Three-Dimensional a = 2 Supergravity Theories: from Superspace To

PHYSICAL REVIEW D 89, 085028 (2014) Three-dimensional N ¼ 2 supergravity theories: From superspace to components

Sergei M. Kuzenko,1 Ulf Lindström,2 Martin Roček,3 Ivo Sachs,4 and Gabriele Tartaglino-Mazzucchelli1 1School of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009, Australia 2Department of Physics and Astronomy, Theoretical Physics, Uppsala University, Box 516, SE-751 20 Uppsala, Sweden 3C.N.Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, New York 11794-3840, USA 4Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians University, Theresienstr. 37, D-80333 München, Germany (Received 31 January 2014; published 11 April 2014) For general off-shell N ¼ 2 supergravity-matter systems in three spacetime dimensions, a formalism is developed to reduce the corresponding actions from superspace to components. The component actions are explicitly computed in the cases of type I and type II minimal supergravity formulations. We describe the models for topologically massive supergravity which correspond to all the known off-shell formulations for three-dimensional N ¼ 2 supergravity. We also present a universal setting to construct supersymmetric backgrounds associated with these off-shell supergravities.

DOI: 10.1103/PhysRevD.89.085028 PACS numbers: 04.65.+e, 11.30.Pb, 12.60.Jv

I. INTRODUCTION dimensional reduction from 4D N ¼ 1 conformal supergravity following the procedure sketched in Sec. 7.2 The simplest way to construct N ¼ 2 locally super- of Superspace [2]. In practice, however, it is more ad- symmetric systems in three spacetime dimensions (3D) is vantageous to follow a manifestly covariant approach and perhaps through dimensional reduction from 4D N ¼ 1 derive the solution from scratch. In this sense the 3D story theories (see [1–3] for reviews). However, not all 3D is similar to that of N 2; 2 supergravity in two theories with four supercharges can be obtained in this ¼ð Þ dimensions [16,17]. way. For instance, N ¼ 2 conformal supergravity [4] and 1 Similarly to N ¼ 1 supergravity in four dimensions (2,0) anti-de Sitter (AdS) supergravity [5] cannot be so (see [2,3,18,19] for more details), different off-shell for- constructed. A more systematic approach to generate 3D mulations for 3D N ¼ 2 Poincaré and AdS supergravity N ¼ 2 supergravity-matter systems is clearly desirable. theories in superspace can be obtained by coupling con- Matter couplings in three-dimensional N ¼ 2 super- formal supergravity to different conformal compensators gravity were thoroughly studied in the 1990s using on- shell component approaches [6–8] (see also [9]). More [10,11]. There are three inequivalent types of conformal compensator: (i) a chiral scalar; (ii) a real linear scalar; and recently, off-shell formulations for general N ¼ 2 supergravity-matter systems have systematically been developed (iii) a (deformed) complex linear scalar. [10,11] purely within the superspace framework, extending Choosing the chiral compensator leads to the type I – minimal supergravity [11] which is the 3D analogue of earlier off-shell constructions [12 14]. One of the main N 1 goals of this paper is to work out techniques to reduce any the old minimal formulation for 4D ¼ supergravity [20]. As in four dimensions, this formulation can be used manifestly N ¼ 2 locally supersymmetric theory presented in [10,11] to components. Upon elimination of the auxiliary to realize both Poincaré and AdS supergravity theories; fields, one naturally reproduces the partial component the latter actually describes the so-called (1,1) AdS results obtained earlier in [6–8]. supergravity, following the terminology of [5]. The prepotential formulation for 3D N ¼ 2 conformal Choosing the real linear compensator leads to the type II minimal supergravity [11] which is a natural extension of supergravity was constructed in [15]. In principle, this N 1 prepotential solution could be obtained by off-shell the new minimal formulation for 4D ¼ supergravity [21]. Unlike the four-dimensional case, the type II formulation is suitable to realize both Poincaré and AdS super- 1In three dimensions, N -extended AdS supergravity exists in gravity theories (the new minimal formulation cannot be N 2 1 N 2 ½ = þ different versions [5],with½ = the integer part of used to describe 4D N ¼ 1 AdS supergravity). The point is N =2. These were called the ðp; qÞ AdS supergravity theories where the non-negative integers p ≥ q are such that N ¼ p þ q. that in three dimensions the real linear superfield is the field These theories are naturally associated with the 3D AdS super- strength of an Abelian vector multiplet, and the corre- groups OSpðpj2; RÞ × OSpðqj2; RÞ. sponding Chern-Simons terms may be interpreted as a

1550-7998=2014=89(8)=085028(40) 085028-1 © 2014 American Physical Society SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) cosmological term [14]. Adding such a Chern-Simons term massive supergravity are given only for the type I and type to the supergravity action results in the action for (2,0) AdS II minimal formulations. supergravity. Recently, supersymmetric backgrounds in the type II Finally, choosing the complex linear compensator leads supergravity have been studied within the component to the nonminimal supergravity presented in [11].Itis approach, both in the Euclidean [34] and Lorentzian analogous to the nonminimal formulation for 4D N ¼ 1 [35] signatures, building on the earlier results in four supergravity [22,23], the oldest off-shell locally super- and five dimensions; see [36–49] and references therein. symmetric theory. Both in three and four dimensions, this Since the authors of [34,35] did not have access to the formulation exists in several versions labeled by a real complete off-shell component actions for type II super- parameter n ≠−1=3, 0 in the 4D case [23] or, more gravity and its matter couplings, their analysis was based conveniently, by w ¼ð1 − nÞ=ð3n þ 1Þ in the 3D case either on the considerations of linearized supergravity [34] [11]. The reason for such a freedom is that the super-Weyl or on the dimensional reduction 4D → 3D of the new transformation of the complex linear compensator is not minimal supergravity [35]. Here we present a universal fixed uniquely [10]. With the standard constraint setting to construct supersymmetric backgrounds associated with all the known off-shell formulations for 3D ðD¯ 2 − 4RÞΣ ¼ 0 (1.1) N ¼ 2 supergravity, that is the type I and type II minimal and the nonminimal supergravity theories.6 Our approach obeyed by the complex linear compensator Σ, the 4D N ¼ will be an extension of the 4D N ¼ 1 formalism to 1 nonminimal formulation is only suitable, for any value of determine (conformal) isometries of curved superspaces n, to describe Poincaré supergravity [2]. The situation in the which was originally developed almost 20 years ago in [3] 3D case is completely similar [11]. However, it was shown and further elaborated in [51].7 in [24] that n ¼ −1 nonminimal supergravity can be used This paper is organized as follows. In Sec. II we review to describe 4D N ¼ 1 AdS supergravity provided the the superspace formulation for the Weyl multiplet of 8 constraint (1.1) is replaced with a deformed one,2 N ¼ 2 conformal supergravity, following [10,11,14]. In Sec. III we present the locally supersymmetric and super- ðD¯ 2 − 4RÞΓ ¼ −4μ ≠ 0; μ ¼ const: (1.2) Weyl invariant action principle which is based on a closed super three-form. The formalism for component reduction, Applying the same ideas in the 3D case gives us the including the important Weyl multiplet gauge, is worked nonminimal formulation for (1,1) AdS supergravity [11]. out in Sec. IV. The component actions for type I and type II All supergravity-matter actions introduced in [10,11] are supergravity-matter systems are derived in Secs. V and VI realized as integrals over the full superspace or over its respectively. In Sec. VII we study the off-shell formulations chiral subspace. The most economical way to reduce such for topologically massive N ¼ 2 supergravity. Section VIII an action to components consists in recasting it as an is devoted to the construction of supersymmetric back- integral of a closed super three-form over spacetime (that is, grounds in all the known off-shell formulations for N ¼ 2 the bosonic body of the full superspace), in the spirit of the supergravity. superform approach3 to the construction of supersymmetric The main body of the paper is accompanied by four invariants [25–28]. The required superform construction is appendixes. In Appendix A we give a summary of the given in Sec. III. notation and conventions used as well as include some In this paper, we work out the component supergravity- technical relations. In Appendix B we give an alternative matter actions in the cases of type I and type II minimal form for the component action of the most general off-shell supergravity formulations.4 The case of nonminimal super- nonlinear σ-model in type I supergravity. Appendix C gravity can be treated in a similar way. As an application, contains the component Lagrangian for the model of an we describe off-shell models for topologically massive Abelian vector multiplet in conformal supergravity. N ¼ 2 supergravity5 which correspond to all the known Appendix D is devoted to the superspace action for N 2 off-shell formulations for three-dimensional N ¼ 2 super- ¼ conformal supergravity; at the component level, gravity. However, the component actions for topologically

2The constraint (1.2) is super-Weyl invariant if and only if 6After our work was completed, there appeared a new paper in n ¼ −1. the hep-th archive [50] which also studied supersymmetric 3It is also known as the rheonomic approach [25] or the backgrounds in type I supergravity. ectoplasm formalism [26,27]. 7This approach has been used to construct rigid supersym- 4Various aspects of the component reduction in 4D N ¼ 1 metric field theories in 5D N ¼ 1 [52],4DN ¼ 2 [53,54], and supergravity theories were studied in the late 1970s [23,29–31]. 3D ðp; qÞ anti-de Sitter [11,55,56] superspaces. More complete presentations were given in the textbooks [1–3]. 8There exists a more general off-shell formulation for N ¼ 2 5Topologically massive N ¼ 1 supergravity was introduced in conformal supergravity [57]. It will be briefly reviewed in [32,33]. Its N ¼ 2 extended version was discussed in [4]. Appendix D.

085028-2 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) ¯ α ¯ α this action reduces to that constructed many years ago by ½J ; Dα¼Dα; ½J ; D ¼−D ; ½J ; Da¼0; č Ro ek and van Nieuwenhuizen [4]. M D ε D M D¯ ε D¯ ½ αβ; γ¼ γðα βÞ; ½ αβ; γ¼ γðα βÞ; M D 2η D (2.3) II. THE WEYL MULTIPLET IN U(1) SUPERSPACE ½ ab; c¼ c½a b: In this section we recall the superspace description of N ¼ 2 conformal supergravity. The results given here are The supergravity gauge group includes local K trans- essential for the rest of the paper. formations of the form

A. U(1) superspace geometry 1 C cd δKD ¼½K; D ; K ¼ ξ D þ K M þ iτJ ; We consider a curved superspace in three spacetime A A C 2 cd dimensions, M3j4, parametrized by local bosonic (xm) and (2.4) μ ¯ M m μ ¯ fermionic ðθ ; θμÞ coordinates z ¼ðx ; θ ; θμÞ, where 0 μ 1 2 θμ m ¼ , 1, 2 and ¼ ; . The Grassmann variables where the gauge parameters obey natural reality conditions, θ¯ and μ are related to each other by complex conjugation: but are otherwise arbitrary. Given a tensor superfield UðzÞ, θμ θ¯μ ¼ . The superspace structure group is chosen to be with its indices suppressed, it transforms as follows: 2 R 1 D SLð ; Þ ×Uð ÞR, and the covariant derivatives A ¼ ¯ α ðDa; Dα; D Þ have the form δKU ¼ KU: (2.5)

DA ¼ EA þ ΩA þ iΦAJ : (2.1) The covariant derivatives obey (anti-)commutation rela- ¯ α M M Here EA ¼ðEa;Eα; E Þ¼EA ðzÞ∂=∂z is the inverse tions of the form superspace vielbein, 1 1 1 D D CD cdM J (2.6) Ω Ω bcM Ω βγM −Ω cM (2.2) ½ A; Bg¼TAB C þ 2 RAB cd þ iRAB ; A ¼ 2 A bc ¼ 2 A βγ ¼ A c Φ 1 where T C is the torsion, and R cd and R constitute the is the Lorentz connection, and A is the Uð ÞR connection. AB AB AB The Lorentz generators with two vector indices curvature tensors. D (Mab ¼ −Mba), one vector index (Ma), and two spinor Unlike the 4D case, the spinor covariant derivatives α D¯ indices (Mαβ ¼ Mβα) are related to each other as follows: and α transform in the same representation of the Lorentz group, and this may lead to misunderstandings. If there is a 1 risk for confusion, we will underline the spinor indices M ¼ ε Mbc; M ¼ −ε Mc; a 2 abc ab abc associated with the covariant derivatives D¯ . For instance, 1 when the index C of the torsion T C takes spinor values, a αβ AB Mαβ ¼ðγ ÞαβM ; M ¼ − ðγ Þ Mαβ: γ a a 2 a we will write the corresponding components as TAB and TABγ. ¯ N 2 The Levi-Civita tensor εabc and the gamma matrices ðγaÞαβ In order to describe ¼ conformal supergravity, the are defined in Appendix A. The generators of SLð2; RÞ × torsion has to obey the covariant constraints given in [14]. 1 Uð ÞR act on the covariant derivatives as follows: The resulting algebra of covariant derivatives is [10,11]

¯ ¯ ¯ fDα; Dβg¼−4RMαβ; fDα; Dβg¼4RMαβ; (2.7a)

¯ c γδ fDα; Dβg¼−2iðγ ÞαβDc − 2CαβJ − 4iεαβSJ þ 4iSMαβ − 2εαβC Mγδ; (2.7b) 1 D D ε γb γCcD γ γSD − γ ¯ D¯ γ − γ γC Mδρ − 2D S D¯ ¯ M ½ a; β¼i abcð Þβ γ þð aÞβ γ ið aÞβγR ð aÞβ γδρ 3 ð β þ i βRÞ a 2 1 1 − ε γb α 2D S D¯ ¯ Mc − γ αγC γ γ 8D S D¯ ¯ J (2.7c) 3 abcð Þβ ð α þ i αRÞ 2 ð aÞ αβγ þ 3 ð aÞβ ð γ þ i γRÞ ;

1 D D¯ − ε γb γCcD¯ γ γSD¯ − γ γ D − γ γC¯ Mδρ − 2D¯ S − D M ½ a; β¼ i abcð Þβ γ þð aÞβ γ ið aÞβ R γ ð aÞβ γδρ 3 ð β i βRÞ a 2 1 1 − ε γb α 2D¯ S − D Mc γ αγC¯ γ γ 8D¯ S − D J (2.7d) 3 abcð Þβ ð α i αRÞ þ 2 ð aÞ αβγ þ 3 ð aÞβ ð γ i γRÞ ;

085028-3 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 4i 2 1 4i 2 D D ε γc αβεγδ − C¯ ε D¯ S ε D D ε γc αβεγδ − C ε D S − ε D¯ ¯ D¯ ½ a; b¼2 abcð Þ i αβδ þ 3 δðα βÞ þ3 δðα βÞR γ þ2 abcð Þ i αβδ þ 3 δðα βÞ 3 δðα βÞR γ 1 1 2 −ε γc αβ γ τδ D C¯ D¯ C D2 D¯ 2 ¯ DαD¯ S −4CcC −4S2 −4 ¯ Md abc 4ð Þ ð dÞ ði ðτ δαβÞ þi ðτ δαβÞÞþ6ð Rþ RÞþ3i α d RR 1 ε γc αβ D D¯ S −εcefD C −4SCc J (2.7e) þi abc 2ð Þ ½ α; β e f ;

C C − D C with αβγ defined by αβγ ¼ i ðα βγÞ. The algebra B. Super-Weyl invariance involves three dimension-1 torsion superfields: a real scalar The algebra of covariant derivatives (2.7) does not S ¯ , a complex scalar R and its conjugate R, and a real vector change under a super-Weyl transformation10 of the covar- C 1 −2 a; the Uð ÞR charge of R is . The torsion superfields iant derivatives [10,11] obey differential constraints implied by the Bianchi identities. The constraints are 0 1σ γ D α ¼ e2 ðDα þðD σÞMγα − ðDασÞJ Þ; (2.11a) ¯ DαR 0; (2.8a) 1 ¼ ¯ 0 σ ¯ ¯ γ ¯ D α ¼ e2 ðDα þðD σÞMγα þðDασÞJ Þ; (2.11b) ðD¯ 2 − 4RÞS ¼ 0; (2.8b) i i D0 σ D − γ γδ D σ D¯ − γ γδ D¯ σ D a ¼ e a 2 ð aÞ ð ðγ Þ δÞ 2 ð aÞ ð ðγ Þ δÞ 1 D C C − ε D¯ ¯ 4 D S (2.8c) α βγ ¼ i αβγ αðβð γÞR þ i γÞ Þ: i 3 ε Dbσ Mc D σ D¯ γσ M þ abcð Þ þ 2 ð γ Þð Þ a Equation (2.8b) means that S is a covariantly linear 3 − i γ γδ D D¯ σ J − i γ γδ D σ D¯ σ J superfield. When doing explicit calculations, it is useful 8 ð aÞ ð½ γ; δ Þ 4 ð aÞ ð γ Þð δ Þ to deal with equivalent forms of the relations (2.7c) and (2.11c) (2.7d) in which the vector index of Da is replaced by a pair of spinor indices. Such identities are given in Appendix A. As an immediate application of the (anti-)commutation accompanied by the following transformation of the torsion relations (2.7), we compute a covariantly chiral tensors: d’Alembertian. Let χ be a covariantly chiral scalar, ¯ 9 D χ 0 1 − J χ − χ 0 σ i ¯ γ α ¼ ,ofUð ÞR charge w, that is ¼ w . The S ¼ e S − DγD σ ; (2.11d) ’ □ 4 covariantly chiral d Alembertian c is defined by 1 1 1 ¯ 2 2 0 σ γδ ¯ γδ ¯ □ χ ≔ D − 4 D χ (2.9) C ¼ e C þ ðγ Þ ½Dγ; Dδσ þ ðγ Þ ðDγσÞDδσ ; c 16 ð RÞ : a a 8 a 4 a (2.11e) □ χ By construction, the scalar c is covariantly chiral and has 1 − Uð ÞR charge w. It is an instructive exercise to evaluate 1 1 □ χ χ 0 σ ¯ 2 ¯ ¯ γ the explicit form of c using the chirality of and the R ¼ e R þ D σ − ðDγσÞD σ : (2.11f) relations (2.7). The result is 4 4 1 1 The gauge group of conformal supergravity is defined to be □ χ DaD D2 − 2 1 − CaD Dα D K c ¼ a þ 2 R ið wÞ a þ 2 ð RÞ α generated by the transformation (2.4) and the super-Weyl transformations. The super-Weyl invariance is the reason 2 1 − D¯ αS D 2 − CaC 4S2 þ ið wÞð Þ α þ wð wÞð a þ Þ why the U(1) superspace geometry describes the Weyl multiplet. − DαD¯ S w D¯ 2 ¯ − D2 χ (2.10) wi α þ 8 ð R RÞ : Using the above super-Weyl transformation laws, it is an instructive exercise to demonstrate that the real symmetric spinor superfield [15] This relation turns out to be useful for the component reduction of locally supersymmetric sigma models to be 10 discussed later on. The super-Weyl transformation (2.11) is uniquely fixed if

one (i) postulates that1 the components of the inverse vielbein EA 0 2σ 0 σ transform as Eα ¼ e Eα and Ea ¼ e Ea þ spinor terms; and 9 1 χ The rationale for choosing the Uð ÞR charge of to be (ii) requires that the transformed covariant derivatives preserve negative is Eq. (2.15). the constraints [14] leading to the algebra (2.7).

