro ek, and W. Siegel, [16] I. Antoniadis, E. Dudas, S. Ferrara, and A. Sagnotti, The Superspace, or One Thousand and One Lessons in Super- Volkov-Akulov-Starobinsky supergravity, Phys. Lett. B symmetry (Benjamin/Cummings, Reading, MA, 1983). 733, 32 (2014). [37] S. M. Kuzenko, U. Lindström, and G. Tartaglino- [17] E. Dudas, S. Ferrara, A. Kehagias, and A. Sagnotti, Proper- Mazzucchelli, Off-shell supergravity-matter couplings in ties of nilpotent supergravity, J. High Energy Phys. 09 three dimensions, J. High Energy Phys. 03 (2011) 120. (2015) 217. [38] P. van Nieuwenhuizen, D ¼ 3 conformal supergravity and [18] E. A. Bergshoeff, D. Z. Freedman, R. Kallosh, and A. Van Chern-Simons terms, Phys. Rev. D 32, 872 (1985). Proeyen, Pure de Sitter supergravity, Phys. Rev. D 92, [39] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, Conformal 085040 (2015); Erratum, Phys. Rev. D93, 069901(E) supergravities as Chern-Simons theories revisited, J. High (2016). Energy Phys. 03 (2013) 113. [19] F. Hasegawa and Y. Yamada, Component action of nilpotent [40] D. Butter, S. M. Kuzenko, J. Novak, and G. Tartaglino- multiplet coupled to matter in 4 dimensional N ¼ 1 super- Mazzucchelli, Conformal supergravity in three dimensions: gravity, J. High Energy Phys. 10 (2015) 106. Off-shell actions, J. High Energy Phys. 10 (2013) 073. [20] S. M. Kuzenko, Complex linear Goldstino superfield and [41] L. Castellani, R. D’Auria, and P. Fre, Supergravity and supergravity, J. High Energy Phys. 10 (2015) 006. superstrings: A Geometric perspective. Vol. 2: Supergravity [21] R. Kallosh and T. Wrase, de Sitter supergravity model (World Scientific, Singapore, 1991), pp. 680–684. building, Phys. Rev. D 92, 105010 (2015). [42] M. F. Hasler, The three-form multiplet in N ¼ 2 superspace, [22] M. Schillo, E. van der Woerd, and T. Wrase, The general de Eur. Phys. J. C 1, 729 (1998). Sitter supergravity component action, Fortschr. Phys. 64, [43] S. J. Gates Jr., in Supersymmetries and Quantum Symmetries, 292 (2016). edited by J. Wess and E. A. Ivanov (Springer, Berlin, 1999),

126015-29 EVGENY I. BUCHBINDER et al. PHYSICAL REVIEW D 96, 126015 (2017) pp. 46–57; J. Wess and E. A. IvanovEctoplasm has no [63] B. M. Zupnik and D. G. Pak, Superfield formulation of the topology, Nucl. Phys. B541, 615 (1999). simplest three-dimensional gauge theories and conformal [44] S. J. Gates Jr., M. T. Grisaru, M. E. Knutt-Wehlau, and W. supergravities, Teor. Mat. Fiz. 77, 97 (1988) [Theor. Math. Siegel, Component actions from curved superspace: Normal Phys. 77, 1070 (1988]. coordinates and ectoplasm, Phys. Lett. B 421, 203 (1998). [64] D. Butter and S. M. Kuzenko, A dual formulation of [45] M. Becker, D. Constantin, S. J. Gates Jr., W. D. Linch III, W. supergravity-matter theories, Nucl. Phys. B854, 1 (2012). Merrell, and J. Phillips, M-theory on Spin(7) manifolds, [65] S. J. Gates, Jr., Super p-form gauge superfields, Nucl. Phys. fluxes and 3D, N ¼ 1 supergravity, Nucl. Phys. B683,67 B184, 381 (1981). (2004). [66] S. J. Gates, Jr. and W. Siegel, Variant superfield representa- [46] S. M. Kuzenko, J. Novak, and G. Tartaglino-Mazzucchelli, tions, Nucl. Phys. B187, 389 (1981). Higher derivative couplings and massive supergravity in [67] I. L. Buchbinder and