085028-4 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) i D¯ L 0 JL −2L δ L 2σL (3.4) W ≔ Dγ D¯ C − D D¯ S − 4SC (2.12) α c ¼ ; c ¼ c; σ c ¼ c; αβ 2 ½ ; γ αβ ½ ðα; βÞ αβ the following chiral action transforms homogeneously, Z Z L 0 2σ 3 2 2 ¯ c 3 2 W ¼ e Wαβ: (2.13) S ¼ d xd θd θE ¼ d xd θEL (3.5) αβ c R c N 2 This superfield is the ¼ supersymmetric generaliza- is locally supersymmetric and super-Weyl invariant. The tion of the Cotton tensor. Using the Bianchi identities, one first representation in (3.5), which is only valid when can obtain an equivalent expression for this super Cotton R ≠ 0, is analogous to that derived by Zumino [29] in 4D tensor: N ¼ 1 supergravity. The second representation in (3.5) 1 involves integration over the chiral subspace of the full W − γ αβW E a ¼ 2 ð aÞ αβ superspace, with the chiral density possessing the 1 properties − γ αβ D D¯ S 2ε DbCc 4SC (2.14) ¼ 2 ð aÞ ½ ðα; βÞ þ abc þ a: JE ¼ 2E; δσE ¼ −2σE: (3.6) An application of this relation will be given in Appendix D. E A curved superspace is conformally flat if and only if The explicit expression for in terms of the supergravity W 0 prepotentials is given in [15]. Complex conjugating αβ ¼ ; see [57] for the proof. ¯ For our subsequent consideration, it is important to recall (3.5) gives the action Sc generated by the antichiral L¯ one of the results obtained in [10]. Let χ be a covariantly Lagrangian c. ¯ chiral scalar, Dαχ ¼ 0, which is primary under the super- The two actions, (3.2) and (3.5), are related to each other Weyl group, δσχ ¼ wσχ. Then its super-Weyl weight w and as follows: 1 its Uð ÞR charge are equal and opposite [10], Z Z 3 2θ 2θ¯ L 3 2θEL d xd d E ¼ d xd c; ¯ 0 wσ Dαχ ¼ 0; J χ ¼ −wχ; χ ¼ e χ: (2.15) 1 L ≔− D¯ 2 − 4 L (3.7) χ ’ □ χ c ð RÞ : Unlike itself, its chiral d Alembertian c , Eq. (2.10),is 4 not a primary superfield under the super-Weyl group. In what follows, we often consider the infinitesimal This relation shows that the chiral action, or its conjugate super-Weyl transformation and denote the corresponding antichiral action, is more fundamental than (3.2). variation by δσ. The chiral action can be reduced to component fields by making use of the prepotential formulation for N 2 conformal supergravity [15] and following III. SUPERSYMMETRIC AND SUPER-WEYL ¼ the component reduction procedure developed in [3] for INVARIANT ACTION N ¼ 1 supergravity in four dimensions. Being concep- There are two (closely related) locally supersymmetric tually straightforward, however, this procedure is tech- and super-Weyl invariant actions in N ¼ 2 supergravity nically rather tedious and time consuming. A simpler L L¯ [10]. Given a real scalar Lagrangian ¼ with the super- way to reduce Sc to components consists in making use Weyl transformation law of the superform approach to the construction of supersymmetric invariants [25–28].Inconjunctionwiththe δσL ¼ σL; (3.1) requirement of super-Weyl invariance, the latter approach turns out to be extremely powerful. As a ¯ the action matter of taste, here we prefer to deal with Sc,becauseit Z turns out that the corresponding closed three-form ¯ 3 2 2 ¯ −1 M involves no one-forms Eα. S ¼ d xd θd θEL;E¼ BerðEA Þ; (3.2) The super-Weyl transformation laws of the components of the superspace vielbein is invariant under the supergravity gauge group. It is also super-Weyl invariant due to the transformation law A M A a α ¯ E ≔ dz EM ¼ðE ;E ; EαÞ; (3.8)

δσE ¼ −σE: (3.3) are Given a covariantly chiral scalar Lagrangian L of super- c δ a −σ a (3.9a) Weyl weight two, σE ¼ E ;

085028-5 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 M3 α α i b αγ ¯ Thus Uj is a field on the spacetime which is the δσE ¼ − σE þ E ðγ Þ Dγσ; 3 4 2 2 b bosonic body of the curved superspace M j . 1 i In a similar way we define the bar projection of the δ ¯ − σ ¯ b γ Dγσ (3.9b) σEα ¼ 2 Eα þ 2 E ð bÞαγ : covariant derivatives: 1 We are looking for a dimensionless three-form, D ≔ M ∂ Ω bc M Φ J (4.2) Ξ L¯ 1 C∧ B∧ AΞ Aj EA j M þ A j bc þ i Aj : ð cÞ¼6 E E E ABC, such that (i) its components 2 Ξ are linear functions of L¯ and covariant derivatives ABC ¯ c thereof; and (ii) ΞðL Þ is super-Weyl invariant, More generally, given a differential operator ¯ c Oˆ ≔ D …D Oˆ δ Ξ L 0 A1 A , we define its bar projection, j, by the σ ð cÞ¼ . Modulo an overall numerical factor, such ˆ n rule ðOjUÞj ≔ ðD …D UÞj, for any tensor superfield U. a form is uniquely determined to be A1 An Of special importance is the bar projection of a vector 1 1 covariant derivative,11 Ξ L¯ γ∧ β∧ aΞ γ∧ b∧ aΞ ð cÞ¼2 E E E aβγ þ 2 E E E abγ 1 1 1 γ ¯ γ c∧ b∧ aΞ (3.10) Daj¼Da − ψ a Dγj − ψ¯ aγD j; (4.3) þ 6 E E E abc; 2 2 where where ¯ Ξ βγ ¼ 4ðγ ÞβγL ; (3.11a) 1 a a c D ω bcM J ≔ m∂ (4.4) a ¼ ea þ 2 a bc þ iba ;ea ea m Ξ − ε γd D¯ δL¯ (3.11b) abγ ¼ i abdð Þγδ c; is a spacetime covariant derivative with Lorentz and Uð1Þ 1 R Ξ ε D¯ 2 − 16 L¯ (3.11c) connections. For some calculations, it will be useful to abc ¼ abcð RÞ c: 1 4 work with a spacetime covariant derivative without Uð ÞR connection, Da, defined by It is easy to check that this three-form is closed,

Ξ L¯ 0 (3.12) d ð cÞ¼ ; Da ¼ Da − ibaJ : (4.5) Ξ L¯ and therefore ð cÞ generates a locally supersymmetric action. A. The Wess-Zumino and normal gauges The locally supersymmetric and super-Weyl invariant The freedom to perform general coordinate and local Ξ L¯ action associated with ð cÞ is Lorentz transformations can be used to choose a Wess- Z Zumino (WZ) gauge of the form 1 i ¯ − 3 D¯ 2 − 4 − γa ψ γD¯ ρ Sc ¼ d xe R ð Þγρ a ∂ ∂ 4 2 D δ μ D¯ α δα (4.6) αj¼ α ∂θμ ; j¼ μ ¯ : 1 ∂θμ εabc γ ψ βψ γ L¯ (3.13) þ 2 ð aÞβγ b c cj; In this gauge, it is easy to see that ≔ a with e detðem Þ. Here we have used definitions intro- 1 1 m m μ − ψ γδ μ ¯ − ψ¯ δγ duced in the next section. Ea j¼ea ;Ea j¼ 2 a γ ; Eaμj¼ 2 aγ μ; (4.7a) IV. COMPONENT REDUCTION

bc bc In this section we develop a simple universal setup to Ωa j¼ωa ; Φaj¼ba: (4.7b) carry out the component reduction of the general N ¼ 2 supergravity-matter systems presented in [10,11].Our The gauge condition (4.6) will be used in what follows. consideration below is very similar to that given in standard In the WZ gauge, we still have a tail of component fields textbooks on four-dimensional N ¼ 1 supergravity [2,3]. which originates at higher orders in the θ, θ¯ expansion of Given a superfield UðzÞ we define its bar projection Uj to E M, Ω bc and Φ and which are pure gauge (that is, they ¯ ¯ A A A be the θ, θ-independent term in the expansion of Uðx; θ; θÞ may be completely gauged away). A way to get rid of such in powers of θ’s and θ¯’s, 11 ≔ θ θ¯ (4.1) The definition of the gravitino agrees with that used in the 4D Uj Uðx; ; Þjθ¼θ¯¼0: case in [1].

085028-6 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) a tail of redundant fields is to impose a normal gauge to the superspace geometric objects by bar projecting the around the bosonic body M3 of the curved superspace (anti-)commutation relations (2.6). A short calculation M3j4; see [58] for more details. This gauge is defined by gives the torsion the conditions i ΘM A Θ ΘMδ A (4.8a) T c ¼ − ðψ¯ γcψ − ψ¯ γcψ Þ: (4.11) EM ðx; Þ¼ M ; ab 2 a b b a

M cd Θ ΩM ðx; ΘÞ¼0; (4.8b) The Lorentz connection is

ΘMΦ Θ 0 (4.8c) 1 Mðx; Þ¼ ; ω ω T − T T (4.12) abc ¼ abcðeÞþ2 ½ abc bca þ cab; where we have introduced where ωabcðeÞ denotes the torsion-free connection, M m μ ¯ μ ¯ Θ ≡ ðΘ ; Θ ; ΘμÞ ≔ ð0; θ ; θμÞ: (4.9) 1 ωabcðeÞ¼− ½Cabc − Cbca þ Ccab; In (4.8) the connections with world indices are defined in 2 Ω cd AΩ cd Φ AΦ c m m c the standard way: M ¼ EM A and M ¼ EM A.It Cab ≔ ðeaeb − ebea Þem : (4.13) can be proved [58] that the normal gauge conditions (4.8) A Θ allow one to reconstruct the vielbein EM ðx; Þ and the For the gravitino field strength defined by cd connections ΩM ðx; ΘÞ and ΦMðx; ΘÞ as Taylor series in Θ, in which all the coefficients [except the leading ψ γ ≔ D ψ γ − D ψ γ − T cψ γ (4.14) Θ-independent terms given by the relations (4.6) and ab a b b a ab c (4.7)] are tensor functions of the torsion, the curvature we read off and their covariant derivatives evaluated at Θ ¼ 0. In principle, there is no need to introduce the normal 4i ψ γ ε γc αβC¯ γ − ε γc γδD¯ S gauge which eliminates the tail of superfluous fields. Such ab ¼ i abcð Þ αβ 3 abcð Þ δ fields (once properly defined) are pure gauge and do not 2 show up in the gauge-invariant action. This is similar to the − ε γc γδD 2 ε γc γδψ Cd 3 abcð Þ δR þ i cd½að Þ bδ concept of double-bar projection; see e.g. [59]. 2 γ γδψ S 2 γ γδψ¯ B. The component field strengths þ ð ½aÞ bδ þ ið ½aÞ bδR : (4.15)

The spacetime covariant derivatives Da defined by (4.3) obey commutation relations of the form This tells us how the gravitino field strength is embedded in the superspace curvature and torsion. A longer calculation 1 is to derive an explicit expression for the Lorentz curvature D D T cD R cdM F J (4.10) ½ a; b¼ ab c þ 2 ab cd þ i ab ; R cd 2 ω cd 2ω cfω d − C fω cd (4.16) ab ¼ e½a b þ ½a bf ab f : c cd where T ab is the torsion, Rab the Lorentz curvature, and F 1 ab the Uð ÞR field strength. These tensors can be related The result is

1 4 cd i e αβ cdf τδ ¯ ¯ c d 2 ¯ 2 ¯ i α ¯ ¯ 2 R ¼ − ε ðγ Þ ε ðγ Þ ðD τCδαβ þ D τCδαβ Þþδ δ ðD R þ D RÞþ D DαS − 8RR − 8S ab 4 abe f ð Þ ð Þ ½a b 3 3 1 4 cdf e β γ cde δρ cd ¯ ¯ ½c d γ ¯ ¯ þ 4ε ε C C þ ψ ðγ Þβ Cγδρε ðγ Þ þ ε ð2DβS þ iDβRÞ − δ ðγ Þβ ð2DγS þ iDγRÞ abe f ½a b e 3 b 3 b 1 4 βγ ¯ cde δρ cd ¯ β β ½c d βγ ¯ þ ψ¯ ðγ Þ Cγδρε ðγ Þ þ ε ð2D S − iD RÞ − δ ðγ Þ ð2DγS − iDγRÞ ½aβ b e 3 b 3 b εcde γ ψ γψ δ ¯ − εcde γ γδψ¯ ψ¯ 2 εcde γ γδψ ψ¯ S 2ψ γψ¯ εcdeC (4.17) þ ð eÞγδ ½a b R ð eÞ ½aγ bδR þ i ð eÞ ½aγ bδ þ ½a bγ e :

085028-7 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 Finally, for the Uð ÞR field strength i i δ M σ M − γ γδ D σ ¯ M − γ γδ D¯ σ M σEa ¼ Ea 2ð aÞ ð ðγ ÞEδÞ 2ð aÞ ð ðγ ÞEδÞ ; c F ab ¼ Dabb − Dbba − T ab bc (4.18) (4.21a) we obtain 1 M M δσEα ¼ σEα ; (4.21b) 1 2 F ε γc αβ D D¯ S − εcefD C − 4SCc ab ¼ abc 2 ð Þ ½ α; β e f i i δ Ω bc σΩ bc − γ γδ D σ Ω¯ bc − γ γδ D¯ σ Ω bc cef γρ cef βγ ¯ σ a ¼ a ð aÞ ð ðγ Þ δÞ ð aÞ ð ðγ Þ δÞ þ iε ðγeÞ ψ fγDρS þ iε ðγeÞ ψ¯ fβDγS 2 2 2 D½bσ δc (4.21c) γ γδψ ψ¯ Cc − 2ψ γψ¯ S (4.19) þ ð Þ a ; þ ið cÞ ½aγ bδ j ½a bγ j:

γ cd 1 It turns out that the expressions for ψab , Rab , and F ab bc bc γ abc δσΩα ¼ σΩα þðD σÞðγ Þγαε ; (4.21d) drastically simplify if we also partially gauge fix the super- 2 a Weyl invariance to choose the so-called Weyl multiplet gauge that will be introduced in Sec. IV D. δ Φ σΦ − i γ γδ D σ Φ¯ − i γ γδ D¯ σ Φ σ a ¼ a 2 ð aÞ ð ðγ Þ δÞ 2 ð aÞ ð ðγ Þ δÞ 1 C. Residual gauge transformations γδ ¯ − ðγaÞ ½Dγ; Dδσ; (4.21e) In the WZ gauge, there remains a subset of gauge 8 transformations which preserve the conditions (4.6).To 1 work out the structure of this residual gauge freedom, we δ Φ σΦ D σ (4.21f) σ α ¼ 2 α þ i α : start from the transformation laws of the inverse vielbein E M and of the connections Ω cd and Φ under the gauge A A A Requiring the WZ gauge to be preserved, ðδK þ δσÞDαj¼ group of conformal supergravity. 0,gives Under the K transformation (2.4), the gauge fields vary as follows: b C b Dαξ j¼ξ TCα j; (4.22a) δ M ξC B M − D ξB M KEA ¼ TCA EB ð A ÞEB 1 1 D ξβ ξC β β τδβ σδβ (4.22b) 1 α j¼ TCα þ Kα þ i α þ α ; cd M B M τ J B M (4.20a) 2 2 þ 2 K ð cdÞA EB þ i ð ÞA EB ; ¯ C Dαξβj¼ξ TCαβj; (4.22c)

δ Ω cd ξC BΩ cd ξB cd − D ξB Ω cd D cd ξB cd γ εacdDγσ (4.22d) K A ¼ TCA B þ RBA ð A Þ B αK j¼ð RBα þð aÞαγ Þj; cd M BΩ cd − D cd τ J BΩ cd þ K ð cdÞA B ð AK Þþi ð ÞA B ; B Dατ ξ R α Dασ : (4.22e) (4.20b) j¼ð B þ i Þj We see that the residual gauge transformations are con- δ Φ ξC BΦ ξB − D ξB Φ strained. More specifically, only the parameters K A ¼ TCA B þ RBA ð A Þ B 1 cd M BΦ τ J BΦ − D τ a ≔ ξa ϵα ≔ ξα ≔ τ ≔ τ (4.23a) þ 2 K ð cdÞA B þ i ð ÞA B A : v j; j;wab Kabj; j (4.20c) are completely unrestricted in the WZ gauge. Here the bosonic parameters correspond to general coordinate (va), 1 Here we have introduced the Lorentz and Uð ÞR generators local Lorentz ðwabÞ and local R-symmetry (τ) transforma- Mcd B J B α ð ÞA and ð ÞA , respectively, defined by tions; the fermionic parameter ϵ generates a local Q- supersymmetry transformation. However, the parameters A cd Mcd D Mcd BD J D J BD Dαξ , DαK and Dατ are fully determined in terms of ½ ; A¼ð ÞA B; ½ ; A¼ð ÞA B: j j j those listed in (4.23a) and the following ones: The super-Weyl transformation (2.11) acts on the gauge σ ≔ σ η ≔ D σ (4.23b) fields as follows: j; α α j:

085028-8 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014)

Here the parameter σ and ηα generate the Weyl and local TABLE I. WZ-gauge choices and the parameters used to S-supersymmetry transformations respectively. It should be achieve them. pointed out that there is no parameter generating a local Gauge choice σ component special conformal transformation. As compared with the α ¯ 3D N ¼ 2 superconformal tensor calculus in superspace Sj¼0 ½D ; Dασj C 0 D D¯ σ [57], our formulation corresponds to a gauge in which αβj¼ ½ ðα; βÞ j the dilatation gauge field is switched off by making use of Rj¼R¯ j¼0 D¯ 2σj; D2σj the local special conformal transformations. D D¯ ¯ 0 D D¯ 2σ D¯ D2σ The relations (4.22) comprise all the conditions on the αRj¼ αRj¼ α j; α j 2 ¯ 2 ¯ 2 ¯ 2 residual gauge transformations, which are implied by D RjþD Rj¼0 fD ; D gσj the WZ gauge. If in addition we also choose the normal gauge (4.8), then all higher-order terms in the Θ expansion of the gauge parameters will be determined in terms of The superspace torsion and curvature transform as tensors those listed in (4.23). under the K-gauge group, Eqs. (2.4) and (2.5). Their super- In what follows, we will be interested in local Q- a Weyl transformations follow from the transformation rules supersymmetry transformations of the gauge fields em , γ γ of the dimension-1 torsion superfields given in the previous ψ ¼ e aψ and b ¼ e ab . Yet we introduce a more m m a m m a section, Eqs. (2.11d)–(2.11f). This allows one to compute general transformation the variations of the component field strengths under the supersymmetry transformation (4.24). δ ≔ δQ þ δS þ δW þ δR (4.24) D. The Weyl multiplet gauge which includes the local Q-supersymmetry (ϵα) and The super-Weyl invariance given by Eq. (2.11) preserves S-supersymmetry (ηα) transformations, as well as the Weyl (σ) and local R-symmetry (τ) transformations. the WZ gauge, so we can eliminate a number of component There is a simple reason for considering this combination fields. We choose the gauge conditions of four transformations. As will be shown in the next two ¯ 2 ¯ 2 ¯ Sj¼0; Cαβj¼0;Rj¼Rj¼0; D RjþD Rj¼0; sections, in any off-shell formulation for Poincaré or AdS supergravity, the Q-supersymmetry transformation has to (4.26) be accompanied by a special S-supersymmetry transformation with parameter ηαðϵÞ and, in some case, by a which constitute the Weyl multiplet gauge. In Table 1,we 1 τ ϵ special Uð ÞR transformation with parameter ð Þ. identify those components of the super-Weyl parameter σ Typically, it will hold that σðϵÞ¼0. However, since δη which have to be used in order to impose the Weyl is part of the super-Weyl transformation, it makes sense multiplet gauge. δ to include W into (4.24). In the gauge (4.26), the super-Weyl gauge freedom is Making use of the relations (4.20), (4.21), (4.22), and not fixed completely. We stay with unbroken Weyl and (4.23), we read off the transformation laws of the gauge local S-supersymmetry transformations corresponding fields under (4.24): to the parameters σ and ηα, η¯α respectively. The only a independent component fields are the vielbein em , a a ψ α ψ¯ α 1 δem ¼ iðϵγaψ¯ m þ ϵγ¯ aψ mÞ − σem ; (4.25a) the two gravitini m and m , and the Uð ÞR gauge field bm. These fields and the associated local symmetries correspond to those describing the N ¼ 2 Weyl δψ α 2D ϵα 2 a ϵβ α ϵ¯ βα m ¼ m þ em ð Taβ jþ βTa jÞ multiplet [4]. 1 In the Weyl multiplet gauge, the explicit expressions γ αβη¯ − τψ α σψ α (4.25b) þ ið mÞ β i m þ 2 m ; for the gravitino field strength and the curvature tensors simplify drastically. The gravitino field strength becomes 1 1 a β γδ γ ¯ ¯ δb ¼ − e ϵ iðγ Þ Cβγδjþ ðγ Þβ ð8iDγSj−DγRjÞ m 2 m a 3 a 4i i ψ γ ε γc αβC¯ γ − ε γc γδD¯ S (4.27) ϵβψ¯ C δ 2δδ S ψ δη ab ¼ i abcð Þ αβ abcð Þ δ : þ mδði β jþ β jÞþ2 m δ þc:c: 3 1 γδ ¯ −D τ − ðγ Þ ½Dγ;Dδσj: (4.25c) m 8 m The Lorentz curvature takes the form:

085028-9 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) i 4i R cd − ε γe αβεcdf γ τδ D C¯ D¯ C δc δd DαD¯ S ab ¼ 4 abeð Þ ð fÞ ð ðτ δαβÞ þ ðτ δαβÞÞþ3 ½a b α 2 8 β γ cde δρ cd ½c d γ þ ψ ðγ Þβ Cγδρε ðγ Þ þ ε DβS − δ ðγ Þβ DγS ½a b e 3 b 3 b 2 8 βγ ¯ cde δρ cd ¯ β ½c d βγ ¯ ψ¯ γ Cγδρε γ ε D S − δ γ DγS : (4.28) þ ½aβ ð bÞ ð eÞ þ 3 b 3 bð Þ

From here we read off the scalar curvature

8 R ψ 4 DαD¯ S ψ β γa γδ γa γD S (4.29) ðe; Þ¼ i α jþ a ð Þ Cβγδ þ 3 ð Þβ γ þ c:c: :

An equivalent form for this result is 1 DαD¯ S R ψ ψ¯ γ ψab ψ γ ψ¯ ab (4.30) i α j¼4 ð ðe; Þþi a b þ i a b Þ: 1 The Uð ÞR field strength becomes 1 F ε γc αβ D D¯ S εcde γ βγ ψ D S ψ¯ D¯ S (4.31) ab ¼ abc 2 ð Þ ½ α; β þ i ð dÞ ½ eβ γ þ eβ γ :

An equivalent form for this result is

1 1 1 D D¯ S γa F ψ bψ¯ − ψ¯ bψ εabc ψ γdψ¯ − ψ¯ γdψ (4.32) ½ ðα; βÞ j¼ð Þαβ a þ 4 ab 4 ab þ 4 ð b cd b cdÞ ;

where F ≔ 1 ε F bc. 1 1 a 2 abc δ − a ϵγbψ¯ ε ϵψ¯ bc − ψ η We need to determine those residual gauge transforma- bm ¼ 2 em ab þ 2 abc i a þ c:c: tions which leave invariant the Weyl multiplet gauge. − Dmτ; (4.34c) Imposing the conditions δCαβj¼δSj¼δRj¼0, with the transformation δ defined by (4.24), we obtain with the γ matrices with world indices defined by γm ≔ aγ γ~ ¯ c ab ab em a and similarly for m. D α; Dβ σ −ε γ ϵψ¯ − ϵ¯ψ ; (4.33a) ½ ð Þ j¼ cabð Þαβð Þ The above description of the Weyl multiplet agrees with that given in [4]. γ ¯ i c ab c ab iD Dγσj¼ ε ðϵγ ψ¯ þ ϵγ¯ ψ Þ; (4.33b) 2 cab E. Alternative gauge fixings There exist different schemes for component reduction 2 ¯ 2 D σj¼D σj¼0: (4.33c) that correspond to alternative choices of fixing the supergravity gauge freedom. Here we mention two possible Using these results in (4.25a)–(4.25c), together with the options that are most useful in the context of type I or type fact that the bar projections of all the dimension-1 curvature II supergravity formulations. 1 superfields vanish, we derive the transformations of the The super-Weyl and local Uð ÞR gauge freedom can be gauge fields in the Weyl multiplet gauge: used to impose the gauge condition [10]

a a a a S ¼ 0; Φα ¼ 0; Φ ¼ C : (4.35) δem ¼ iðϵγ ψ¯ m þ ϵγ¯ ψ mÞ − σem ; (4.34a) a a Since the resulting U 1 connection is a tensor superfield, 1 ð ÞR α α α α α ∇ δψ ¼ 2D ϵ þ iðγ~ η¯Þ − iτψ − σψ ; (4.34b) we may equally well work with covariant derivatives A m m m m 2 m 1 without Uð ÞR connection, which are defined by

085028-10 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) ∇ ∶ D ∇ ≔ D − C J (4.36) 1 1 α ¼ α; a a i a : D¯ Φ 0 J Φ − Φ δ Φ σΦ (5.1) α ¼ ; ¼ 2 ; σ ¼ 2 ; The gauge condition (4.35) does not fix completely the ¯ 1 and its conjugate Φ. super-Weyl and local Uð ÞR gauge freedom. The residual transformation is generated by a covariantly chiral scalar ¯ parameter λ, ∇αλ ¼ 0, and has the form [11] A. Pure supergravity

0 1ð3λ¯−λÞ γ As a warm-up exercise, we first analyze the action for ∇ α 2 ∇α ∇ λ Mγα ; (4.37a) ¼ e ð þð Þ Þ pure type I supergravity with a cosmological term. It is obtained from (5.15) by switching off the matter sector, that ¯ i i ∇0 λþλ ∇ − γ αβ ∇ λ ∇¯ − γ αβ ∇¯ λ¯ ∇ is by setting K ¼ 0 and W ¼ μ ¼ const, a ¼ e a 2 ð aÞ ð α Þ β 2 ð aÞ ð α Þ β Z Z i 3 2 2 3 2 4 ε ∇b λ λ¯ Mc − ∇γλ ∇¯ λ¯ M S ¼ −4 d xd θd θ¯EΦΦ¯ þ μ d xd θEΦ þ abcð ð þ ÞÞ 2 ð Þð γ Þ a : SG Z (4.37b) 3 2 ¯ 4 þ μ¯ d xd θ¯ E Φ¯ : (5.2) The dimension-1 torsion superfields transform as The second and third terms in the action generate a ¯ i 1 supersymmetric cosmological term, with the parameter C0 λþλ C − ∇ λ − λ¯ γ αβ ∇ λ ∇¯ λ¯ 2 a ¼ e a 2 að Þþ4 ð aÞ ð α Þ β ; jμj being proportional to the cosmological constant. The dynamics of this theory was analyzed in superspace in [11]. (4.38a) Here we reduce the action (5.2) to components. In the Weyl multiplet gauge, the super-Weyl gauge 1 3λ ¯ 2 −λ¯ R0 ¼ − e ð∇ − 4RÞe : (4.38b) freedom is not fixed completely. We can use the residual 4 1 Weyl and local Uð ÞR symmetries to impose the gauge condition This formulation is very similar to the old minimal 4D N ¼ 1 supergravity, see e.g. [3] for a review. It is best Φ 1: (5.3a) suited when dealing with type I minimal supergravity- j¼ matter systems. The super-Weyl freedom can be used to impose the In addition, the local S-supersymmetry invariance can be gauge condition [10] used to make the gauge choice D Φ 0 (5.3b) R ¼ 0; (4.39) α j¼ :

1 The only surviving component field of Φ may be defined as with the local Uð ÞR group being unbroken. This superspace geometry is most suitable for the type II minimal 2 supergravity. The gauge condition (4.39) does not M ≔ D Φj: (5.4) completely fix the super-Weyl group. The residual super- Weyl transformation is generated by a real superfield σ To perform the component reduction of the kinetic term constrained by D2e−σ ¼ D¯ 2e−σ ¼ 0. in (5.2), the first step is to associate with it, by applying the Each of the two restricted superspace geometries relation (3.7), the equivalent antichiral Lagrangian L¯ D2 − 4 ¯ ΦΦ¯ considered, (4.35) and (4.39), is suitable for describing c ¼ð RÞð Þ. After that we can use (3.13) to the Weyl multiplet of conformal supergravity. In each reduce the action to components. The antichiral Lagrangian μ¯ L¯ μ¯Φ¯ 4 case, we can define a Wess-Zumino gauge and a Weyl corresponding to the term in (5.2) is c ¼ . Finally, multiplet gauge. the component version of the μ term in (5.2) is the complex Some alternative gauge conditions will be used in conjugate of the μ¯ term. Sec. VIII. Direct calculations lead to the supergravity Lagrangian 1 i 1 R ψ εabc ψ¯ ψ ψ¯ ψ − ¯ a LSG ¼ 2 ðe; Þþ4 ð ab c þ a bcÞ 4MMþb ba V. TYPE I MINIMAL SUPERGRAVITY 1 1 −μ¯ ¯ − εabcψ γ ψ −μ εabcψ¯ γ ψ¯ This off-shell supergravity theory and its matter cou- M 2 a b c M þ2 a b c ; plings are analogous to the old minimal formulation for 4D N ¼ 1 supergravity; see [1–3] for reviews. Its specific (5.5) feature is that its conformal compensators are a covariantly chiral superfield Φ of super-Weyl weight w ¼ 1=2, where the gravitino field strength is defined as

085028-11 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) i ψ ≔ D ψ − D ψ − T cψ ; (5.6) δ ψ α 2D ϵα − ϵα aε b γ~cϵ α ¯ γ~ ϵ¯ α ab a b b a ab c ϵ m ¼ m ibm þ iem abcb ð Þ þ 2 Mð m Þ ; which differs from (4.14). We recall that the covariant (5.12b) D 1 derivative a, Eq. (4.5), has no Uð ÞR connection. It is natural to use D since the local U 1 symmetry has been 1 1 a ð ÞR δ − a ϵγbψ¯ ε ϵψ¯ bc ε bϵψ¯ c fixed. The type I supergravity multiplet consists of the ϵbm ¼ 2 em ab þ 2 abc þ i abcb following fields: the dreibein e a, the gravitini ψ α and m ¯ m i ψ¯ α, and the auxiliary fields M, M and b . ϵγbψ¯ − 2 ϵγbψ¯ − ϵψ m m þ iðba b bb aÞ 2 M a þ c:c: Upon elimination of the auxiliary fields, the Lagrangian becomes (5.12c)

1 i R ψ εabc ψ¯ ψ ψ¯ ψ The supergravity multiplet also includes the auxiliary LSG ¼ ðe; Þþ ð ab c þ a bcÞ 2 4 scalar M ¼ D2Φj. Due to (5.9) and since D2σj¼0, μ¯ μ 4¯μμ εabcψ γ ψ − εabcψ¯ γ ψ¯ (5.7) Eq. (4.33c), the supersymmetry transformation of M is þ þ 2 a b c 2 a b c: β 2 ¯ β 2 ¯ β 2 δϵM ¼ ϵ DβD Φjþϵ¯βD D Φj¼ϵ¯β½D ; D Φj: (5.13) This Lagrangian describes (1,1) anti-de Sitter supergravity for μ ≠ 0 [5]. Making use of the algebra of covariant derivatives gives

B. Supersymmetry transformations c ab a a δϵM ¼ −εcabϵ¯γ~ ψ¯ − iMϵ¯γ~ ψ a − 2ibaϵ¯ψ¯ : (5.14) The gauge conditions (5.3a) and (5.3b) completely fix the Weyl, local U 1 and S-supersymmetry invariances. ð ÞR C. Matter-coupled supergravity However, performing just a single Q-supersymmetry transformation, with ϵα and ϵ¯α the only nonzero parameters We consider a general locally supersymmetric nonlinear in (4.34), does not preserve these gauge conditions. To σ-model restore the gauge defined by (5.3a) and (5.3b), the Z Z Q-supersymmetry transformation has to be accompanied S ¼ −4 d3xd2θd2θ¯EΦ¯ e−K=4Φ þ d3xd2θEΦ4W by a compensating S-supersymmetry transformation. Z Φ Indeed, applying the transformation (4.24) to j gives 3 2 ¯ 4 þ d xd θ¯ E Φ¯ W:¯ (5.15)

β 1 1 δΦj¼ϵ DβΦjþ ðσ − iτÞΦj¼ ðσ − iτÞ; (5.8) ¯ 2 2 Here the Kähler potential, K ¼ KðφI; φ¯ JÞ, is a real function of the covariantly chiral superfields φI and their conjugates where we have used Eqs. (5.3a) and (5.3b). Setting δΦj¼0 I¯ ¯ I I φ¯ , Dαφ ¼ 0. The superpotential, W ¼ Wðφ Þ, is a hol- gives omorphic function of φI alone. The matter superfields φI φ¯ J¯ σ τ 0 (5.9) and are chosen to be inert under the super-Weyl and ¼ ¼ : 1 local Uð ÞR transformations. This guarantees the super- Weyl invariance of the action. In Appendix B, we describe a On the other hand, the transformation of DαΦj is different parametrization of the nonlinear σ-model (5.15) in 1 which the dynamical variables Φ and φI are replaced by δD Φ ϵβD D Φ ϵ¯ D¯ βD Φ − η J Φ − Φ ϕi ϕ0 ϕI α j¼ β α jþ β α j α 2 covariantly chiral superfields ¼ð ; Þ of super-Weyl weight w ¼ 1=2 that parametrize a Kähler cone. 1 a The action (5.15) is also invariant under a target-space ¼ − ϵαM þðγ ϵ¯Þαb þ ηα; (5.10) 2 a Kähler transformation where here we have used (5.3a) and (5.3b). Setting ¯ Kðφ; φ¯Þ → Kðφ; φ¯ÞþΛðφÞþΛðφ¯Þ; (5.16a) δDαΦj¼0 gives

1 WðφÞ → e−ΛðφÞWðφÞ; (5.16b) η ϵ ϵ − γaϵ¯ (5.11) αð Þ¼2 αM ð Þαba: provided the compensator changes as Using these results in (4.34), we obtain the supersymmetry transformation laws of the gauge fields: Φ → eΛðφÞ=4Φ; (5.16c) δ a ϵγaψ¯ ϵγ¯ aψ (5.12a) ϵem ¼ ið m þ mÞ; with ΛðφIÞ an arbitrary holomorphic function.

085028-12 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) We first compute the component form of (5.15) in the The auxiliary scalar fields contained in Φ and Φ¯ may be special case W ¼ 0, defined in a manifestly Kähler-invariant way as Z 3 2 2 − φ φ¯ 4 2 −1 2 −1 −4 θ θ¯ Φ¯ Kð ; Þ= Φ (5.17) M ≔ D Φ¯ 4KΦ M¯ ≔ D¯ Φ¯ 4KΦ (5.20) Skinetic ¼ d xd d E e : ð e Þj; ð e Þj:

Associated with Skinetic is the antichiral Lagrangian To make the gauge condition (5.19c) Kähler invariant, the Kähler transformation generated by a parameter Λ has L¯ D2 − 4 ¯ Φ¯ −K=4Φ (5.18) 1 c ¼ð RÞð e Þ; to be accompanied by a special Uð ÞR transformation with i ¯ parameter τ ¼ 4 ðΛ − ΛÞ such that the component vector which has to be used for computing the component action field ba, which belongs to the Weyl multiplet and is defined using the general formula (3.13). by Eq. (4.4), transforms as Our consideration in this and the next sections is similar to that in 4D N ¼ 1 supergravity [60,61]. To reduce the → i D Λ − Λ¯ (5.21) action to components, we impose the following Weyl and ba ba þ 4 að Þ: local S-supersymmetry gauge conditions: We define the component fields of φI as follows: ðΦ¯ e−K=4ΦÞj ¼ 1; (5.19a)

I I ¯ −K=4 X ≔ φ ; (5.22a) DαðΦe ΦÞj ¼ 0: (5.19b) j

Both gauge conditions are manifestly Kähler invariant. It I I λα ≔ Dαφ j; (5.22b) turns out that the condition (5.19a) leads to the correct Einstein-Hilbert gravitational Lagrangian at the component 1 level. On the other hand, the condition (5.19b) guarantees I 2 I I α J K α ¯ F ≔− ½D φ þ Γ ðD φ ÞDαφ j: (5.22c) that no cross terms D SjDαKj are generated at the 4 JK component level. See Appendix B for more details. 1 Finally we fix the local Uð ÞR symmetry by imposing Under a holomorphic reparametrization, XI → fIðXÞ,of I I the gauge condition the target Kähler space, the fields λα and F transform as holomorphic vector fields. Direct calculations lead to the Φ Φ¯ K=8 (5.19c) j¼ j¼e : following component Lagrangian:

1 i 1 R ψ εabc ψ~¯ ψ ψ¯ ψ~ − MM¯ BaB Lkinetic ¼ 2 ðe; Þþ4 ð ab c þ a bcÞ 4 þ a ↔ ¯ ¯ i ¯ 1 ¯ I ¯ J − Da I D ¯ J − λIγaD~ λ¯J λIλ¯J εabcψ¯ γ ψ − ψ¯ aψ þ gIJ¯ F F ð X Þ aX 4 a þ 8 ð a b c aÞ 1 1 1 − λIγaλ¯J¯ ψ¯ bγ ψ ε ψ¯ bψ c − ψ aγ γ~ λIDb ¯ J¯ − λ¯J¯ γ~ γ ψ¯ aDb I 8 ð a b þ abc Þ 2 b a X 2 a b X 1 1 λIλJλ¯K¯ λ¯L¯ − λIλ¯J¯ 2 (5.23) þ 16 RIKJ¯ L¯ 64 ðgIJ¯ Þ ;

~ where the auxiliary vector field Ba is defined by the rule where the Kähler-covariant derivative Da is defined (similarly to the 4 D case; see e.g. [1]) as follows:

1 ¯ i ¯ B ≔ − λIγ λ¯J − D I − D ¯ I (5.24) 1 ¯ a ba gIJ¯ a ðKI aX KI¯ aX Þ; D~ ψ ≔ D ψ D J − D ¯ J ψ (5.26a) 8 4 a b a b þ 4 ðKJ aX KJ¯ aX Þ b; 1 and is invariant under the Kähler transformations, in D~ λI ≔ D λI − D J − D ¯ J¯ λI λJΓI D K a a ðKJ aX KJ¯ aX Þ þ JK aX : accordance with (5.21). The gravitino field strength in 4 (5.23) differs from that introduced earlier in (5.6): (5.26b)

In (5.23), as usual g ¯ denotes the Kähler metric, g ¯ K ¯ , ~ ~ c IJ IJ ¼ IJ ψ~ ab ¼ Daψ b − Dbψ a − T ab ψ c; (5.25) and RIKJ¯ L¯ the Riemann curvature,

085028-13 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) ¯ ¯ ¯ ¯ ¯ − ¯ ΓMΓN (5.27) rule (3.13) in conjunction with the relations (5.19) and RIKJL ¼ KIJK L gMN IJ K¯ L¯ ; (5.22) which define the component fields of Φ and φI. The ΓI IL¯ with JK ¼ g KJKL¯ the Christoffel symbol. second term in (5.15) is just the complex conjugate of the We now turn to the third term in (5.15). The correspond- third term. L¯ Φ¯ 4 ¯ φ¯ ing antichiral Lagrangian is c ¼ Wð Þ. To reduce this The component Lagrangian corresponding to the second functional to components, we again make use of the general and third terms in (5.15) is

1 1 ¯ i ¯ i − K=2 M¯ − εabcψ γ ψ ¯ M εabcψ¯ γ ψ¯ − ¯ I ψ γaλ¯I ∇ ¯ − I ψ¯ γaλI ∇ Lpotential ¼ e 2 a b c W þ þ 2 a b c W F þ 2 a I¯W F þ 2 a IW 1 1 λ¯I¯λ¯J¯ ∇ ∇ ¯ λIλJ∇ ∇ (5.28) þ 4 I¯ J¯ W þ 4 I JW :

Here we have introduced the Kähler-covariant derivatives

∇IW ≔ WI þ KIW; (5.29a)

∇ ∇ ≔ 2 ∇ −ΓL ∇ (5.29b) I JW WIJ þ KðI JÞW IJ LW þKIJW þKIKJW:

The component Lagrangian corresponding to the supergravity-matter system (5.15) is L ¼ Lkinetic þ Lpotential. Putting together (5.23) and (5.28) gives

↔ 1 i 1 ¯ ¯ i ¯ R ψ εabc ψ~¯ ψ ψ¯ ψ~ − MM¯ BaB I ¯ J − Da I D ¯ J − λIγaD~ λ¯J L ¼ 2 ðe; Þþ4 ð ab c þ a bcÞ 4 þ a þ gIJ¯ F F ð X Þ aX 4 a 1 1 1 1 λIλ¯J¯ εabcψ¯ γ ψ − ψ¯ aψ − λIγaλ¯J¯ ψ¯ bγ ψ ε ψ¯ bψ c − ψ aγ γ~ λIDb ¯ J¯ − λ¯J¯ γ~ γ ψ¯ aDb I þ 8 ð a b c aÞ 8 ð a b þ abc Þ 2 b a X 2 a b X 1 1 1 1 λIλJλ¯K¯ λ¯L¯ − λIλ¯J¯ 2 − K=2 M¯ − εabcψ γ ψ ¯ M εabcψ¯ γ ψ¯ þ 16 RIKJ¯ L¯ 64 ðgIJ¯ Þ e 2 a b c W þ þ 2 a b c W ¯ i ¯ i 1 ¯ ¯ 1 − ¯ I ψ γaλ¯I ∇ ¯ − I ψ¯ γaλI ∇ λ¯Iλ¯J∇ ∇ ¯ λIλJ∇ ∇ (5.30) F þ 2 a I¯W F þ 2 a IW þ 4 I¯ J¯ W þ 4 I JW :

Upon elimination of the auxiliary fields, the Lagrangian turns into

↔ 1 i 1 ¯ ¯ 1 ¯ ¯ i ¯ R ψ εabc ψ~¯ ψ ψ¯ ψ~ λIλJλ¯Kλ¯L − λIλ¯J 2 − Da I D ¯ J − λIγaD~ λ¯J L ¼ 2 ðe; Þþ4 ð ab c þ a bcÞþ16 RIKJ¯ L¯ 64 ðgIJ¯ Þ þ gIJ¯ ð X Þ aX 4 a 1 1 1 1 λIλ¯J¯ εabcψ¯ γ ψ − ψ¯ aψ − λIγaλ¯J¯ ψ¯ bγ ψ ε ψ¯ bψ c − ψ aγ γ~ λIDb ¯ J¯ − λ¯J¯ γ~ γ ψ¯ aDb I þ 8 ð a b c aÞ 8 ð a b þ abc Þ 2 b a X 2 a b X 1 i ¯ i 1 ¯ ¯ 1 K=2 εabc ψ γ ψ ¯ − ψ¯ γ ψ¯ ψ γaλ¯I∇ ¯ ψ¯ γaλI∇ − λ¯Iλ¯J∇ ∇ ¯ − λIλJ∇ ∇ þ e 2 ð a b cW a b cWÞþ2 a I¯W þ 2 a IW 4 I¯ J¯ W 4 I JW − K IJ¯ ∇ ∇¯ ¯ − 4 ¯ (5.31) e ðg IW J¯ W WWÞ:

K IJ¯ ∇ ∇¯ ¯ − 4 ¯ The potential generated, P3D ¼ e ðg IW J¯ W WWÞ, fields will differ from those given in Sec. VB. To preserve differs slightly from the famous four-dimensional result the gauge condition Φj¼eK=8, we have to choose K IJ¯ ∇ ∇¯ ¯ − 3 ¯ P4D ¼ e ðg IW J¯ W WWÞ, see e.g. [1]. i ¯ σ ϵ 0 τ ϵ − ϵλI − ϵ¯λ¯I (5.32) ð Þ¼ ; ð Þ¼ 4 ðKI KI¯ Þ: D. Supersymmetry transformations in Einstein frame ¯ −K=4 In matter coupled supergravity, the gauge conditions To preserve the gauge condition DαðΦe ΦÞj ¼ 0,we (5.19) depend on the matter fields. As a consequence, the have to apply the compensating S-supersymmetry trans- supersymmetry transformation laws of the supergravity formation with parameter

085028-14 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) 1 i ¯ δ G σG (6.2) η ϵ ϵ M ϵ¯β − D I − D ¯ I σ ¼ : αð Þ¼2 α þ bαβ þ 4 ðKI αβX KI¯ αβX Þ 1 I ¯J¯ I ¯J¯ þ g ¯ ðεαβλ λ þ 2λ λ Þ : (5.33) A. Real linear scalar 8 IJ ðα βÞ A general solution of the off-shell constraint (6.1) is

Making use of the parameters τ ϵ and ηα ϵ in (4.34), ð Þ ð Þ G D¯ αD DαD¯ (6.3) one may derive the supersymmetry transformations of the ¼ i αG ¼ i αG; a γ supergravity fields em and ψ m and bm. These expressions are not illuminating, and here we do not give them. We only where the real unconstrained scalar G is defined modulo comment upon the derivation of the supersymmetry trans- gauge transformations of the form formation of M. Its transformation follows from the fact δ Λ Λ¯ J Λ 0 D¯ Λ 0 (6.4) that M is defined to be the lowest component of the scalar G ¼ þ ; ¼ ; α ¼ : 2 ¯ −1 superfield D ðΦe 4KΦÞ. Making use of (5.32) and (5.33), after some algebra we get This gauge freedom allows us to interpret G as the gauge prepotential for an Abelian massless vector multiplet, 12 δ M −ε ϵ¯γ~cψ¯ ab − ϵ¯γ~aψ M − 2 ϵ¯ψ¯ a Iϵ¯λ¯J¯ and G as the gauge invariant field strength. The prepo- ϵ ¼ cab i a iba þ gIJ¯ F 1 tential can be chosen to be inert under the super-Weyl − ϵ¯γ~aλID ¯ J¯ − ϵ¯γ~aγbψ¯ D ¯ I¯ − D I transformations,13 igIJ¯ aX 2 aðKI¯ bX KI bX Þ 2 τ ϵ M (5.34) þ i ð Þ : δσG ¼ 0: (6.5)

In conclusion, we give the transformation rules of the Then the field strength G, defined by Eq. (6.3), transforms component fields of φI: according to (6.2). Making use of the constraint (6.1), we deduce the I I δϵX ¼ ϵλ ; (5.35a) important identity

αβ α ¯ ¯ α D Gαβ ¼ 8fðD SÞDα − ðD SÞDαgG 1 1 4 Dα D D¯ α ¯ D¯ G (6.6) δ λI 2ϵ I ΓI λJλK 2 γaϵ¯ D I − ψ λI þ ifð RÞ α þð RÞ αg ; ϵ α ¼ α F þ 4 JK þ ið Þα aX 2 a I where we have denoted þ iτðϵÞλα; (5.35b) G γa G ≔ D D¯ G 4C G (6.7) αβ ¼ð Þαβ a ½ ðα; βÞ þ αβ : 1 I J I K I I a I In the Weyl multiplet gauge (4.26), it follows from (6.6) δϵF ¼−ϵλ Γ F þ λ ηðϵÞþ2iτðϵÞF þiϵγ¯ ðD þib Þλ JK 2 a a that 1 − IL¯ ϵ¯λ¯P¯ λJλK ϵγ¯ aλJΓI D K − ϵγ¯ aψ I 4g RJLK¯ P¯ þi JK aX i aF Ga Ha − εabcψ¯ ψ G − εabc ψ γ DG ψ¯ γ D¯ G j¼ b c j i ð b c jþ b c jÞ; 1 −ϵγ¯ aγ~bψ¯ D I − ψ λI (5.35c) (6.8) a bX 2 b : Ha These can be derived by using the definition of the where denotes the Hodge-dual of the field strength of a components of φI (5.22). U(1) gauge field aa, 1 Ha εabcH H D − D − T c VI. TYPE II MINIMAL SUPERGRAVITY ¼ 2 bc; ab ¼ aab baa ab ac: This supergravity theory is a 3D analogue of the new (6.9) minimal formulation for 4D N ¼ 1 supergravity [21] (see [2,3] for reviews). Its conformal compensator is a real The other independent component fields of G may be G covariantly linear scalar , chosen as follows: ðD2 − 4R¯ ÞG ¼ðD¯ 2 − 4RÞG ¼ 0; (6.1) 12In four dimensions, the real linear superfield is naturally G ≠ 0 interpreted as the gauge invariant field strength of a massless chosen to be nowhere vanishing, . The super-Weyl tensor multiplet [62]. transformation of G is uniquely fixed by the constraint (6.1) 13The transformation law (6.5) is consistent with the require- to be ment that the gauge parameter Λ in (6.4) be super-Weyl inert.