S. M. Kuzenko, Quantization of the three dimensions, J. High Energy Phys. 09 (2015) 081. classically equivalent theories in the superspace of simple [47] S. M. Kuzenko, U. Lindström, and G. Tartaglino- supergravity and quantum equivalence, Nucl. Phys. B308, Mazzucchelli, Three-dimensional (p,q) AdS superspaces 162 (1988). and matter couplings, J. High Energy Phys. 08 (2012) 024. [68] I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of [48] I. A. Batalin and G. A. Vilkovisky, Quantization of gauge Supersymmetry and Supergravity, Or a Walk Through theories with linearly dependent generators, Phys. Rev. D Superspace (IOP, Bristol, 1995) (revised ed. 1998). 28, 2567 (1983). [69] P. Binetruy, F. Pillon, G. Girardi, and R. Grimm, The 3-form [49] M. F. Sohnius and P. C. West, An alternative minimal off- multiplet in supergravity, Nucl. Phys. B477, 175 (1996). shell version of N ¼ 1 supergravity, Phys. Lett. 105B, 353 [70] B. A. Ovrut and D. Waldram, Membranes and three-form (1981). supergravity, Nucl. Phys. B506, 236 (1997). [50] M. F. Sohnius and P. C. West, in Quantum Structure of [71] S. M. Kuzenko and S. A. McCarthy, On the component Space and Time, edited by M. J. Duff and C. J. Isham structure of N ¼ 1 supersymmetric nonlinear electrodynam- (Cambridge University Press, Cambridge, England, 1982), ics, J. High Energy Phys. 05 (2005) 012. pp. 187–222. [72] P. Binetruy, G. Girardi, and R. Grimm, Supergravity cou- [51] M. Sohnius and P. C. West, The tensor calculus and matter plings: AGeometric formulation, Phys. Rep. 343, 255 (2001). coupling of the alternative minimal auxiliary field formu- [73] F. Farakos, S. Lanza, L. Martucci, and D. Sorokin, Three- lation of N ¼ 1 supergravity, Nucl. Phys. B198, 493 (1982). forms in supergravity and flux compactifications, Eur. Phys. [52] A. Achúcarro and P. K. Townsend, A Chern-Simons action J. C 77, 602 (2017). for three-dimensional AdS supergravity theories, Phys. Lett. [74] T. Dereli and S. Deser, Fermionic Goldstone-Higgs effect in B 180, 89 (1986). (2 þ 1)-dimensional supergravity, J. Phys. A 11, L27 (1978). [53] P. S. Howe, J. M. Izquierdo, G. Papadopoulos, and P. K. [75] Z. Komargodski and N. Seiberg, From linear SUSY to Townsend, New supergravities with central charges and constrained superfields, J. High Energy Phys. 09 (2009) Killing spinors in 2 þ 1 dimensions, Nucl. Phys. B467, 183 066. (1996). [76] S. M. Kuzenko and S. J. Tyler, On the Goldstino actions and [54] M. Roček and P. van Nieuwenhuizen, N ≥ 2 supersym- their symmetries, J. High Energy Phys. 05 (2011) 055. metric Chern-Simons terms as d ¼ 3 extended conformal [77] E. Bergshoeff, D. Freedman, R. Kallosh, and A. Van supergravity, Classical Quantum Gravity 3, 43 (1986). Proeyen, Construction of the de Sitter supergravity, arXiv: [55] S. M. Kuzenko, Prepotentials for N ¼ 2 conformal super- 1602.01678. gravity in three dimensions, J. High Energy Phys. 12 (2012) [78] S. M. Kuzenko and S. J. Tyler, Relating the Komargodski- 021. Seiberg and Akulov-Volkov actions: Exact nonlinear field [56] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, Different redefinition, Phys. Lett. B 698, 319 (2011). representations for the action principle in 4D N ¼ 2 super- [79] M. Roček, Linearizing the Volkov-Akulov