085028-15 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) G D G D¯ G DαD¯ G (6.10) j; α j; α j; i α j: i D¯ αD2 ψ¯ bγ γ~ α a ψ¯ aγ~ α (6.17b) Gj¼2 ð a bÞ a þð aÞ ; B. Poincaré supergravity The off-shell action for type II supergravity without a 1 i 1 i 1 − D¯ 2D2Gj¼ D aa þ ψ¯ bγ ψ aa þ ψ¯ aψ þ Z: cosmological term [10,11] is 4 2 a 4 a b 2 a 2 Z (6.17c) 3 2 2 ¯ S ́¼ 4 d xd θd θEðG ln G − 4GSÞ: (6.11) Poincare The only independent component fields of G are 1 The action can be written in a different but equivalent form: D D¯ (6.18a) ½ ðα; βÞGj¼2 aαβ; Z G 3 2 2 ¯ α ¯ S ́¼ 4i d xd θd θEGD Dα ln ; (6.12) ¯ α 2 Poincare ΦΦ¯ ðD DαÞ Gj¼−Z: (6.18b) where Φ is a nowhere vanishing covariantly chiral super- By construction, the scalar Z is invariant under the gauge field of the type Eq. (5.1). One may see that the variables Φ transformations (6.4). and Φ¯ are purely gauge degrees of freedom. The component supergravity Lagrangian is The theory (6.11) was shown in [11] to be classically 1 i equivalent to type I supergravity without a cosmological R ψ εabc ψ¯ ψ ψ¯ ψ LPoincaré¼ 2 ðe; Þþ4 ð ab c þ a bcÞ term, the latter being defined by Eq. (5.2) with μ ¼ 0. The 1 1 above action can equivalently be described by the antichiral a ~ ~ a 2 þ aaF − HaH − Z ; (6.19) Lagrangian 4 4

L¯ − D2 − 4 ¯ G G − 4 S (6.13) where we have introduced the combination c ¼ ð RÞð ln G Þ; H~ a ≔ Ha − εabcψ¯ ψ (6.20) which has to be used to carry out the component reduction b c: of (6.11) by applying the general rule (3.13). Component reduction is often greatly simplified if The gravitino field strength is defined as in (4.14), with D D ψ suitable gauge conditions are imposed. Making use of a ¼ aðe; ;bÞ the covariant derivative containing the 1 the Weyl and local S-supersymmetry transformations allow Uð ÞR connection ba. In this formulation, the supergravity a us to choose the gauge conditions multiplet consists of the following fields: the dreibein em , α ¯ the gravitini ψ m and ψ mα, the two gauge fields am and bm, Gj¼1; (6.14a) and the auxiliary scalar Z. It is not difficult to demonstrate that the vector fields aa and ba have no propagating degrees of freedom for the DαGj¼0: (6.14b) dynamical system (6.19). To see this, let us work out 1 the equation of motion for the Uð ÞR gauge field ba. In the The compensator also contains a real scalar component supergravity Lagrangian (6.19), this field appears both in field that can be defined as the Rarita-Schwinger and Chern-Simons terms. We note that ≔ DαD¯ G (6.15) Z i α j: Z Z 3 F a 3 Ha (6.21) It is also useful to choose a WZ gauge for the U(1) gauge d xeaa ¼ d xeba ; symmetry (6.4). A standard choice is modulo a total derivative. Another relevant observation is Gj¼0; (6.16a) that the Rarita-Schwinger Lagrangian depends on ba only abc ¯ via the linear term −ε baψ bψ c. As a result, the equation DαG 0; (6.16b) j¼ of motion for ba is

D2 0 (6.16c) a Gj¼ : H~ ¼ 0: (6.22) It then follows from (6.14) and (6.16) that This equation tells us that aa has no independent degrees of 2 ¯ freedom on the mass shell. Now, varying (6.19) with D DαGj¼0; (6.17a) respect to aa and making use of (6.22) gives

085028-16 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) F a ¼ 0; (6.23) The component Lagrangian for off-shell (2,0) AdS supergravity is 1 and therefore the Uð ÞR connection ba is flat and may completely be gauged away. 1 i R ψ εabc ψ¯ ψ ψ¯ ψ F a The off-shell Lagrangian (6.19) does not coincide with LAdS ¼ 2 ðe; Þþ4 ð ab c þ a bcÞþaa that proposed in [14] to describe (2,0) Poincaré super- 1 1 1 i gravity (in our terminology, type II supergravity without a − H~ H~ a − 2 ξ Ha − εabcψ¯ γ ψ 4 a 4Z þ Z þ4aa 2 a b c : cosmological term),; see Eq. (4.1) in [14]. In particular, the a Lagrangian given in [14] contains no HaH term. The two (6.30) Lagrangians are actually equivalent modulo a total deriva- 14 tive and a redefinition of the ba field. Indeed, making use In this theory, the equation of motion for the Uð1Þ gauge of (6.21) and defining R field ba is still given by (6.22). As concerns the equation of motion for a , it becomes 1 a 0 ~ ba → ba ¼ ba − Ha; (6.24) 4 1 F a ξHa 0 (6.31) þ 2 ¼ : the Lagrangian (6.19) takes the form We see that the local Uð1Þ gauge freedom can be 1 i R L R e; ψ εabc ψ¯ 0 ψ ψ¯ ψ0 completely fixed by imposing the condition a ¼ − 2 b . Poincaré¼ 2 ð Þþ4 ð ab c þ a bcÞ a ξ a Dynamics described by the off-shell theory (6.30) is 1 0 Ha − 2 (6.25) equivalent to that generated by þ ba 4 Z ; 1 i where the gravitino field strength ψ0 is defined as (4.14) ~ R ψ εabc ψ¯ ψ ψ¯ ψ − 2ξψ¯ γ ψ ab LAdS ¼ 2 ðe; Þþ4 ð ab c þ a bc a b cÞ but with the U 1 connection b replaced by b0 . The ð ÞR a a 1 Lagrangian (6.25) is equivalent to the one given in [14]. ξ2 − F a (6.32) þ ξ ba :

C. (2,0) anti-de Sitter supergravity One can recognize (6.32) to be the standard on-shell The main difference between type II supergravity and the Lagrangian for (2,0) AdS supergravity [5] (see also [7]). N 1 new minimal formulation for ¼ supergravity in four The third term in the parentheses in (6.32) may be absorbed dimensions is that the action (6.11) can be deformed by into the gravitino field strength by introducing a modified adding a gauge-invariant cosmological term covariant derivative Z S ¼ −4ξ d3xd2θd2θ¯EGG: (6.26) 1 cosm Dˆ ψ β D ψ β − ξ γ βψ γ (6.33) a b ¼ a b 2 ð aÞγ b : To evaluate its component form, we have to make use of the supersymmetric action principle (3.13) with D. Supersymmetry transformations The gauge conditions (6.14) completely fix the Weyl and L¯ ξ D2 −4 ¯ G ξGD2 2ξ DαG D (6.27) c ¼ ð RÞðG Þ¼ Gþ ð Þ αG: local S-supersymmetry freedom. To preserve the condition Gj¼1, no residual Weyl invariance remains, σ ¼ 0. A short calculation that makes use of (6.17c) leads to However, each Q-supersymmetry transformation has to be accompanied by a compensating S-supersymmetry 1 i ξ Ha − εabcψ¯ γ ψ (6.28) transformation in order to preserve the condition Lcosm ¼ Z þ 4 aa 2 a b c : DαGj¼0. Indeed, the field DαGj transforms as The superfield action for (2,0) AdS supergravity is δ δ D G ϵβD D G ϵ¯ D¯ βD G η G Z ð Q þ SÞ α j¼ β α jþ β α jþ α j 3 2 2 ¯ 1 i S ¼ 4 d xd θd θEðG ln G − 4GS − ξGGÞ: (6.29) H~ γaϵ¯ − ϵ¯ η (6.34) AdS ¼ 2 að Þα 2 Z α þ α;

– 14G. T.-M. is grateful to Daniel Butter for pointing out the same where here we have used the identities (6.6) (6.8), (6.14), δ δ D G 0 situation in the new minimal formulation for 4D N ¼ 1 super- and (6.15). We have to require ð Q þ SÞ α j¼ , and gravity (see, e.g., [63,64] for the relevant discussions). therefore

085028-17 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 i η ϵ − H~ γaϵ¯ ϵ¯ (6.35) Making use of (4.33b), we then derive αð Þ¼ 2 að Þα þ 2 Z α: 1 1 i a abc ~ ~ a Choosing σ ¼ τ ¼ 0 and ηα ¼ ηαðϵÞ in (4.34), we obtain δϵZ ¼ − ϵγ ψ¯ aZ − ε ϵγaψ¯ bHc þ ϵψ¯ aH a 2 2 2 the supersymmetry transformations of the gauge fields em , γ i ψ and b : εabcϵγ ψ¯ (6.41) m m þ 2 a bc þ c:c: a a a δϵem ¼ iðϵγ ψ¯ m þ ϵγ¯ ψ mÞ; (6.36a) For completeness, let us also work out the supersym- ~ metry transformation of the field strength Ha. Making use H~ 1 of the definition of a gives α α i ~ α i a ~ c b α α δϵψ m ¼ 2Dmϵ þ Hmϵ − em εabcH ðγ~ ϵÞ þ Zðγ~mϵÞ ; 2 2 2 1 δ H~ a − γa αβ ϵγD D D¯ G ϵ¯ D¯ γ D D¯ G (6.36b) ϵ ¼ 2 ð Þ f γ½ α; β jþ γ ½ α; β j ¯ þð½Dα; DβσÞGjg: (6.42) 1 δ − a ε ϵψ¯ bc 2ϵγbψ¯ − ϵγbψ¯ H~ − ϵψ¯ With the aid of (4.33a) we obtain ϵbm ¼ 4 em f abc þ ab i a b aZg þ c:c: (6.36c) ~ a i abc ~ ½a ~ b δϵH ¼ − ε ϵψ¯ bHc þ iϵγ ψ¯ bH The supergravity multiplet also includes the fields a 2 m 1 and Z. The supersymmetry transformation of am follows εabcϵγ ψ¯ − εabcϵψ¯ (6.43) a þ b cZ bc þ c:c: from its definition am ¼ em aa, with aa originating as a 2 component field of G, Eq. (6.18a). Note that, in order to preserve the WZ gauge (6.16), in computing the super- E. Matter-coupled supergravity symmetry transformations of am it is necessary to include a compensating ϵ-dependent U(1) gauge transformation (6.4) The action for a locally supersymmetric σ-model with parameter ΛðϵÞ such that coupled to type II supergravity is Z Λ ϵ 0; (6.37a) ð Þj ¼ 3 2θ 2θ¯ G φ φ¯ (6.44) Smatter ¼ d xd d E Kð ; Þ: 1 i D Λ ϵ − ϵγ¯ b ϵ¯ (6.37b) α ð Þj ¼ 4 ð Þαab þ 2 α; Here the Kähler potential Kðφ; φ¯Þ and the matter super- i fields are the same as in Sec. V. In particular, the covariantly D2ΛðϵÞj ¼ ϵγ¯ aγ~bψ¯ a − ϵγ¯ aψ¯ : (6.37c) φI 1 2 a b a chiral superfields are super-Weyl and Uð ÞR neutral, I I δσφ ¼ J φ ¼ 0. The action is invariant under the Kähler We then obtain transformations (5.16a) due to the identity

a a αβ Z δϵam ¼ðδϵem Þaa − em ðγaÞ d3xd2θd2θ¯EGΛ φ 0: (6.45) γ ¯ ð Þ¼ × ðϵ Dγ½Dα; DβGjþ2iDαβΛðϵÞj þ c:c:Þ a αβ γ ¼ iϵγ ψ¯ maa þðγmÞ ϵ Dγ In order to carry out the component reduction of Smatter, ¯ α × fDα; DβgGjþ4iψ m DαΛðϵÞj þ c:c: (6.38) we associate with (6.44) the antichiral Lagrangian

Evaluating this variation gives 1 1 1 L¯ − D2 − 4 ¯ G − GD2 − DαG D c ¼ 4 ð RÞð KÞ¼ 4 K 2 ð Þ αK: δϵa −2 ϵψ¯ ϵψ¯ : (6.39) m ¼ ð m þ mÞ (6.46) The scalar field Z originates as a component field of G, Eq. (6.15), and therefore its supersymmetry transformation The component fields of φI are defined as in (5.22). Unlike is the type I supergravity case, now we do not have to modify the gauge conditions on the compensator in the i 2 ¯ α i ¯ 2 α α ¯ presence of matter. Direct calculations lead to the following δϵZ ¼ ϵαD D Gjþ ϵ¯αD D GjþiðD DασÞGj: (6.40) 2 2 component Lagrangian:

085028-18 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) ↔ ¯ ¯ i ¯ 1 ¯ 1 ¯ I ¯ J − D I Da ¯ J − λIγaD~ λ¯J − ψ¯ aλ¯JD I ψ λIDa ¯ J Lmatter ¼ gIJ¯ F F ð aX Þ X 4 a 2 aX þ 2 a X 1 1 1 − εabc ψ γ λID ¯ J¯ − ψ¯ γ λ¯J¯ D I ψ γbψ¯ aλIγ λ¯J¯ − ψ ψ¯ aλIλ¯J¯ 2 ð a b cX a b cX Þþ8 a b 8 a 1 ¯ ¯ 1 ¯ i ¯ εabc ψ ψ¯ λIγ λ¯J ψ γ ψ¯ λIλ¯J λIγ λ¯JH~ a − λIλ¯J þ 8 ð a b c þ a b c Þþ8 a 8 Z

i ¯ 1 ¯ ¯ Ha D ¯ I − D I λIλJλ¯Kλ¯L (6.47) þ 4 ½KI¯ aX KI aX þ16 RIKJ¯ L¯ :

1 Here we have introduced the Kähler-covariant derivative J a εabc D R − D R − T dR top ¼ 2 ð b c c b bc dÞ; ~ D λI ≔ D λI þ λJΓI D XK R ≔ D ¯ I¯ − D I (6.50) a a JK a a iðKI¯ aX KI aX Þ; D λI λI λJΓI D K (6.48) ¼ a þ iba þ JK aX : which is identically conserved. These properties were The σ-model action generated by the Lagrangian (6.47) pointed out in [14]. proves to be invariant under the Kähler transformations. Now, we consider a complete supergravity-matter The first term in the fourth line of (6.47) is the only one system described by the action [11] which varies under the Kähler transformations. The cor- Z 1 responding contribution to theR action is indeed Kähler 3 2 2 ¯ 3 a S ¼ 4 d xd θd θE G ln G þ Kðφ; φ¯Þ − 4GS : invariant due to the identity d xeH DaΛ ¼ 0. 4 As may be seen from (6.47), the gauge fields ba and aa (6.51) couple to conserved currents of completely different types. 1 1 The Uð ÞR gauge field couples to the Uð ÞR Noether current It describes Poincaré supergravity coupled to the locally supersymmetric σ-model. As shown in [11], this theory is 1 ¯ ¯ dual to the type I supergravity-matter system (5.17).To J a εabcψ¯ ψ g ¯ λIγaλJ: (6.49) Noether ¼ b c þ 2 IJ compute the corresponding component Lagrangian, we combine Lmatter given by (6.47) with the type II super- As regards the gauge field aa, it couples to the topological gravity Lagrangian without cosmological term, Eq. (6.19). current The result is

1 i 1 1 R ψ εabc ψ¯ ψ ψ¯ ψ F a − H~ H~ a − Z2 L ¼ 2 ðe; Þþ4 ð ab c þ a bcÞþaa 4 a 4 ↔ ¯ ¯ i ¯ 1 ¯ 1 ¯ I ¯ J − D I Da ¯ J − λIγaD~ λ¯J − ψ¯ aλ¯JD I ψ λIDa ¯ J þ gIJ¯ F F ð aX Þ X 4 a 2 aX þ 2 a X 1 1 1 − εabc ψ γ λID ¯ J¯ − ψ¯ γ λ¯J¯ D I ψ γbψ¯ aλIγ λ¯J¯ − ψ ψ¯ aλIλ¯J¯ 2 ð a b cX a b cX Þþ8 a b 8 a 1 1 εabc ψ ψ¯ λIγ λ¯J¯ ψ γ ψ¯ λIλ¯J¯ λIγ λ¯J¯ H~ a þ 8 ð a b c þ a b c Þþ8 gIJ¯ a

i ¯ 1 ¯ ¯ 1 ¯ Ha D ¯ I − D I λIλJλ¯Kλ¯L − λIλ¯J 2 (6.52) þ 4 ½KI¯ aX KI aX þ16 RIKJ¯ L¯ 64 ðgIJ¯ Þ ; where we have defined Z ¼ 0: (6.54)

i ¯ The equation of motion for the gauge field b is Z ≔ λIλ¯J (6.53) a Z þ 4 gIJ¯ : 1 Ha ≔ H~ a − λIγaλ¯J¯ gIJ¯ Let us show that the dynamical system (6.52) is 2 equivalent to the type I supergravity-matter system 1 ¯ ¯ ¼ Ha − εabcψ¯ ψ − g ¯ λIγaλJ ¼ 0: (6.55) (5.31) with W ¼ 0. Integrating out Z gives b c 2 IJ

085028-19 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) Let us consider the equation of motion for the gauge field Making use of the equations (6.54), (6.55), and (6.57) aa. It can be represented in the form reduces the supergravity-matter system (6.52) to that described by the Lagrangian (5.31) with W ¼ 0. c DaBb − DbBa − T ab Bc ¼ 0; (6.56) To preserve the gauge condition (6.57), any Kähler transformation generated by a parameter Λ has to be B 1 where a is defined in (5.24). This equation tells us that the accompanied by a special Uð ÞR transformation with 1 i ¯ local Uð ÞR gauge freedom can be completely fixed by parameter τ ¼ 4 ðΛ − ΛÞ; see also Eq. (5.21). choosing the condition Finally, we generalize the supergravity-matter system (6.51) to include a cosmological term. The manifestly 1 ¯ i ¯ λIγ λ¯J D I − D ¯ I (6.57) supersymmetric action is ba ¼ 8 gIJ¯ a þ 4 ðKI aX KI¯ aX Þ:

Z 1 4 3 2θ 2θ¯ G G φ φ¯ − 4 S − ξ G (6.58) S ¼ d xd d E ln þ 4 Kð ; Þ G G :

The corresponding component Lagrangian is obtained from the supergravity-matter Lagrangian (6.52) by adding the cosmological term (6.28). The result is

1 i 1 1 1 i R ψ εabc ψ¯ ψ ψ¯ ψ F a − H~ H~ a − Z − 2ξ 2 ξ Ha − ξεabcψ¯ γ ψ ξ2 L ¼ 2 ðe; Þþ4 ð ab c þ a bcÞþaa 4 a 4 ð Þ þ 4 aa 2 a b c þ ↔ ¯ ¯ i ¯ 1 ¯ 1 ¯ 1 ¯ ¯ I ¯ J − D I Da ¯ J − λIγaD~ λ¯J − ψ¯ aλ¯JD I ψ λIDa ¯ J − εabc ψ γ λID ¯ J − ψ¯ γ λ¯JD I þ gIJ¯ F F ð aX Þ X 4 a 2 aX þ 2 a X 2 ð a b cX a b cX Þ 1 ¯ 1 ¯ 1 ¯ ¯ 1 ¯ i ¯ ψ γbψ¯ aλIγ λ¯J − ψ ψ¯ aλIλ¯J εabc ψ ψ¯ λIγ λ¯J ψ γ ψ¯ λIλ¯J λIγ λ¯JH~ a − ξλIλ¯J þ 8 a b 8 a þ 8 ð a b c þ a b c Þþ8 gIJ¯ a 4

i ¯ 1 ¯ ¯ 1 ¯ Ha D ¯ I − D I λIλJλ¯Kλ¯L − λIλ¯J 2 (6.59) þ 4 ½KI¯ aX KI aX þ16 RIKJ¯ L¯ 64 ðgIJ¯ Þ :

We conclude this section by giving the supersymmetry transformations of the component field of φI:

I I δϵX ¼ ϵλ ; (6.60a) 1 1 δ λI 2ϵ I ΓI λJλK 2 γaϵ¯ D I − ψ λI (6.60b) ϵ α ¼ α F þ 4 JK þ ið Þα aX 2 a ;

1 1 ¯ ¯ i δ I −ϵλJΓI K λIη ϵ ϵγ¯ aD λI − IL ϵ¯λ¯PλJλK ϵγ¯ aλJΓI D K − ϵγ¯ aψ I ϵF ¼ JKF þ 2 ð Þþi a 4 g RJLK¯ P¯ þ i JK aX 2 aF 1 − ϵγ¯ aγ~bψ¯ D I − ψ λI (6.60c) a bX 2 b :

¯ I I I I I It is a useful exercise for the reader to derive these Dαϕ ¼ 0; J ϕ ¼ −rIϕ ; δσϕ ¼ rIσϕ : (6.61) transformation laws.

We introduce a supergravity-matter system of the form F. R-invariant sigma models Type II minimal supergravity admits more general matter Z couplings [11] than those we have so far studied. In S ¼ 4 d3xd2θd2θ¯E σ particular, it can be coupled to R-invariant -models, similarly to the new minimal N ¼ 1 supergravity in four 1 ¯ G G K ϕI GrI ϕ¯ J GrJ − 4 S dimensions (see, e.g., [65] for more details). Here we × ln þ 4 ð = ; = Þ G briefly discuss such theories. Z I We consider a system of covariantly chiral scalars ϕ of þ d3xd2θEWðϕIÞþc:c: : (6.62) super-Weyl weights rI,

085028-20 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) This action is super-Weyl invariant if the superpotential motion determine the dynamics of the supergravity-matter WðϕIÞ obeys the homogeneity equation system. X ϕIW 2W (6.63) rI I ¼ : A. Properties of the supercurrent I The fundamental properties of the super Cotton tensor The action is invariant under the local Uð1Þ transforma- are (i) its super-Weyl transformation law (2.13); and (ii) the ¯ R tions if the Kähler potential KðϕI; ϕ¯ JÞ obeys the equation transversality condition [57] X X ¯ β ¯ β ϕIK ϕ¯ IK (6.64) D Wαβ ¼ D Wαβ ¼ 0: (7.4) rI I ¼ rI I¯: I I¯ The matter supercurrent must have analogous properties. In a flat superspace limit, the theory (6.62) reduces to a Specifically, it is characterized by the super-Weyl trans- general R-invariant nonlinear σ-model. formation law The action (6.62) may be reduced to components using 0 2σ the formalism developed above. In general, however, the J αβ ¼ e J αβ (7.5) Weyl and S-supersymmetry gauge conditions (6.14) have to be replaced with matter-dependent ones [similar to the and obeys the conservation equation gauge conditions (5.19) in type I supergravity] if we want β ¯ β the gravitational action to be given in Einstein frame. We D J αβ ¼ D J αβ ¼ 0: (7.6) will not give such an analysis here. These must hold when the matter fields are subject to their VII. TOPOLOGICALLY MASSIVE equations of motion. Of course, the relations (7.5) and (7.6) SUPERGRAVITY may be viewed as the consistency conditions for the N 2 equation of motion (7.2). However, there is an independent Consider ¼ conformal supergravity (CSG) coupled way to justify (7.5) and (7.6) that follows from the to matter supermultiplets. The supergravity-matter action is definition of J αβ as the covariantized variational derivative αβ 1 with respect to H . Here we only sketch the proof. For a (7.1) S ¼ SCSG þ Smatter; more complete derivation, it is necessary to develop a g background-quantum formalism for 3D N ¼ 2 supergravity similar to that given by Grisaru and Siegel for N ¼ 1 where SCSG denotes the conformal supergravity action [4,66] and S the matter action [10,11]. Both terms supergravity in four dimensions [67,68] (see [3] for a matter pedagogical review). in (7.1) must be super-Weyl invariant. As regards SCSG, the formulation given in [4] is purely component, and the As demonstrated in [15], in complete analogy with the 4D case [69], the gravitational superfield originates via concept of super-Weyl transformations is not defined −2 within this approach. However, the super-Weyl invariance expð iHÞ, where S of CSG is manifest in the superspace formulation given ¯ m∂ μ ¯ ¯ μ (7.7) recently in [66]; see Appendix D for a review. Requiring the H ¼ H ¼ H m þ H Dμ þ HμD super-Weyl invariance of Smatter is equivalent to the fact that ¯ μ and Dμ and D are the spinor covariant derivatives of this action will describe an N ¼ 2 superconformal field Minkowski superspace. By construction, the superfields theory in a flat superspace limit. M m μ ¯ H H ;H ; Hμ The equation of motion for conformal supergravity is ¼ð Þ are super-Weyl invariant. The supergravity gauge group can be used to gauge away Hμ and its 4 conjugate, leaving us with the only unconstrained prepo- − Wαβ þ J αβ ¼ 0; (7.2) Hm g tential . This prepotential possesses a highly nonlinear gauge transformation where Wαβ is the N ¼ 2 super Cotton tensor, Eq. (2.12), δ ¯ − ¯ (7.8) and J αβ is the matter supercurrent. This equation is LHαβ ¼ DðαLβÞ DðαLβÞ þ OðHÞ; obtained by varying S with respect to the real vector prepotential Hαβ ¼ Hβα of conformal supergravity [15], where the gauge parameter Lα is an unconstrained complex spinor. Due to the nonlinear nature of this transformation, δ δ the gravitational superfield is not a tensor object, and Wαβ ∝ S ; J αβ ∝ S ; (7.3) δHαβ CSG δHαβ matter special care is required in order to represent the variation of the action induced by a variation Hm → Hm þ δHm in a with δ=δHαβ a covariantized variational derivative with covariant way. This is what the background-quantum respect to Hαβ. Equation (7.2) and the matter equations of splitting in supergravity [67,68] is about.

085028-21 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) δ 0 It turns out that giving the gravitational superfield a finite Smatter ¼ if the matter equations of motion hold. Since displacement is equivalent to a deformation of the covariant Lα is completely arbitrary, this is possible if and only if the derivatives that can be represented, in a chiral representa- conservation equation (7.6) holds. tion, as follows: B. Topologically massive minimal supergravity: Type I D¯ α → F¯ D¯ α þ; (7.9a) Let us choose Smatter to be the superconformal sigma model (B2). The corresponding supercurrent proves to be −2iH β 2iH β Dα → e ðN α FDβ þÞe ; detðN α Þ¼1; (7.9b) 1 i ¯ ¯ j¯ ¯ J αβ N ¯D αϕ Dβ ϕ − D α; Dβ N − CαβN: (7.16) where ¼ ij ð Þ 4 ½ ð Þ 1 H − HαβD − i D Hαβ D¯ − i D¯ Hαβ D The matter equations of motion are ¼ 2 αβ 6 ð β Þ α 6 ð β Þ α þ 1 (7.10) − D¯ 2 − 4 ¯ 0 (7.17) 4 ð RÞNi þ Pi ¼ : The ellipses in these expressions denote all terms with 1 Lorentz and Uð ÞR generators. The deformed covariant The relative coefficients in (7.16) are uniquely fixed if one derivatives must obey the same constraints as the original demands the transversality condition (7.6) to hold on the mass ones DA. This can be shown to imply that the complex scalar shell, Eq. (7.17). Alternatively, it may be shown that the F and the unimodular 2 × 2 matrix N are determined in relative coefficients in (7.16) are uniquely fixed if one requires terms of Ha. The vector superfield Ha describes the finite thesuper-Weyltransformationlaw (7.5). In the flatsuperspace deformation of thegravitational superfield. A crucial property limit, the supercurrent (7.16) reduces to the one given in [11]. of the first-order operator H is that it is super-Weyl invariant We now turn to considering topologically massive type I ¯ when acting on any super-Weyl inert real scalar U ¼ U, supergravity. It is described by the action

δσH · U ¼ 0; (7.11) 1 S ¼ S − S ; (7.18) TMSG g CSG SG provided Hαβ transforms as where S is the action for type I supergravity with a δ H −σH (7.12) SG σ αβ ¼ αβ: cosmological term, Eq. (5.2). In topologically massive gravity [70] and its supersymmetric extensions [32,33], The superfield Hαβ proves to be defined modulo gauge the Einstein term appears with the “wrong" sign. In the transformations of the form context of the σ-model action (B2), the matter sector in ¯ 4 ¯ ¯ (7.18) corresponds to the choice N ¼ 4ΦΦ and P ¼ −μΦ . δ Hαβ ¼ D αLβ − D αLβ þ OðHÞ; (7.13) L ð Þ ð Þ The equation of motion for Φ is which are compatible with the super-Weyl transformation 1 D¯ 2 − 4 Φ¯ μΦ3 0 (7.19) (7.12) provided the gauge parameter is endowed with the 4 ð RÞ þ ¼ : properties 3 The equation of motion for the gravitational superfield (7.2) J δ − σ (7.14) Lα ¼ Lα; σLα ¼ 2 Lα: becomes 4 Giving the gravitational superfield an infinitesimal i γ ¯ ¯ H δH − ½D ; DγCαβ − ½D α; Dβ S − 4SCαβ displacement, a ¼ a, the matter action changes as g 2 ð Þ Z ¯ ¯ ¯ ¯ ¯ 4D αΦDβ Φ − D α; Dβ ΦΦ − 4CαβΦΦ 0: δ 3 2θ 2θ¯ δHaJ þ ð Þ ½ ð Þð Þ ¼ Smatter ¼ d xd d E a Z (7.20) δ ≡ d3xd2θd2θ¯EδHa S : (7.15) δHa matter As shown in [10], the freedom to perform the super-Weyl 1 and local Uð ÞR transformations can be used to impose the This functional must be super-Weyl invariant. Due to gauge15 Eqs. (3.3) and (7.12), and since the matter equations of motion hold, we conclude that the super-Weyl transforma- 15Upon gauge-fixing Φ to become constant, there still remain tion of the supercurrent is given by Eq. (7.5). Since Smatter is 1 rigid scale and Uð ÞR transformations that allow us to make f in invariant under the supergravity gauge transformations, (7.21) have any givenvalue. The choice f ¼ 1 leads to a canonically δH D¯ − D ¯ choosing αβ ¼ ðαLβÞ ðαLβÞ in (7.15) should give normalized Einstein-Hilbert term at the component level.

085028-22 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) pffiffiffi Φ ¼ f ¼ const; (7.21) i Dγ D¯ C C 0 (7.24) 2 ½ ; γ αβ þ gf αβ ¼ : which implies the conditions (4.35). Then, the matter equation of motion (7.19) turns into C 0 R ¼ μ ¼ const: (7.22) Equations (7.23) and (7.24) have a solution a ¼ , which corresponds to (i) a flat superspace for μ ¼ 0,or β 1 ¯ ¯ (ii) (1,1) anti-de Sitter superspace if μ ≠ 0. In the case Using the identity D Cαβ ¼ − 2 DαR − 2iDαS,which μ ¼ 0, we can linearize Eq. (7.24) around Minkowski follows from (2.8c),wealsoobtain 2 superspace. Its obvious implication is ð□ − m ÞCa ¼ 0, β ¯ β 1 D Cαβ ¼ D Cαβ ¼ 0: (7.23) where m ¼ 2 fg. Combining the Lagrangians (5.5) and (D12), we obtain Now, the conformal supergravity equation (7.20) drastically the component Lagrangian for topologically massive type I simplifies supergravity

1 2 1 i L ¼ εabc½R ω fg þ ω gω hω f − 4F b þ iψ¯ γ γ~ εdefψ − Rðe; ψÞ − εabcðψ¯ ψ þ ψ¯ ψ Þ TMSG 4g bcfg a 3 af bg ch ab c bc d a ef 2 4 ab c a bc 1 1 1 ¯ − a μ¯ ¯ − εabcψ γ ψ μ εabcψ¯ γ ψ¯ (7.25) þ 4 MM b ba þ M 2 a b c þ M þ 2 a b c :

1 The Lagrangian is computed in the Weyl, local Uð ÞR and Instead of (7.20), now the equation of motion for the S-supersymmetry gauge (5.3). However, it is possible to gravitational superfield is avoid the use of (5.3). To achieve this the component form 4 i γ ¯ ¯ of SSG has to be computed using the results of Appendix B. − ½D ; DγCαβ − ½D α; Dβ S − 4SCαβ g 2 ð Þ 4 C. Topologically massive minimal supergravity: Type II D GD¯ G − D D¯ G − 4C G 0 (7.29) þ G ðα βÞ ½ ðα; βÞ αβ ¼ : Topologically massive type II supergravity is described by the action As shown in [10], the freedom to perform the super-Weyl transformations can be used to impose the gauge 1 S ¼ S − S ; (7.26) TMSG g CSG AdS G ¼ f ¼ const; (7.30) where SAdS is the action for (2,0) AdS supergravity, which implies the constraint (4.39). Then the equation of Eq. (6.29). We can think of the theory with action (6.29) motion (7.27) tells us that as a model for the vector multiplet coupled to background ξ supergravity. Then, the equation of motion for G is S − (7.31) ¼ 2 ¼ const: α ¯ iD Dα ln G − 4S − 2ξG ¼ 0: (7.27) These properties lead to the constraint (7.23). As a result, the conformal supergravity equation (7.29) turns into The supercurrent corresponding to the action Smatter ¼ − SAdS is i Dγ D¯ C 2ξ C 0 (7.32) 4 2 ½ ; γ αβ þðgf þ Þ αβ ¼ : J D GD¯ G − D D¯ G − 4C G (7.28) αβ ¼ G ðα βÞ ½ ðα; βÞ αβ : Equations (7.23) and (7.32) have a solution Ca ¼ 0, which It is an instructive exercise to show that J αβ possesses the corresponds either to a flat superspace for ξ ¼ 0 or (2,0) super-Weyl transformation law (7.5) and obeys the con- anti-de Sitter superspace if ξ ≠ 0. servation equation (7.6) provided (7.27) holds. In the flat Combining the Lagrangians (6.30) and (D12), we obtain superspace limit, the supercurrent (7.28) reduces to the one the component Lagrangian for topologically massive type given in [11]. II supergravity

085028-23 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 2 L ¼ εabc R ω fg þ ω gω hω f − 4F b þ iψ¯ γ γ~ εdefψ TMSG 4g bcfg a 3 af bg ch ab c bc d a ef 1 i 1 1 1 i − R ψ − εabc ψ¯ ψ ψ¯ ψ − F a H~ H~ a 2 − ξ Ha − εabcψ¯ γ ψ (7.33) 2 ðe; Þ 4 ð ab c þ a bcÞ aa þ 4 a þ 4 Z Z þ 4 aa 2 a b c :

The Lagrangian is computed in the Weyl and local where Φ is a chiral scalar of super-Weyl weight 1/2 under S-supersymmetry gauge 6.14. However, one can avoid the equation of motion (7.19). the use of 6.14. To achieve this the component form of SAdS has to be computed using the results of Appendix C. VIII. SYMMETRIES OF CURVED SUPERSPACE In this section we derive the conditions for a curved D. Topologically massive nonminimal supergravity superspace to possess (conformal) isometries. After that Topologically massive nonminimal supergravity is we concentrate on a discussion of curved backgrounds described by the action admitting conformal and rigid supersymmetries.

1 A. Conformal isometries S ¼ S − S ; (7.34) 3 4 TMSG g CSG AdS Consider some background superspace M j such that its geometry is of the type described in Sec. II A. In order to where S denotes the action for nonminimal (1,1) AdS formulate rigid superconformal or rigid supersymmetric AdS 3 4 supergravity [11] field theories on M j , it is necessary to determine all Z (conformal) isometries of this superspace. This can be done similarly to the case of 4D N ¼ 1 supergravity described in −2 3 2θ 2θ¯ ΓΓ¯ −1=2 (7.35) SAdS ¼ d xd d Eð Þ : detail in [3] and reviewed in [51]. In this subsection we study the infinitesimal conformal isometries of M3j4. ξ ξA M3j4 Γ Let ¼ EA be a real supervector field on , The dynamical variable is a deformed complex linear A a α ¯ Γ ξ ≡ ðξ ; ξ ; ξαÞ. It is called conformal Killing if one scalar obeying the constraint (1.2). If we think of (7.35) ξ 16 as the action describing the dynamics of matter superfields can associate with a supergravity gauge transformation Γ and Γ¯ in a background curved superspace, then this (2.4) and an infinitesimal super-Weyl transformation (2.11) theory is dual to the type I minimal model (5.2); see [11] for such that their combined action does not change the covariant derivatives, more details. As a result, topologically massive nonminimal supergravity is dual to that constructed in Sec. VII B. ðδK þ δσÞD ¼ 0: (8.1) To relate the two theories, it suffices to note that when Γ A and Γ¯ are subject to their equations of motion, we can Since the vector covariant derivative D is given in terms of represent a an anticommutator of two spinor ones, it suffices to analyze the implications of (8.1) for the case A ¼ α. A short Γ Φ−3Φ¯ (7.36) ¼ ; calculation gives

1 1 1 δ δ D σ 2 τ ε D ξ ξ γC − ξ S − Dβ − D ξ¯ ξ ¯ D¯ β D ξ − 2 ε ξ¯ Dβγ ð K þ σÞ α ¼ 2 ð þ i Þ αβ þ α β þ i ðα βÞγ αβ 2 Kαβ f α β þ i αβRg þ 2 α βγ i αðβ γÞ 1 − ε D σ D − 4ε ξ ¯ − 4 ε ξ¯ S − 2ξ¯ C ξ δC αðβð γÞ Þþ2 αKβγ αðβ γÞR i αðβ γÞ α βγ þ α βγδ 2 1 − ε ξ 2DδS D¯ δ ¯ 2D S D¯ ¯ ξ Mβγ 3 αðβ γÞδð þ i RÞþ6 ð α þ i αRÞ βγ 1 1 − D σ τ − 2ξ¯βC − 4 ξ¯ S ξβγC − ξ 8DγS D¯ γ ¯ J (8.2) αð þ i Þ αβ i α þ 2 αβγ 6 αγð þ i RÞ :

16Strictly speaking, the parameters of gauge transformations are usually restricted to have compact support in spacetime; see e.g. [71]. The (conformal) Killing vector and spinor fields do not have this property.

085028-24 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) The right-hand side of (8.2) is a linear combination of the i i Dβ D¯ β Dβγ Mβγ ξ ξA ξA ≡ ξa ξα ξ¯ ξa − D¯ ξβα − Dβξ five linearly independent operators , , , and ¼ EA; ð ; ; αÞ¼ ; 6 β ; 6 βα J . Therefore, demanding ðδK þ δσÞDα ¼ 0 gives five different equations. Let us first consider the equations (8.8) associated with the operators Dβ and Dβγ in the right-hand side of (8.2), which obeys the master equation (8.5). If ξ1 and ξ2 are two conformal Killing supervector fields, 1 1 ξ ξ D ξ − ε σ 2 τ − ξ γC ξ S (8.3a) their Lie bracket ½ 1; 2 is a conformal Killing supervector α β ¼ αβð þ i Þ i ðα βÞγ þ αβ þ Kαβ; 2 2 field. It is obvious that, for any real c numbers r1 and r2, the linear combination r1ξ1 þ r2ξ2 is a conformal Killing D ξ 4 ε ξ¯ (8.3b) α βγ ¼ i αðβ γÞ; supervector field. Thus the set of all conformal Killing supervector fields is a super Lie algebra. The conformal as well as their complex conjugate equations. These Killing supervector fields generate the symmetries of a relations imply, in particular, that the parameters M3j4 α ¯ superconformal field theory on . ξ ; ξα;Kαβ; σ and τ are uniquely expressed in terms of Making use of (8.2), the condition δK δσ Dα 0 ξa ð þ Þ ¼ and its covariant derivatives as follows: leads to several additional relations which can be represented in the form α i ¯ βα ¯ i β ξ ¼ − Dβξ ; ξα ¼ − D ξβα; (8.4a) 6 6 ¯ ¯ Dαξβ ¼ −iξαβR; (8.9a) 1 α ¯ α ¯ σ ¼ ðDαξ þ D ξαÞ; (8.4b) 1 2 D 4C ξ¯ − 2C ξ δ − D¯ ¯ 2D S ξ αKβγ ¼ ðαβ γÞ δðαβ γÞ 3 ði ðαR þ ðα Þ βγÞ i α ¯ α ¯ 8 τ ¼ − ðDαξ − D ξαÞ; (8.4c) ε −2D σ 8 ¯ ξ 8 Sξ¯ C ξ¯δ 4 þ αðβ γÞ þ R γÞ þ i γÞ þ 3 γÞδ D ξ − D¯ ξ¯ − 2ξ S (8.4d) 4 10 Kαβ ¼ ðα βÞ ðα βÞ αβ : − C ξδρ ξ D¯ δ ¯ 2DδS (8.9b) 3 γÞδρ þ 9 γÞδði R þ Þ ; This is why we may also use the notation K ¼ K½ξ and σ ¼ σ½ξ. In accordance with (8.3b), the remaining vector ¯ ¯δ i δρ Dατ ¼ iDασ þ 4Sξα − 2iCαδξ þ Cαδρξ parameter ξa satisfies the equation17 2 1 ¯ β ¯ β D ξ 0 (8.5) þ ðD R − 8iD SÞξαβ: (8.9c) ðα βγÞ ¼ 6 and its complex conjugate. Immediate corollaries of (8.5) Actually these relations have nontrivial implications. are Equations (8.3) and (8.9) tell us that the spinor covariant derivatives of the parameters Υ ≔ ðξB;Kβγ; τÞ can be D2 4R¯ ξ D¯ 2 4R ξ 0; (8.6a) represented as linear combinations of Υ, σ, Dασ and ð þ Þ a ¼ð þ Þ a ¼ ¯ Dασ. It turns out that the vector covariant derivative of c Υ can be represented as a linear combination of Υ, σ and Daξb ¼ ηabσ − εabcK : (8.6b) DAσ. In order to prove this assertion, the key observation is C The latter relation implies the conformal Killing equation that, because of (8.1), the torsion tensor TAB , the Lorentz 1 cd and Uð ÞR curvature tensors RAB and RAB, all defined by 2 Eq. (2.6), as well as their covariant derivatives are invariant D ξ þ D ξ ¼ η Dcξ : (8.7) a b b a 3 ab c under the transformation δ ¼ δK þ δσ generated by the conformal Killing supervector field. In particular, the – If Eq. (8.5) holds and the conditions (8.4a) (8.4d) are dimension-1 torsion tensors S, R and Ca are invariant, adopted, it can be shown that the conditions (8.1) are and therefore satisfied identically. Therefore, (8.5) is the fundamental equation containing all the information about the conformal i − DβD¯ σ ξBD σ S (8.10a) Killing supervector fields. As a consequence, we can 4 β ¼ð B þ Þ ; give an alternative definition of the conformal Killing 1 supervector field. It is a real supervector field − D¯ 2σ ξBD σ − 2 τ (8.10b) 4 ¼ð B þ ÞR i R; 17 Equation (8.5) is analogous to the conformal Killing 1 D 0 βγ ¯ B b equation, ðαβVγδÞ ¼ , on a (pseudo)Riemannian three- − ðγ Þ ½Dβ; Dγσ ¼ðξ D þ σÞC þ K C : (8.10c) dimensional manifold. 8 a B a a b