model, Phys. gravity, J. High Energy Phys. 04 (2009) 007. Rev. Lett. 41, 451 (1978). [57] W. Siegel, Solution to constraints in Wess-Zumino super- [80] I. Antoniadis and C. Markou, The coupling of non-linear gravity formalism, Nucl. Phys. B142, 301 (1978). supersymmetry to supergravity, Eur. Phys. J. C 75, 582 [58] B. Zumino, in Recent Developments in Gravitation, (2015). Carg`ese 1978, edited by M. L´evy and S. Deser (Plenum [81] N. Cribiori, G. Dall’Agata, F. Farakos, and M. Porrati, Press, New York, 1979), pp. 405–459. Minimal constrained supergravity, Phys. Lett. B 764, 228 [59] S. M. Kuzenko, U. Lindström, M. Roček, I. Sachs, and G. (2017). Tartaglino-Mazzucchelli, Three-dimensional N ¼ 2 super- [82] E. Ivanov and A. Kapustnikov, General relationship be- gravity theories: From superspace to components, Phys. tween linear and nonlinear realizations of supersymmetry, Rev. D 89, 085028 (2014). J. Phys. A 11, 2375 (1978). [60] J. Wess and B. Zumino, Superfield Lagrangian for super- [83] E. Ivanov and A. Kapustnikov, The nonlinear realization gravity, Phys. Lett. 74B, 51 (1978). structure of models with spontaneously broken supersym- [61] K. S. Stelle and P. C. West, Minimal auxiliary fields for metry, J. Phys. G 8, 167 (1982). supergravity, Phys. Lett. 74B, 330 (1978). [84] R. Casalbuoni, S. De Curtis, D. Dominici, F. Feruglio, and R. [62] S. Ferrara and P. van Nieuwenhuizen, The auxiliary fields of Gatto, Non-linear realization of supersymmetry algebra from supergravity, Phys. Lett. 74B, 333 (1978). supersymmetric constraint, Phys. Lett. B 220, 569 (1989).

126015-30 TWO-FORM SUPERGRAVITY, SUPERSTRING COUPLINGS, … PHYSICAL REVIEW D 96, 126015 (2017) [85] S. Deser and J. H. Kay, Topologically massive supergravity, [91] M. J. Duff, The cosmological constant is possibly zero, but Phys. Lett. 120B, 97 (1983). the proof is probably wrong, Phys. Lett. B 226, 36 (1989); [86] S. Deser, in Quantum Theory of Gravity, edited by S. M. Conf. Proc. C 8903131, 403 (1989). Christensen (Adam Hilger, Bristol, 1984), pp. 374–381. [92] M. J. Duncan and L. G. Jensen, Four-forms and the vanish- [87] R. Andringa, E. A. Bergshoeff, M. de Roo, O. Hohm, E. ing of the cosmological constant, Nucl. Phys. B336, 100 Sezgin, and P. K. Townsend, Massive 3D supergravity, (1990). Classical Quantum Gravity 27, 025010 (2010). [93] K. Groh, J. Louis, and J. Sommerfeld, Duality and couplings [88] E. A. Bergshoeff, O. Hohm, J. Rosseel, E. Sezgin, and P. K. of 3-form-multiplets in N ¼ 1 supersymmetry, J. High Townsend, More on massive 3D supergravity, Classical Energy Phys. 05 (2013) 001. Quantum Gravity 28, 015002 (2011). [94] F. Farakos and A. Kehagias, Emerging potentials in higher- [89] G. Alkac, L. Basanisi, E. A. Bergshoeff, M. Ozkan, and E. derivative gauged chiral models coupled to N ¼ 1 super- Sezgin, Massive N ¼ 2 supergravity in three dimensions, gravity, J. High Energy Phys. 11 (2012) 077. J. High Energy Phys. 02 (2015) 125. [95] F. Farakos, S. Ferrara, A. Kehagias, and M. Porrati, [90] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, New nil- Supersymmetry breaking by higher dimension operators, potent N ¼ 2 superfields, arXiv:1707.07390. Nucl. Phys. B879, 348 (2014).

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