085028-25 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) To complete the proof, it only remains to make use supervector fields. Given such a supervector field ξ, it can of Eq. (2.7b). be represented in two different forms It is an instructive exercise to derive the following identity A Â ˆ ξ ¼ ξ EA ¼ ξ EA; (8.14) 2 D D σ ε 2 C Dδσ 4SD σ 3 ¯ D¯ σ ˆ α βγ ¼ αðβ i γÞδ þ γÞ þ iR γÞ where EA is the inverse vielbein associated with the 3 ˆ A Â covariant derivatives DA. The parameters ξ and ξ are − i D¯ D2σ − i D DδD¯ σ related to each others as follows: γÞð Þ γÞð δ Þ 4 2 i â −ω a ˆα −1ω α i βα ¯ − D D D¯ σ (8.11) ξ ¼ e ξ ; ξ ¼ e 2 ξ þ ξ Dβω : (8.15) 2 ðαð½ β; γÞ Þ 2 and its complex conjugate. In conjunction with Eqs. (8.10), One may prove that the following identities hold: they tell us that DADBσ can be represented as a linear combination of Υ, σ and D σ. We have already established C σ½ξˆ¼σ½ξ − ξω; (8.16a) that DAΥ is a linear combination of Υ, σ and DCσ. These properties mean that the super Lie algebra of the conformal M3j4 ˆ α ¯ ¯ α Killing vector fields on is finite dimensional. The τ½ξ¼τ½ξ − iξ Dαω þ iξαD ω number of its even and odd generators cannot exceed those 1 1 in the N 2 superconformal algebra osp 2 4 . ξαβ D D¯ ω − ξαβ D ω D¯ ω (8.16b) ¼ ð j Þ þ 8 ½ α; β 4 ð α Þ β ; To study supersymmetry transformations at the component level, it is useful to spell out one of the implications of ξˆ ξ − 2ξ D ω 2ξ¯ D¯ ω (8.1) with A ¼ a. Specifically, we consider the equation Kαβ½ ¼Kαβ½ ðα βÞ þ ðα βÞ δ δ D 0 ð K þ σÞ a ¼ and read off its part proportional to a i D εabc γ ξ D ω ξ Dγω D¯ ω (8.16c) linear combination of the spinor covariant derivatives β. þ ð cÞαβ a b þ 2 αβð Þ γ : The result is These identities imply the following important relation: i β ¯ b β c 0 ¼ Daξα þ ðγaÞα Dβσ − iεabcðγ Þα C ξβ 2 1 β ¯ K ξˆ ≔ ξÂDˆ cd ξˆ M τ ξˆ J K ξ (8.17) − ðγ Þα ðξβS þ ξβRÞ ½ A þ K ½ cd þ i ½ ¼ ½ : a 2 1 4 2 b c βγ ¯ i ¯ − ε ξ γ iCαβγ − εα βDγ S − εα βDγ R : 2 abc ð Þ 3 ð Þ 3 ð Þ C. Isometries (8.12) In order to describe N ¼ 2 Poincaré or anti-de Sitter supergravity theories, the Weyl multiplet has to be coupled B. Conformally related superspaces to a certain conformal compensator Ξ and its conjugate. In 3 4 Consider a curved superspace Mˆ j that is conformally general, the latter is a scalar superfield of super-Weyl M3j4 ≠ 0 1 related to . This means that the covariant derivatives weight w and Uð ÞR charge q, ˆ 3j4 ˆ 3j4 DA and DA, which correspond to M and M respectively, are related to each other in accordance with (2.11), δσΞ ¼ wσΞ; J Ξ ¼ qΞ; (8.18)

1ω γ Dˆ 2 D D ω M − D ω J (8.13a) α ¼ e ð α þð Þ γα ð α Þ Þ; chosen to be nowhere vanishing, Ξ ≠ 0. It is assumed that q ¼ 0 if and only if Ξ is real, which is the case for type II ˆ ω i γδ ¯ i γδ ¯ supergravity. Different off-shell supergravity theories cor- D ¼ e D − ðγ Þ ðD γωÞDδ − ðγ Þ ðD γωÞDδ a a 2 a ð Þ 2 a ð Þ respond to different superfield types of Ξ. i Once Ξ and its conjugate have been fixed, the off-shell D ω D¯ γω M ε Dbω Mc þ 2 ð γ Þð Þ a þ abcð Þ supergravity multiplet is completely described in terms of 3 the following data: (i) the U(1) superspace geometry − i γ γδ D D¯ ω J − i γ γδ D ω D¯ ω J 8 ð aÞ ð½ γ; δ Þ 4 ð aÞ ð γ Þð δ Þ ; described earlier; (ii) the conformal compensator and its conjugate. Given a supergravity background, its isometries (8.13b) should preserve both of these inputs. This leads us to the concept of Killing supervector fields. A 3j4 for some super-Weyl parameter ω. The two superspaces A conformal Killing supervector field ξ ¼ ξ EA on M 3 4 M3j4 and Mˆ j prove to have the same conformal Killing is said to be Killing if the following conditions hold:

085028-26 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) 1 defined by D → ∇ ≔ D − iΦ J.18 The original U 1 ξBD bc ξ M τ ξ J D δ D 0 A A A A ð ÞR B þ K ½ bc þ i ½ ; A þ σ½ξ A ¼ ; 2 connection ΦA turns into a tensor superfield. (8.19a) D. Charged conformal Killing spinors We wish to look for those curved superspace back- ðξBD þ iqτ½ξþwσ½ξÞΞ ¼ 0; (8.19b) B grounds which admit at least one conformal supersym- bc ξ τ ξ σ ξ metry. By definition, such a superspace possesses a with the parameters K ½ , ½ and ½ defined as in (8.4). conformal Killing supervector field ξA with the property The set of all Killing supervector fields on M3j4 is a super Lie algebra. The Killing supervector fields generate the ξaj¼0; ϵα ≔ ξαj ≠ 0: (8.25) symmetries of rigid supersymmetric field theories on M3j4. The Killing equation (8.19) are super-Weyl invariant. All other bosonic parameters will also be assumed to Specifically, if ðD ; ΞÞ and ðDˆ ; Ξˆ Þ are conformally related A A vanish, σj¼τj¼Kαβj¼0. Our analysis will be restricted supergravity backgrounds, to U(1) superspace backgrounds without covariant fermionic fields, that is ˆ 1ω γ ˆ wσ Dα ¼ e2 ðDα þðD ωÞMγα − ðDαωÞJ Þ; Ξ ¼ e Ξ; (8.20) DαSj¼0; DαRj¼0; DαCβγj¼0: (8.26)

These conditions mean that the gravitini can completely be then Eq. (8.19) imply that ξ ¼ ξBE ¼ ξˆBEˆ is also a B Bˆ ˆ gauged away such that the projection (4.3) becomes Killing supervector field with respect to ðDA; ΞÞ.In particular, it holds that α Daj¼Da ⟺ ψ m ¼ 0: (8.27) ξˆBDˆ τ ξˆ σ ξˆ Ξˆ 0 (8.21) ð B þ iq ½ þw ½ Þ ¼ : In what follows, we always assume that the gravitini have been gauged away. 1 The super-Weyl and local Uð ÞR symmetries allow us to The above definitions provide a superspace realization choose a useful gauge for what is usually called a “supersymmetric spacetime.” For instance, according to [14], it is a supergravity back- Ξ ¼ 1 (8.22) ground “for which all fermions and their supersymmetry variations vanish for some nonzero supersymmetry which characterizes the off-shell supergravity formulation parameter.” chosen. If q ≠ 0, there remains no residual super-Weyl and We introduce scalar and vector fields associated with the 1 local Uð ÞR freedom in this gauge. Otherwise, the local superfield torsion: 1 Uð ÞR symmetry remains unbroken while the super-Weyl freedom is completely fixed. s ≔ Sj;r≔ Rj;ca ≔ Caj: (8.28) In the gauge (8.22), the Killing equation (8.19b) becomes We also recall that the S-supersymmetry parameter is ηα ≔ Dασj. Bar projecting Eq. (8.12) gives B iqðξ ΦB þ τ½ξÞ þ wσ½ξ¼0: (8.23) α i α b c α α α 0 ¼ Daϵ þ ðγ~aη¯Þ þ iεabcc ðγ~ ϵÞ − sðγ~aϵÞ − irðγ~aϵ¯Þ : Hence, the isometry transformations are generated by those 2 conformal Killing supervector fields which respect the (8.29) conditions This is equivalent to the following two equations: σ½ξ¼0; (8.24a) 0 D − ϵ (8.30a) ¼ð ðαβ icðαβÞ γÞ; τ ξ −ξBΦ ≠ 0 ½ ¼ B;q: (8.24b) 2i η¯ − γaD ϵ 2 γaϵ 3 ϵ 3 ϵ¯ α ¼ 3 ðð a Þα þ ið Þαca þ s α þ ir αÞ: (8.30b) These properties provide the main rationale for choosing the gauge condition (8.22) which is for any off-shell Equation (8.30a) tells us that the supersymmetry parameter supergravity formulation; the isometry transformations is a charged conformal Killing spinor, since (8.30a) can be are characterized by the condition σ½ξ¼0. rewritten in the form ≠ 0 1 Since for q the local Uð ÞR symmetry is completely fixed in the gauge (8.22), it is reasonable to switch to new 18This is similar to the 4D procedure of degauging introduced 1 covariant derivatives without Uð ÞR connection which are by Howe [18] and reviewed in [2].

085028-27 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) D~ ϵ 0 D~ ϵ ≔ D ϵ − ϵ (8.31) D ϵ ϵ − 2 ε ρ ðαβ γÞ ¼ ; αβ γ αβ γ iðbαβ þ cαβÞ γ: αβ γ ¼ icðαβ γÞ i γðα βÞ; 2 a ρα ≔ c ðγ ϵÞα − isϵα þ rϵ¯α. (8.36) Let us choose ϵα to be a bosonic (commuting) spinor. Then 3 a it follows from (8.31) that the real vector field Va ≔ αβ ðγaÞ ϵ¯αϵβ has the following properties: (i) Va is a This relation shows that, in a neighborhood of any given D 0 ϵ conformal Killing vector field, ðαβVγδÞ ¼ ; and (ii) Va point x0, the supersymmetry parameter γðxÞ is determined a α 2 is null or timelike, since V Va ¼ðϵ¯ ϵαÞ ≤ 0. This vector by its value at x0. As a result, any nonzero solution of field is null if and only if ϵ¯α ∝ ϵα. As a result, we have Eq. (8.35) or, equivalently, (8.36), is nowhere vanishing if 3 19 reproduced two of the main results of [35]. the spacetime M is a connected manifold. By construction, the conditions (8.26) are supersymmetric, that is F. Supersymmetric backgrounds with four supercharges ðδK þ δσÞDαS ¼ 0; ðδK þ δσÞDαR ¼ 0; The existence of rigid supersymmetries imposes nontrivial restrictions on the background fields. For simplicity, ðδK þ δσÞDαCβγ ¼ 0: (8.32) here we work out these restrictions in the case of four supercharges. Since σ ¼ 0, one may deduce from (8.33) the Evaluating the bar projection of these variations gives, following conditions: respectively, 2 γ ¯ ¯ ¯ ¯ D Rj¼D DγSj¼½D α; Dβ Sj¼ðD αCβγδ þ D αCβγδ Þj D2D¯ σ ϵ¯β 8D − 4 D D¯ S − 4 ε DγD¯ S ð Þ ð Þ ð Þ α j¼ ð αβs i½ ðα; βÞ j i αβ γ jÞ ¼ 0: (8.37) þ 16iϵαrs¯ − 8iηαs þ 6η¯αr;¯ (8.33a) It is an instructive exercise to demonstrate that these 2 ¯ β D Dασj¼ϵ ð8iðDαβ þ 2ibαβ − 2icαβÞr¯ − 32iεαβsr¯Þ conditions constrain the background fields s, r and ca as follows: ¯ 2 ¯ β þ 2ϵ¯αD Rj − 4iDαβη þ 4iηαs − 6η¯αr;¯ (8.33b)

Das ¼ 0; (8.38a) 0 ¼ ϵ αðDβγ þ 2ibβγ þ 4icβγ Þr¯ ð Þ Þ Þ D 2 (8.38b) 1 ar ¼ ibar; ϵ¯δ D − D C¯ D¯ C þ ðαβcγδÞ ð ðα βγδÞ þ ðα βγδÞÞj 2 D 2ε c (8.38c) acb ¼ abcc s; 3 ¯ cab εδ α Dβ; Dγ S − iDβγ s ε γ D c þ ð ½ Þ j Þ þ 2 ð cÞβγÞ a b rs ¼ 0; (8.38d) 1 3i 6 − D η − η (8.33c) rc 0: (8.38e) þ cβγÞs 2 ðαβ γÞ 2 cðαβ γÞ: a ¼

It follows from (8.38c) that ca is a Killing vector field, E. Supersymmetric backgrounds D c þ D c ¼ 0: (8.39) In order to describe a rigid supersymmetry transfor- a b b a mation, the structure equations given in the previous The Uð1Þ field strength proves to vanish, subsection have to be supplemented by the additional R condition F ab ¼ 0: (8.40)

σ½ξ¼0 ⟹ ηα ¼ 0; (8.34) The Einstein tensor (A12) is uniquely fixed to be

2 in accordance with (8.24a). Here we do not specify any Gab ¼ 4½cacb þ ηabðs þ rr¯ Þ: (8.41) particular compensator. However, we assume that some compensator has been chosen and the gauge condition 19 This can be proved as follows. Let us assume that ϵγðxÞ (8.22) has been imposed. 3 vanishes at some point x0 ∈ M . We can expand ϵγðxÞ in a Because of (8.34), Eq. (8.29) turns into covariant Taylor series centered at x0 (see, e.g., [58]) in an open neighborhood U of x0. Then, due to (8.36), ϵγðxÞ is equal to zero ϵ ¯ D ϵα ¼ −iε cbðγ~cϵÞα þ sðγ~ ϵÞα þ irðγ~ ϵ¯Þα: (8.35) on U. It is also clear that γðxÞ vanishes on the closure U of U. a abc a a Now we can introduce the subset W ∈ M3 consisting of all zeros of ϵγðxÞ. It follows that this subset is open and closed, and In the spinor notation, this equation reads therefore it coincides with M3 since the latter is connected.

085028-28 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) We recall that in three dimensions the Riemann tensor is D c ¼ 0: (8.49c) determined in terms of the Einstein tensor according to a b Eq. (A12). For the Cotton tensor (A14) we obtain The Einstein tensor is 1 W −24s c c − η cdc : (8.42) G 4 η ¯ (8.50) ab ¼ a b 3 ab d ab ¼ ½cacb þ abrr: Such a spacetime is necessarily conformally flat, The spacetime is conformally flat if sca ¼ 0. So far we have not specified any compensator. We now W 0 (8.51) turn to considering the known off-shell supergravity ab ¼ : formulations [11]. The solution with ca ¼ 0 corresponds to the (1,1) AdS G. Type I minimal backgrounds with four supercharges superspace [11]. The Killing spinor equation (8.48) is equivalent to the In type I supergravity, the conformal compensators are a condition that the gravitino variation (5.12b) vanishes, covariantly chiral superfield Φ of super-Weyl weight w ¼ 1 2 Φ¯ = and its complex conjugate . We recall that the 1 i i Φ − δ ψ α −D ϵα ϵα − aε b γ~cϵ α properties of are given by Eq. (5.1). The freedom to 2 ϵ m ¼ m þ 2 bm 2 em abcb ð Þ 1 perform the super-Weyl and local Uð ÞR transformations can be used to impose the gauge − i ¯ γ~ ϵ¯ α 0 (8.52) 4 Mð m Þ ¼ ; Φ ¼ 1: (8.43) provided we replace Such a gauge fixing is accompanied by the consistency 1 1 → → − ¯ (8.53) conditions [10] ca 2 ba;r 4 M:

0 D Φ − i Φ ¼ α ¼ 2 α; H. Type II minimal backgrounds with four ¯ supercharges 0 ¼fDα; DβgΦ ¼ −Φαβ þ Cαβ − 2iεαβS; (8.44) Type II minimal supergravity is obtained by coupling the and therefore Weyl multiplet to a real linear compensator G with the super-Weyl transformation law given by Eq. (6.2). S ¼ 0; Φα ¼ 0; Φαβ ¼ Cαβ: (8.45) The super-Weyl invariance allows us to choose the gauge 1 Since the local Uð ÞR invariance is completely fixed in this G ¼ 1: (8.54) gauge, it is more convenient to make use of covariant 1 1 derivatives without Uð ÞR connection, Because the compensator is real, its Uð ÞR charge (8.18) is equal to zero, and thus the local Uð1Þ group remains ∇ ≔ D − Φ J (8.46) R A A i A ; unbroken in the gauge chosen. The consistency condition for (8.54) is which satisfy the anticommutation relations ¯ 0 (8.55) ¯ ¯ ¯ R ¼ R ¼ : f∇α; ∇βg¼−4RMαβ; f∇α; ∇βg¼4RMαβ; (8.47a) Then, the anticommutators of spinor covariant derivatives ¯ γδ f∇α; ∇βg¼−2i∇αβ − 2εαβC Mγδ: (8.47b) become ¯ ¯ The Killing spinor equation (8.35) becomes fDα; Dβg¼fDα; Dβg¼0; (8.56a)

α α b c α α Daϵ ¼ icaϵ − iεabcc ðγ~ ϵÞ þ irðγ~aϵ¯Þ : (8.48) ¯ fDα; Dβg¼−2iDαβ − 2CαβJ − 4iεαβSJ γδ The supersymmetric backgrounds with four supercharges þ 4iSMαβ − 2εαβC Mγδ: (8.56b) are characterized by the properties The Killing spinor equation for type II minimal super- rca ¼ 0; (8.49a) gravity is

α b c α α Dar ¼ 0; (8.49b) Daϵ ¼ −iεabcc ðγ~ ϵÞ þ sðγ~aϵÞ : (8.57)

085028-29 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) All supersymmetric backgrounds with four supercharges Σ ¼ 1: (8.65) are characterized by the conditions In this gauge, some restrictions on the geometry occur [10]. Das ¼ 0; (8.58a) To describe them, it is useful to split the covariant derivatives as D c ¼ 2ε ccs: (8.58b) a b abc ¯ ¯ ¯ Dα ¼ ∇α þ iTαJ ; Dα ¼ ∇α þ iTαJ ; (8.66) The Einstein tensor is 1 Φ where the original Uð ÞR connection α has been renamed 2 D¯ 2 − 4 Σ 0 Gab ¼ 4½cacb þ ηabs ; (8.59) as Tα. In the gauge (8.65), the constraint ð RÞ ¼ turns into and the Cotton tensor is given by Eq. (8.42). The solution 1 − with c ¼ 0 corresponds to the (2,0) AdS superspace [11]. w ∇¯ ¯ α ¯ ¯ α (8.67) a R ¼ 4 ði αT þ wTαT Þ: In the case ca ≠ 0, the traceless Ricci tensor is ¯ 1 1 We see that R becomes a descendant of Tα. Equation (8.67) R − η R 4 − η 2 (8.60) ab 3 ab ¼ cacb 3 abc : is not the only consistency condition implied by the gauge fixing (8.65). Evaluating explicitly fDα; DβgΣ and ¯ fDα; DβgΣ and then setting Σ ¼ 1 gives From this we conclude (see, e.g., Table 1 in [72]) that spacetime is of type N (for ca null), type Ds (for ca 1 ¯ α α ¯ α ¯ ∇ αTβ 0; S ∇ Tα −∇ Tα 2 T Tα ; (8.68a) spacelike) or Dt (for ca timelike) in the Petrov-Segre ð Þ ¼ ¼ 8ð þ i Þ classification. For Dt and Ds it is shown in [72] that spacetime is necessarily biaxially squashed AdS3. Φ C i ∇ ¯ i ∇¯ ¯ (8.68b) The Killing spinor equation (8.57) is equivalent to αβ ¼ αβ þ ðαTβÞ þ ðαTβÞ þ TðαTβÞ: α 2 2 the condition that, in the gauge ψ m ¼ 0, the gravitino variation (6.36b) vanishes, If we define a new vector covariant derivative ∇a by Da ¼ ∇a þ iΦaJ , then the algebra of the covariant deriv- 1 α α i α i α ¯ − δ ψ −D ϵ − H ϵ aε Hc γ~bϵ atives ∇ ¼ð∇ ; ∇α; ∇αÞ proves to be 2 ϵ m ¼ m 4 m þ 4 em abc ð Þ A a 1 ∇ ∇ −2 ∇ − − 1 ∇γ γ M − γ~ ϵ α 0 (8.61) f α; βg¼ iTðα βÞ iðw Þð Tγ þ iwT TγÞ αβ; 4 Zð m Þ ¼ ; (8.69a) provided we make the replacements ¯ ¯ ¯ γδ f∇α; ∇βg¼−2i∇αβ − iTβ∇α þ iTα∇β − 2εαβC Mγδ 1 1 1 → − H → − H → − (8.62) i ba ba a;ca a;s Z: ∇¯ γ − ∇γ ¯ 2 γ ¯ M (8.69b) 4 2 4 þ 2 ð Tγ Tγ þ iT TγÞ αβ:

I. Nonminimal backgrounds with four supercharges The Killing spinor equation in this case is Nonminimal supergravity in three dimensions was stud- D ϵα Φ ϵα − ε cb γ~cϵ α s γ~ ϵ α r γ~ ϵ¯ α: ied in [10,11]. It is obtained by coupling the Weyl multiplet a ¼ i aj i abc ð Þ þ ð a Þ þ i ð a Þ to a complex linear compensator Σ and its conjugate. Here (8.70) Σ obeys the constraint It should be kept in mind that R, S and Φa are now ¯ 2 ¯ ðD − 4RÞΣ ¼ 0 (8.63) composite superfields constructed in terms of Tα, Tα and their covariant derivatives, in accordance with Eqs. (8.67), and is subject to no reality condition. By definition, the (8.68a) and (8.68b) respectively. Supersymmetric back- compensator Σ is chosen to be nowhere vanishing and grounds with four supercharges are very constrained in the transforms as a primary field of weight w ≠ 0; 1 under the nonminimal case. Indeed, the requirement that Tαj¼0 be 1 Σ super-Weyl group. Then, the Uð ÞR charge of is uniquely invariant under the isometry transformations leads to the determined [10], condition

γ γ ¯ δσΣ ¼ wσΣ ⟹ J Σ ¼ð1 − wÞΣ: (8.64) 0 ¼ ϵ ∇γTαj − ϵ¯ ∇γTαj; (8.71)

1 ∇ ∇¯ 0 – The super-Weyl and local Uð ÞR symmetries can be used which implies αTβ ¼ αTβ ¼ . Due to (8.67) (8.68b), to impose the gauge condition we deduce that

085028-30 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) 0 0 Φ (8.72) ∇ ∇ −2 ∇ − 4¯μM 2 ∇γ − γ M r ¼ ;s¼ ; aj¼ca; f α; βg¼ iTðα βÞ αβ þ ið Tγ iT TγÞ αβ; (8.79a) and then cb is covariantly constant,

D 0 (8.73) ¯ ¯ ¯ γδ acb ¼ : f∇α; ∇βg¼− 2i∇αβ − iTβ∇α þ iTα∇β − 2εαβC Mγδ The Einstein tensor becomes i ∇¯ γ − ∇γ ¯ 2 γ ¯ M (8.79b) þ 2 ð Tγ Tγ þ iT TγÞ αβ: G ¼ 4c c : (8.74) ab a b The Killing spinor equation coincides with (8.70). Unlike the nonminimal formulation studied in the previous sub- Such a spacetime is necessarily conformally flat, Wab ¼ 0. Nonminimal supergravity is the only off-shell super- section, the scalar R is now given by Eq. (8.78). This gravity formulation which does not allow for anti-de Sitter modified expression for R leads to different backgrounds backgrounds. However, there exists an alternative non- with four supercharges. Due to the presence of the μ minimal formulation in the case w ¼ −1 [11], inspired by parameter in (8.78), demanding the existence of four the 4D construction in [24], which admits an anti-de Sitter supersymmetries gives solution. Sj¼0;Rj¼μ; Φaj¼Caj¼ca: (8.80) J. Nonminimal AdS backgrounds with four supercharges Moreover, one also finds the condition CcjRj¼0. Since In the case w ¼ −1, the complex linear constraint (8.63) Rj¼μ ≠ 0, we conclude that admits a nontrivial deformation. We introduce a new Γ conformal compensator that has the transformation ca ¼ 0: (8.81) properties The Einstein tensor is δσΓ ¼ −σΓ; J Γ ¼ 2Γ (8.75)

20 G ¼ 4η μμ¯ : (8.82) and obeys the improved linear constraint ab ab 1 This background corresponds to the (1,1) anti-de Sitter − D¯ 2 − 4 Γ μ (8.76) 4 ð RÞ ¼ ¼ const; superspace [11]. After this work was completed, there appeared a new with the complex parameter μ ≠ 0 inducing a cosmological paper [50] which has some overlap with our results in constant. This constraint is super-Weyl invariant. Secs. VII B and VIII G. 1 The super-Weyl and local Uð ÞR symmetries allow us to impose the gauge condition ACKNOWLEDGMENTS Γ ¼ 1: (8.77) S. M. K. acknowledges the generous hospitality of the Arnold Sommerfeld Center for Theoretical Physics, As in the previous subsection, this gauge condition implies Munich and the INFN, Padua where part of this work some restrictions on the geometry. Indeed, the constraint was done. M. R. and I. S. acknowledge the generous ¯ 2 ðD − 4RÞΓ ¼ μ turns into hospitality and stimulating atmosphere of the physics department at Harvard University, where this work was i μ ∇¯ ¯ α ¯ ¯ α (8.78) begun. U. L., M. R. and I. S. acknowledge the stimulating R ¼ þ 2 ð αT þ iTαT Þ: atmosphere of the “2012 Summer Simons Workshop in ¯ ” We see that R becomes a descendant of Tα. Next, Mathematics and Physics where part of this work was ¯ evaluating the expressions fDα; DβgΓ and fDα; DβgΓ done. The work of S. M. K. and U. L. was supported in part and then setting Γ ¼ 1, we again obtain the relations by the Australian Research Council, Grant (8.68a) and (8.68b). As in the previous subsection, we No. DP1096372. M. R. acknowledges NSF Grant 1 No. phy 0969739. I. S. was supported by the DFG can introduce covariant derivatives without Uð ÞR con- ¯ Transregional Collaborative Research Centre TRR 33, nection, ∇A ¼ð∇a; ∇α; ∇αÞ. Their algebra proves to be the DFG cluster of excellence Origin and Structure of the Universe as well as the DAAD Project 20In the case w ¼ −1, there exists a more general deformation, ðD¯ 2 − 4RÞΓ ¼ −4WðφÞ, where WðφIÞ is a matter superpotential No. 54446342. The work of G. T.-M. was supported by depending on super-Weyl inert chiral superfields φI. This super- the Australian Research Council’s Discovery Early Career Weyl invariant constraint reduces to (8.76) for W ¼ μ. Award (DECRA), Project No. DE120101498.

085028-31 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) α α APPENDIX A: NOTATION, CONVENTIONS AND ψχ ≔ ψ χα; ψχ¯ ≔ ψ χ¯α; SOME TECHNICAL DETAILS α α ψχ¯ ≔ ψ¯ χα; ψ¯ χ¯ ≔ ψ¯ αχ¯ ; (A5a) Our 3D notation and conventions follow those used in γ ψ ≔ γ ψ β ψγ [10]. In particular, the vector indices are denoted by ð a Þα ð aÞαβ ¼ð aÞα; lower case Latin letters from the beginning of the alphabet, α αβ α ðγ~aψÞ ≔ ðγaÞ ψ β ¼ðψγ~aÞ ; (A5b) for instance a; b ¼ 0; 1; 2. The Minkowski metric is α β αβ ηab ¼ diagð−1; 1; 1Þ, and the Levi-Civita tensor εabc is ψγ χ ≔ ψ γ χ ψγ~ χ ≔ ψ γ χ (A5c) 012 a ð aÞαβ ; a αð aÞ β: normalized by ε012 ¼ −1, and hence ε ¼ 1. The spinor indices are denoted by small Greek letters from the In particular, contractions of two spinor covariant deriva- α β 1 2 beginning of the alphabet, for instance ; ¼ ; . tives are defined as To deal with spinors, we introduce a basis of real 2 2 2 α ¯ 2 ¯ ¯ α symmetric × matrices D ≔ D Dα; D ¼ DαD : (A6)

γ ¼ðγ Þαβ ¼ðγ Þβα ¼ð1; σ1; σ3Þ; (A1a) a a a Any three-vector Fa can equivalently be realized as a symmetric spinor Fαβ ¼ Fβα. The relationship between Fa and also define and Fαβ is as follows:

αβ βα αγ βδ γ~ ¼ðγ Þ ¼ðγ Þ ≔ ε ε ðγ Þγδ; (A1b) 1 a a a a ≔ γa − γ αβ (A7) Fαβ ð ÞαβFa ¼ Fβα;Fa ¼ 2 ð aÞ Fαβ: with σ1 and σ3 two of the three Pauli matrices. The spinor indices are raised and lowered using the SLð2; RÞ invariant We can also describe the one-form Fa in terms of its tensors Hodge-dual two-form Fab ¼ −Fba, 0 −1 01 1 αβ ≔−ε c ε bc (A8) εαβ ¼ ; ε ¼ (A2) Fab abcF ;Fa ¼ abcF : 10 −10 2

Then, the symmetric spinor Fαβ Fβα, which is associated as follows: ¼ with Fa, can equivalently be defined in terms of Fab: α αβ β ψ ¼ ε ψ β; ψ α ¼ εαβψ : (A3) 1 ≔ γa γa ε bc (A9) Fαβ ð ÞαβFa ¼ 2 ð Þαβ abcF : The 3D Dirac γ matrices are

These three algebraic objects, F , F and Fαβ, are in one- γˆ γ β ≔ εβγ γ γˆ γˆ η 1 ε γˆc (A4) a ab ¼ð aÞα ð aÞαγ; a b ¼ ab þ abc : to-one correspondence to each other. Their inner products are related as follows: In this representation of the γ matrices, the Majorana spinors are real. 1 1 − a ab αβ (A10) In N ¼ 2 supersymmetry, we usually deal with complex F Ga ¼ 2 F Gab ¼ 2 F Gαβ: spinors. Only in the case of complex spinors, we use throughout this paper the following types of index An equivalent form of the commutation relations (2.7c) contraction: and (2.7d) is

4 D D − ε C Dδ C D − 2ε SD − 2 ε ¯ D¯ 2ε C Mδρ − 2D S D¯ ¯ M ½ αβ; γ¼ i γðα βÞδ þ i γðα βÞ γðα βÞ i γðαR βÞ þ γðα βÞδρ 3 ð ðα þ i ðαRÞ βÞγ 1 1 2D S D¯ ¯ M C ε 8D S D¯ ¯ J (A11a) þ 3 ð γ þ i γRÞ αβ þ αβγ þ 3 γðαð βÞ þ i βÞRÞ ;

4 D D¯ ε C D¯ δ − C D¯ − 2ε SD¯ 2 ε D 2ε C¯ Mδρ − 2D¯ S − D M ½ αβ; γ¼i γðα βÞδ i γðα βÞ γðα βÞ þ i γðαR βÞ þ γðα βÞδρ 3 ð ðα i ðαRÞ βÞγ 1 1 2D¯ S − D M − C¯ ε 8D¯ S − D J (A11b) þ 3 ð γ i γRÞ αβ αβγ þ 3 γðαð βÞ i βÞRÞ :

These relations are very useful for actual calculations.

085028-32 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) X X In three dimensions, the Weyl tensor is identically zero, ¯ ¯ ϕiN ¼ ϕiN¯ ¼ N; (B3a) and the Riemann tensor R is related to the Einstein i i abcd i i¯ tensor by the simple rule X 1 1 ϕiP ¼ 4P: (B3b) εacdεbefR ¼ Gab ≔ Rab − ηabR; i 4 cdef 2 i ef Rabcd ¼ εabeεcdfG : (A12) Equation (B3a) means that the σ-model target space is a Kähler cone [74]. As a consequence, the Riemann tensor is expressed in Before reducing the action to components, we introduce terms of the Ricci tensor R ≔ Rc and the scalar ab acb several standard σ-model definitions. As usual, multiple curvature R ≔ ηabR as follows: ab derivatives of the Kähler potential are denoted as

R ¼ η R − η R þ η R − η R abcd ac bd ad bc bd ac bc ad ∂ðpþqÞ 1 ≔ (B4) Ni1…ip ¯ …¯ ¯ ¯ N: − η η − η η R (A13) j1 jq ∂ϕi1 …∂ϕip ∂ϕ¯ j1 …∂ϕ¯ jq 2 ð ac bd ad bcÞ :

21 The Kähler metric N ¯ ¼ N ¯ is assumed to be non- The Cotton tensor is defined as follows ij ji ¯ ¯ singular, with its inverse being denoted Nij ¼ Nji, 1 W ≔ ε Wcd W ¯j j ¯k ¯ ab acd b ¼ ba; ¯ k δ i ¯ δi (B5) 2 NikN ¼ i ;NNkj ¼ j¯: 1 W ¼ 2D R þ η D R: (A14) abc ½a bc 2 c½a b γk The Christoffel symbols ij are

A spacetime is conformally flat if and only if Wab ¼ 0 [73] ¯ ¯ ¯ γk ≔ N ¯Nlk; γk ≔ N ¯ ¯Nlk; (B6) (see [57] for a modern proof). ij ijl i¯ j¯ li j

APPENDIX B: SUPERCONFORMAL and the Riemann curvature R ¯ ¯ is SIGMA MODEL ikjl

p In this appendix we consider an alternative parametriza- R ¯ ¯ ¼ R ¯ pN ¯ ¼ð∂ ¯γ ÞN ¯: (B7) tion of the supergravity-matter system (5.15) and reduce it ikjl ikj pl k ij pl 1 to components without gauge fixing the Weyl, local Uð ÞR and S-supersymmetry transformations. We define the component fields of ϕi as follows: In the new parametrization, the matter sector of the theory is described in terms of several covariantly chiral i i ρα ≔ Dαϕ j; (B8a) superfields ϕi ¼ðϕ0; ϕIÞ of super-Weyl weight w ¼ 1=2,

1 1 1 ¯ i 2 i i α j k D ϕi 0 J ϕi − ϕi δ ϕi σϕi (B1) F ≔− ½D ϕ þ γ ðD ϕ ÞDαϕ j: (B8b) α ¼ ; ¼ 2 ; σ ¼ 2 : 4 jk

The action is defined to be The physical scalar ϕij will be denoted by the same symbol Z as the chiral superfield ϕi itself. 3 2 2 ¯ To reduce the kinetic term in (B2) to components, we S ¼ d xd θd θ¯ENðϕi; ϕ¯ jÞ associate with it the antichiral Lagrangian Z 3 2 i þ d xd θ E Pðϕ Þþc:c: 1 ¯ 2 ¯ Lc ¼ − ðD − 4RÞN (B9) ≡ (B2) 4 Skinetic þ Spotential and may naturally be interpreted as a locally supersym- and make use of the action principle (3.13). The resulting σ 1 component Lagrangian is metric -model. For the action to be super-Weyl and Uð ÞR invariant, the Kähler potential N and the superpotential P should obey the homogeneity conditions 21We do not assume the Kähler metric to be positive definite.

085028-33 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 i ¯ ¯ i ¯ i ¯ i ¯ − R εabc ψ ψ¯ ψ¯ ψ F iF¯ j − Daϕi D ϕ¯ j − ρ¯jγaD~ ρi − ρiγaD~ ρ¯j ψ ρiDaϕ¯ j Lkinetic ¼ 8 þ2 ð a bc þ a bcÞ N þNij¯ ð Þ a 4 a 4 a þ2 a

i ¯ i ¯ ¯ 1 ¯ 1 ¯ − ψ¯ ρ¯jDaϕi εabc ψ¯ γ ρ¯jD ϕi −ψ γ ρiD ϕ¯ j − ψ aψ¯ ρiρ¯j ψ aγ ψ¯ ρiγbρ¯j 2 a þ2 ð a b c a b c Þ 8 a þ8 b a 1 εabc ψ γ ψ¯ ρiρ¯j¯ ψ ψ¯ ρiγ ρ¯j¯ þ8 ð a b c þ a b c Þ

1 ¯ i ¯ 1 ¯ ¯ εabc ψ¯ γ ρ¯i −ψ γ ρi εabcψ ψ¯ D ϕi − D ϕ¯ i R ρiρjρ¯kρ¯l (B10) þ8 ½ ab c Ni¯ ab c Niþ4 a b½Ni c Ni¯ c þ16 ikj¯ l¯ ;

! where we have introduced the target-space covariant − 1 1 − 1 L 1 J ¯ 1 4 4 K KL 4Φ K derivative Nij ¼ e4K ; (B16a) 1 I¯ 1 IJ¯ 4Φ¯ K ΦΦ¯ K D~ ρi ≔ D ρi γi ρj D ϕk (B11) a α a α þ jk α a : where we have denoted A short calculation of the component Lagrangian corre- I ≔ IJ¯ I¯ ≔ IJ¯ (B16b) sponding to Spotential gives K g KJ¯ ;Kg KJ:

1 i γi L F iP − P − γi P ρjρk ψ¯ γaρjP For the Christoffel symbols kl we read off potential ¼ i 4 ð jk jk iÞ þ 2 a j 1 00 − εabcψ¯ γ ψ¯ P þ c:c: (B12) γ0 2 a b c kl ¼ 0 1 Φ ΓI − − 1 ; 4 KLKI KKL 4 KKKL Both Lagrangians (B10) and (B12) are quite compact. 1 I 0 Φ δ γI L (B17) Now, we relate the theory under consideration to the kl ¼ 1 1 ; 0 δI ΓI − K δI σ-model (5.15). We assume that the chiral scalar ϕ from Φ K KL 2 ðK LÞ the set ϕi ¼ðϕ0; ϕIÞ is nowhere vanishing, ϕ0 ≠ 0, and ΓI therefore it may be chosen to play the role of conformal where KL is the Christoffel symbol for the Kähler metric ∂ γk 0 compensator. We introduce a new parametrization of the gIJ¯ . Since 0¯ ij ¼ , the Riemann tensor is characterized dynamical chiral superfields defined by by the properties

0 R R R R 0 (B18) ϕ ¼ Φ; ϕI ¼ ΦφI: (B13) 0kj¯ l¯ ¼ i0¯jl¯ ¼ ik¯0l¯ ¼ ikj¯ 0¯ ¼ :

Here the chiral scalars φI are neutral under the super-Weyl Thus the only nonzero components of the Riemann tensor 1 Φ are and Uð ÞR transformations. Since is nowhere vanishing, Nðϕ; ϕ¯Þ and PðϕÞ may be represented in the form 1 ¯ −1 R ¯ ¯ ¼ ΦΦe 4K R ¯ ¯ − ðK ¯ K ¯ þ K ¯ K ¯ Þ ; ¯ ¯ −1K φ;φ¯ 4 IKJL IKJL 4 IK JL JK IL Nðϕ; ϕÞ¼−4Φe 4 ð ÞΦ;PðϕÞ¼Φ WðφÞ: (B14) ∂ ΓP (B19) RIKJ¯ L¯ ¼ gPL¯ K¯ IJ: We assume that Kðφ; φ¯Þ is the Kähler potential of a Kähler ≔ manifold with positive definite metric gIJ¯ KIJ¯ . Our next step is to express the auxiliary fields F i, Let us express the geometric objects in terms of the new ~ i Eq. (B8b), and the spinor fields Daρα, Eq. (B11), in terms coordinates introduced. A short calculation gives of the component fields of Φ and φI. We recall that the component fields of φI are defined in (5.22). We do not −4 Φ¯ −1 KJ¯ introduce special names for the component fields of Φ;we N ¯ ¼ e 4K ; (B15a) ij Φ ΦΦ¯ ˆ Φ D Φ D2Φ KI KIJ¯ simply write them as , α and , with the bar projection being always assumed here and in what follows. where we have denoted For the auxiliary fields F i we get ∂ 1 1 1 1 K ˆ 0 2 K ≔ ; K ¯ ≔ g ¯ − K K ¯ : (B15b) F − D Φ − Φ ΓI K − K − K K λKλL; I ∂φI IJ IJ 4 I J ¼ 4 16 KL I KL 4 K L (B20a) It follows from (B15a) that the conditions detðN ¯Þ ≠ 0 and ij 1 1 detðg ¯ Þ ≠ 0 are equivalent. For the inverse metric we I I αI I J IJ F ¼ F − λ DαΦ þ λ λ K : (B20b) obtain 2Φ 8 J

085028-34 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) ~ i For the spinor fields Daρα we derive 1 1 D~ D Φ D~ ρ0 D D Φ Φ ΓI − − λKD L (B21a) a α ¼ a α ¼ a α þ 4 KLKI KKL 4 KKKL α aX ;

2 1 D~ λI D λI ΓI λJD K D Φ D I − λðID JÞ (B21b) a α ¼ a α þ JK α aX þ Φ ð α Þ aX 2 KJ α aX :

Using the above results, for the kinetic term (B10) we obtain 1 i −1 0 0¯ i R εabc ψ ψ¯ ψ¯ ψ Φ¯ 4KΦ − 4 F F¯ − DaΦ D Φ¯ − D¯ Φ¯ γaD~ DΦ DΦ γaD~ D¯ Φ Lkinetic ¼ 2 þ 4 ð a bc þ a bcÞ e ð Þ a 4 ðð Þ a þð Þ a Þ 1 1 1 ψ DΦ DaΦ¯ ψ¯ D¯ Φ¯ DaΦ εabc ψ¯ γ D¯ Φ¯ D Φ − ψ γ DΦ D Φ¯ þ 2 að Þ þ 2 að Þ þ 2 ð a bð Þ c a bð Þ c Þ 1 1 1 −1 εabc ψ γ ψ¯ DΦ D¯ Φ¯ ψ ψ¯ DΦ γ D¯ Φ¯ − ψ aψ¯ DΦ D¯ Φ¯ ψ aγcψ¯ DΦ γ D¯ Φ¯ 4K þ 8 ð a b cð Þ þ a bð Þ c Þ 8 að Þ þ 8 að Þ b e ¯ ¯ i ¯ ¯ 1 ¯ 1 ¯ F 0F¯ I − DaΦ D ¯ I − λ¯IγaD~ DΦ DΦ γaD~ λ¯I ψ DΦ Da ¯ I ψ¯ λ¯IDaΦ þ ð Þ aX 4 ð a þð Þ a Þþ2 að Þ X þ 2 a 1 1 εabc ψ¯ γ λ¯I¯D Φ − ψ γ DΦ D φ¯ I¯ εabc ψ γ ψ¯ λ¯I¯DΦ − ψ ψ¯ λ¯I¯γ DΦ þ 2 ð a b c a bð Þ c Þþ8 ð a b c a b c Þ 1 ¯ 1 ¯ −1 0¯ i − ψ aψ¯ λ¯IDΦ − ψ aγbψ¯ λ¯Iγ DΦ 4KΦ¯ F IF¯ − Da I D Φ¯ − λIγaD~ D¯ Φ¯ D¯ Φ¯ γaD~ λI 8 a 8 a b e KI¯ þ ð X Þ a 4 ð a þð Þ a Þ 1 1 1 ψ λIDaΦ¯ ψ¯ D¯ Φ¯ Da I εabc ψ¯ γ D¯ Φ¯ D I − ψ γ λID Φ¯ þ 2 a þ 2 að Þ X þ 2 ð a bð Þ cX a b c Þ 1 1 1 −1 εabc ψ γ ψ¯ λID¯ Φ¯ ψ ψ¯ λIγ D¯ Φ¯ − ψ aψ¯ λID¯ Φ¯ ψ aγbψ¯ λIγ D¯ Φ¯ 4KΦ þ 8 ð a b c þ a b c Þ 8 a þ 8 a b e KI ¯ ¯ i ¯ ¯ 1 ¯ 1 ¯ F IF¯ J − Da I D ¯ J − λ¯IγaD~ λI λIγaD~ λ¯J ψ λIDa ¯ J ψ¯ λ¯JDa I þ ð X Þ aX 4 ð a þ a Þþ2 a X þ 2 a X 1 1 εabc ψ¯ γ λ¯J¯ D I − ψ γ λID ¯ J¯ εabc ψ γ ψ¯ λIλ¯J¯ ψ ψ¯ λIγ λ¯J¯ þ 2 ð a b cX a b cX Þþ8 ð a b c þ a b c Þ 1 ¯ 1 ¯ −1 − ψ aψ¯ λIλ¯J ψ aγbψ¯ λIγ λ¯J 4KΦΦ¯ ˆ 8 a þ 8 a b e KIJ¯ 1 1 −1 − εabc Φ¯ ψ ψ¯ 4D Φ − Φ D I − ψ γ 4DΦ − Φ λI 4K 4 i a bð c KI cX Þ 2 ab cð KI Þ þ c:c: e 1 1 − λIλJλ¯K¯ λ¯L¯ (B22) þ 16 RIKJ¯ L¯ 2 gIK¯ gJL¯ :

The potential term (B12) becomes 1 Φ4 I − Φ−1 D2Φ − 3Φ−2 DΦ DΦ − 2Φ−1 λIDΦ − λIλJ − ΓK Lpotential ¼ F WI W Wð Þ WI 4 ðWIJ IJWKÞ i 1 ψ¯ γa 4Φ−1 DΦ λI εabcψ¯ γ ψ¯ (B23) þ 2 a ð W þ WI Þþ2 W a b c þ c:c:

The sum of the expressions (B22) and (B23) constitutes the component Lagrangian of the theory (5.15) with no gauge condition on the chiral compensator Φ imposed. Looking at the explicit form of (B22), it is easy to understand why the gauge conditions (5.19) have been chosen. First of all, it is seen from the first line of (B22) the canonically normalized Hilbert-Einstein gravitational Lagrangian corresponds to the Weyl gauge condition (5.19a). Secondly, consider the terms in (B22) which involve the gravitino field strength coupled to the matter fermions. These consist of

085028-35 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) 1 ¯ −1K −1 I ¯ −1K We define the component fields of the vector multiplet as Φψ γ e 4 DαΦ − Φ e 4K λα ¼ ψ γ DαðΦ e 4 ΦÞ ab c 4 I ab c follows: (B24) Y ≔ Gj; (C2) and its complex conjugate. To eliminate these cross terms, we have to impose the S-supersymmetry gauge condition 1 (5.19b). Finally, the Uð ÞR gauge condition (5.19c) elim- υα ≔ DαGj; (C3) inates an overall phase factor in the superpotential (B23).In the gauge (5.19), the only remaining field in Φ occurs at the Z ≔ DαD¯ G (C4) θ2 component. It can be defined in the Kähler invariant i α j; way (5.20). In the gauge (5.19), the following useful relations hold 1 Ba ≔− γa αβ D D¯ G 2 ð Þ ½ ðα; βÞ j 1 1 D Φ 8KλL (B25a) Ha − εabcYψ¯ ψ − εabc ψ γ υ ψ¯ γ υ¯ (C5) α ¼ 4 e α KL; ¼ b c i ð b c þ b c Þ:

a 0 1 −1 1 1 As in Sec. VI, H denotes the Hodge dual of the component F ¼ − e 8KM þ e8KFIK ; (B25b) 4 4 I field strength,

F I ¼ FI: (B25c) 1 Ha εabcH H D − D − T c (C6) ¼ 2 bc; ab ¼ aab baa ab ac: As usual, the bar projection is assumed here. Using these relations one may obtain the component Lagrangians (5.23) We also choose the WZ gauge (6.16) for the vector and (5.28) from (B22) and (B23). multiplet. Then, the other component fields of G are

APPENDIX C: VECTOR MULTIPLET MODEL 1 D D¯ (C7) In this appendix we present the component Lagrangian ½ ðα; βÞGj¼2 aαβ; for the model of an Abelian vector multiplet coupled to conformal supergravity. As in Sec. VI A, we denote by G 2 ¯ D DαGj¼2iυα; (C8) the gauge prepotential of the vector multiplet, and by G the corresponding guage-invariant field strength. The vector ¯ 2 2 b a a multiplet action is D D Gj¼−2iDba − 2ðψ aγ υ þ ψ¯ aγ υ¯Þ Z − 2iYψ¯ ψ a − ψ¯ γbψ aa − 2Z: (C9) −4 3 2θ 2θ¯ G G − 4 S − κ G (C1) a a b SVM ¼ d xd d Eð ln G G Þ; The component Lagrangian corresponding to the with κ a constant parameter. action (C1) is

1 1 1 1 Y−1BaB − F a − Y−1 Dˆ aY Dˆ Y − YR Y−1Z2 − Y−1 υγ¯ aDˆ υ υγaDˆ υ¯ − Y−2B υγ¯ aυ LVM ¼ 4 a aa ð Þ a 2 þ 4 i ð a þ a Þ 2 a i 1 1 i 1 − Y−2Zυυ¯ − Y−3υ2υ¯2 κ −YZ − Bb − υυ¯ εabcY2ψ¯ γ ψ − εabcY ψ¯ ψ 2 2 þ 4 aa i þ 2 a b c 4 ac a b 1 1 i 1 i − εabc 2υγ¯ ψ¯ Yψ ψ¯ − Y−1ψ γaγ~bυ Dˆ Y − B Y−1Zψ γaυ − Y−2ψ γaυυ¯ 2 þ 4 ð a bc þ i a bcÞ 2 a b 2 b þ 4 a 4 a 1 i Y−1εabcψ γ ψ υ2 κ Yψ γaυ − εabc ψ γ υ (C10) þ 4 a b c þ a 4 aa b c þ c:c: :

Here we have introduced new covariant derivatives 1 1 Dˆ Y ≔ D Y − ψ υ − ψ¯ υ¯ (C11) a a 2 a 2 a ; i i i Dˆ υα ≔ D υα − ψ¯ αZ ψ¯ γ~b α Dˆ Y − B (C12) a a 2 a þ 2 ð a Þ b 2 b :

085028-36 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) APPENDIX D: THE ACTION FOR N ¼ 1, and a general method of constructing conformal CONFORMAL SUPERGRAVITY supergravity actions for N > 1 was outlined. However, it N N 2 turns out that SOð Þ superspace [10,14] is not an optimum In the family of ¼ locally supersymmetric theories setting to carry out this program; see [57] for a detailed in three dimensions, conformal supergravity [4] is one discussion. From a technical point of view, the derivation of of the oldest. Originally it was constructed by gauging the N 2 osp 2 4 the conformal supergravity actions greatly simplifies if one 3D ¼ superconformal algebra, ð j Þ, in ordinary makes use of the so-called N -extended conformal superspace spacetime, as a direct generalization of the formulation for of [57], which is a novel formulation for conformal super- N 1 ¼ conformal supergravity [75] (the latter theory being a gravity. The SO N superspace of [10,14] is obtained from the N 1 ð Þ naturalreformulationoftopologicallymassive ¼ super- N -extended conformal superspace by gauge fixing certain gravity [32,33]). The construction in [4] was soon general- local symmetries; see [57] for more details. In conformal N ized to the case of -extended conformal supergravity [76]. superspace, the action for N ¼ 2 conformal supergravity is N In accordance with [76], the action for -extended con- simply the Chern-Simons term associated with ospð2j4Þ [66]. formal supergravity is a locally supersymmetric completion Below we reformulate this action in SO(2) superspace. of the gravitational and SOðN Þ gauge Chern-Simons terms. This action is on shell for N ≥ 3, and therefore its applica- 1. Conformal superspace and SO(2) superspace tions are rather limited.22 As concerns the off-shell N ¼ 1 Conceptually, the N ¼ 2 conformal superspace of [57] and N ¼ 2 component actions [4,75], it appears useful to corresponds to a certain gauging of the superconformal recast them in a superfield form, simply because all N ¼ 1 algebra ospð2j4Þ in superspace [57]. The corresponding and N ¼ 2 locally supersymmetric matter systems are covariant derivatives ∇ include two types of connections: naturally formulated in superspace. A (i) the Lorentz and Uð1Þ connections [as in SO(2) As mentioned in the Introduction, Refs. [10,11] described R superspace]; and (ii) those associated with the dilatation the most general matter couplings to conformal supergravity ¯α (D), special conformal (K ) and S-supersymmetry ðSα; S Þ in the cases 1 ≤ N ≤ 4, including the off-shell formulations a generators of the N ¼ 2 superconformal algebra. To for Poincaré and AdS supergravity theories. But no conformal emphasize this grouping, the covariant derivatives ∇A supergravity action was considered in these publications, due 24 can be written in the form to the fact that an alternative action principle is required in order to describe pure conformal supergravity. Building on ˆ b β ~ ¯β ∇A ¼ DA þ BAD þ FA Kb þ FA Sβ þ FAβS ; (D1a) the earlier incomplete results in [2,12,79], the action for N ¼ 1 conformal supergravity has recently been constructed in where we have denoted terms of the superfield connection as a superspace integral ˆ M ˆ a ˆ [80]. However, such a construction becomes impossible DA ¼ EA ∂M − ΩA Ma þ iΦAJ : (D1b) starting at N ¼ 2.23 This is because (i) the spinor and vector ∇ parts of the superfield connection have positive dimension By construction, the operators A are subject to certain equal to 1=2 and 1 respectively; and (ii) the dimension of the covariant constraints [57] such that the entire algebra of full superspace measure is (N − 3). As a result, it is not covariant derivatives is expressed in terms of a single — W possible to construct contributions to the action that are cubic primary superfield the super Cotton tensor αβ. in the superfield connection for N ≥ 2. As demonstrated in [57], the conformal superspace is Nevertheless, it was argued in [80] that off-shell con- intimately related to the SO(2) superspace via a degauging formal supergravity actions (assuming their existence) may procedure.Thecrucialobservationhereisthatthelocalspecial be realized in terms of the curved superspace geometry conformal and S-supersymmetry gauge freedom can be used 0 given in [10,14] [also known as SOðN Þ superspace] to switch off the dilatation connection, BA ¼ .Inthisgauge, provided one makes use of the superform approach for there remains no residual special conformal and S-supersym- the construction of supersymmetric invariants. Such a metry gauge freedom, but the covariant derivatives (D1a) still realization was explicitly worked out in [80] for the case include the connections associated with the generators Kb, Sβ and S¯β. These connections are tensor superfields with respect 1 to the remaining local Lorentz and Uð ÞR symmetries. From 22Recently, off-shell conformal supergravity actions have been N 3 N 6 the constraints obeyed by the conformal covariant derivatives, constructed for the cases ¼ ,4,5[66] and ¼ [77,78]. Dˆ Upon elimination of the auxiliary fields, these actions reduce to one may deduce that the operators A look like those proposed in [76] only for N ¼ 3, 4, 5. In the case N ¼ 6, ˆ ˆ ˆ¯ α ¯ α however, the on-shell version of the off-shell action in [77,78] Da ¼ Da þ i CaJ ; Dα ¼ Dα; D ¼ D ; (D2) contains an additional U(1) gauge Chern-Simons term as compared with [76]. where DA are the covariant derivatives of the SO(2) super- 23 If a prepotential formulation is available, the conformal space, as defined in Sec. II A,andCa is one of the supergravity action may be written as a superspace integral in terms of the prepotentials. Currently, the prepotential formulations are known only for the cases N ¼ 1 [2] and N ¼ 2 [15]. 24The connections in D1differ in sign from those used in [57].

085028-37 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) corresponding torsion superfields. The connections F’sare dealing with Da, one may work equally well with a uniquely determined as functionals of the torsion superfields deformed covariant derivative Da defined by of the SO(2) superspace. In terms of the one-forms Fα ≔ B α ~ B ~ E F and Fα ≔ E F α, one obtains B B Da ¼ Da þ λSMa þ ρiCaJ ; (D4) 1 1 Fα b − γ Cαβγ γ αβ D¯ ¯ D S where λ and ρ are real parameters. A natural question is ¼ E 2 ð bÞβγ þ 6 ð bÞ ði βR þ β Þ the following: What is special about the deformation (D2)? α ¯ ¯ βα βα − E R þ Eβ½C þ iε S; (D3a) Here we answer this question. Let us introduce the torsion and curvature tensors for the 1 1 covariant derivatives (D2), F~ b − γ βγC¯ γ Dβ − D¯ βS α ¼ E 2 ð bÞ αβγ þ 6 ð bÞαβði R Þ ˆ ˆ ˆ C ˆ ˆ c ˆ β ¯ α ½D ; D g¼T D − R M þ iR J : (D5) − E ½Cβα þ iεβαS − E R: (D3b) A B AB C AB c AB 1 Associated with the Lorentz and Uð ÞR curvature tensors 2. Curvature two-forms ˆ c 1 B A ˆ c are the following two-forms: R ¼ 2 E ∧E RAB and ˆ 1 B A ˆ In SO(2) superspace, there exists a two-parameter free- R ¼ 2 E ∧E RAB. The explicit expressions for these dom to define the vector covariant derivative. Instead of two-forms are

1 1 ˆ c β∧ α 4 ¯ γc ¯ ∧ α −4 S γc β − 4δβCc ¯ ∧ ¯ −4 γc αβ R ¼ 2 E E ½ Rð ÞαβþEβ E ½ i ð Þα α þ2 Eβ Eα½ Rð Þ 1 β∧ a γ γ γc δρC δγ δc 2ε c γb γ 2D S D¯ ¯ þ E E ð aÞβ ð Þ γδρ þ 3 ð β a þ ab ð Þβ Þð γ þ i γRÞ 1 ¯ ∧ a γ βγ γc δρC¯ εβγδc 2ε c γb βγ 2D¯ S − D þ Eβ E ð aÞ ð Þ γδρ þ 3 ð a þ ab ð Þ Þð γ i γRÞ 1 1 b∧ aε γd αβ γc τδ D C¯ D¯ C − 4CdCc þ 2 E E abd 4 ð Þ ð Þ ði ðτ δαβÞ þ i ðτ δαβÞÞ 2i 1 ηcd DαD¯ S D2 D¯ 2 ¯ − 4S2 − 4 ¯ (D6a) þ 3 ð α Þþ6 ð R þ RÞ RR ; 1 ˆ ¯ ∧ α 4 C β 4δβS β∧ a γ γδC − γ γ D¯ ¯ − 2 D S R ¼ Eβ E ½ i α þ α þE E ið aÞ βγδ 3 ð aÞβ ð γR i γ Þ 1 1 ¯ ∧ a γ C¯ βγδ − γ β Dγ 2 D¯ γS − b∧ a ε Wc (D6b) þ Eβ E ið aÞγδ 3 ð aÞ γð R þ i Þ 2 E E ½ abc ;

with Wc the super Cotton tensor, Eq. (2.14). For com- The unique feature of the deformation (D2) is that the top pleteness, we also reproduce the expressions for the two- component of the U(1) curvature two-form (D6b) is a c 1 B A c 1 B A 25 forms R ¼ 2 E ∧E RAB and R ¼ 2 E ∧E RAB, where the primary superfield equivalent to the super-Cotton tensor. curvature tensors are those which appear in (2.6). Direct calculations give 3. Closed three-form Rˆ c ¼ Rc; (D7a) In N ¼ 2 conformal superspace, the Chern-Simons three-form Σ is characterized by the following properties i CS ˆ ¯ ∧ α 2 C β β∧ a γ γδC [66]: (i) it is closed, dΣ ¼ 0; and (ii) under the gauge R ¼ R þ Eβ E ½ i α þE E 2 ð aÞ βγδ CS transformations, it is invariant modulo exact terms. This 1 three-form generates the off-shell action for N 2 con- − γ γ D¯ ¯ 4 D S ¼ 6 ð aÞβ ð γR þ i γ Þ formal supergravity. In this subsection, we follow the degauging procedure of [57] to obtain an expression i 1 ¯ ∧ a γ C¯ βγδ − γ β Dγ − 4 D¯ γS for this closed three-form in SO(2) superspace, þ Eβ E 2 ð aÞγδ 6 ð aÞ γð R i Þ

1 25 − Eb∧Ea½ε εcefD C : (D7b) S. M. K. and G. T.-M. are grateful to Joseph Novak for this 2 abc e f observation.

085028-38 THREE-DIMENSIONAL N ¼ 2 SUPERGRAVITY … PHYSICAL REVIEW D 89, 085028 (2014) J ≔ Σ J CSjde-gauged. The calculations are straightforward, and By construction, the closed three-form is invariant we present only the final result. under the super-Weyl transformations modulo exact terms. The three-form J turns out to be In fact, the relative coefficient between the Lorentz and 1 Uð ÞR Chern-Simons terms in (D10) is fixed by the 1 J J ¼ −Rˆ a∧Ωˆ þ Ωˆ c∧Ωˆ b∧Ωˆ aε − 2Rˆ ∧Φˆ condition that be super-Weyl invariant modulo a 6 abc exact terms. a α ¯ β − 8iE ∧F ∧FβðγaÞα : (D8) The covariant derivatives (D2) and the closed three-form (D9) constitute the unique solution to the N ¼ 2 problem The expression for J is naturally written in terms of the posed in [80]. ˆ deformed covariant derivatives DA. Making use of (D2),it is a simple exercise to rewrite (D8) in terms of the original 4. Conformal supergravity action covariant derivatives D . A J 1 C∧ B∧ AJ It is interesting to note that the closed three-form J can Using the three-form ¼ 3! E E E ABC ¼ 1 P∧ N∧ MJ be written as 3! dz dz dz MNP, we can write down the locally supersymmetric and super-Weyl invariant action (εmnp ≔ εabc m n p J Σˆ − Σ (D9a) ea eb ec ) ¼ CS T; Z Z 1 where we have introduced J 3 J J εmnpJ S ¼ ¼ d xe jθ¼0; ¼ mnp: M3 3! a α ~ β ΣT ¼ −8iE ∧F ∧FβðγaÞα ; (D9b) (D11) 1 Σˆ ˆ a∧Ωˆ − Ωˆ c∧Ωˆ b∧Ωˆ aε 2 ˆ ∧Φˆ (D9c) Upon implementing the component and gauge fixing CS ¼ R a abc þ R : 6 reduction described in Sec. IV, the action becomes Σˆ 1 Z The three-form CS is a sum of the Lorentz and Uð ÞR 1 2 Chern-Simons three-forms associated with the covariant S ¼ d3xeεabc R ω fg þ ω gω hω f Dˆ Σ 4 bcfg a 3 af bg ch derivatives A. The components of T are functions of the torsion tensor and its covariant derivatives only; this is why ¯ def − 4F abbc þ iψbcγdγ~aε ψef : (D12) ΣT was called the torsion induced three-form in [80]. The Σˆ Σ three-forms CS and T satisfy the equations This is the component action for N ¼ 2 conformal super- Σ Σˆ ˆ a∧ ˆ 2 ˆ ∧ ˆ (D10) d T ¼ d CS ¼ R Ra þ R R: gravity of [4].

[1] J. Wess and J. Bagger, Supersymmetry and Supergravity [8] N. S. Deger, A. Kaya, E. Sezgin, and P. Sundell, Nucl. Phys. (Princeton University Press, Princeton, 1983) (Second B573, 275 (2000). Revised and Expanded Edition: 1992). [9] S. Deger, A. Kaya, E. Sezgin, P. Sundell, and Y. Tanii, Nucl. [2] S. J. Gates, Jr., M. T. Grisaru, M. Roček, and W. Siegel, Phys. B604, 343 (2001). Superspace, or One Thousand and One Lessons in Super- [10] S. M. Kuzenko, U. Lindström, and G. Tartaglino- symmetry (Benjamin/Cummings, Reading, MA, 1983). Mazzucchelli, J. High Energy Phys. 03 (2011) 120. [3] I. L. Buchbinder and S. M. Kuzenko, Ideas and Methods of [11] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, J. High Supersymmetry and Supergravity or a Walk Through Energy Phys. 12 (2011) 052. Superspace (IOP, Bristol, 1995) (Revised Edition: 1998). [12] B. M. Zupnik and D. G. Pak, Teor. Mat. Fiz. 77, 97 (1988) [4] M. Roček and P. van Nieuwenhuizen, Classical Quantum [Theor. Math. Phys. 77 (1988) 1070]. Gravity 3, 43 (1986). [13] H. Nishino and S. J. Gates, Jr., Int. J. Mod. Phys. A 08, 3371 [5] A. Achúcarro and P. K. Townsend, Phys. Lett. B 180,89 (1993). (1986). [14] P. S. Howe, J. M. Izquierdo, G. Papadopoulos, and P. K. [6] B. de Wit, H. Nicolai, and A. K. Tollsten, Nucl. Phys. B392, Townsend, Nucl. Phys. B467, 183 (1996). 3 (1993). [15] S. M. Kuzenko, J. High Energy Phys. 12 (2012) 021. [7] J. M. Izquierdo and P. K. Townsend, Classical Quantum [16] M. T. Grisaru and M. E. Wehlau, Int. J. Mod. Phys. A 10, Gravity 12, 895 (1995). 753 (1995).

085028-39 SERGEI M. KUZENKO et al. PHYSICAL REVIEW D 89, 085028 (2014) [17] M. T. Grisaru and M. E. Wehlau, Nucl. Phys. B457, 219 [49] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. (1995). Komargodski, J. High Energy Phys. 01 (2014) 124. [18] P. S. Howe, Nucl. Phys. B199, 309 (1982). [50] N. S. Deger, A. Kaya, H. Samtleben, and E. Sezgin, [19] B. de Wit and M. Roček, Phys. Lett. 109B, 439 (1982). arXiv:1311.4583. [20] J. Wess and B. Zumino, Phys. Lett. 74B, 51 (1978);K.S. [51] S. M. Kuzenko, J. High Energy Phys. 03 (2013) 024. Stelle and P. C. West, ibid. 74B, 330 (1978); S. Ferrara and [52] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, Nucl. Phys. P. van Nieuwenhuizen, ibid. 74B, 333 (1978). B785, 34 (2007). [21] M. F. Sohnius and P. C. West, Phys. Lett. 105B, 353 (1981). [53] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, J. High [22] P. Breitenlohner, Phys. Lett. 67B, 49 (1977); Nucl. Phys. Energy Phys. 10 (2008) 001. B124, 500 (1977). [54] D. Butter and S. M. Kuzenko, J. High Energy Phys. 11 [23] W. Siegel and S. J. Gates, Jr., Nucl. Phys. B147, 77 (1979). (2011) 080; D. Butter, S. M. Kuzenko, U. Lindström, and G. [24] D. Butter and S. M. Kuzenko, Nucl. Phys. B854, 1 (2012). Tartaglino-Mazzucchelli, J. High Energy Phys. 05 (2012) [25] L. Castellani, R. D’Auria, and P. Fre, Supergravity and 138. Superstrings: A Geometric Perspective. Vol. 2: Supergrav- [55] S. M. Kuzenko, U. Lindström, and G. Tartaglino- ity (World Scientific, Singapore, 1991), pp. 680–684. Mazzucchelli, J. High Energy Phys. 08 (2012) 024. [26] S. J. Gates, Jr., in Supersymmetries and Quantum Sym- [56] D. Butter, S. M. Kuzenko, and G. Tartaglino-Mazzucchelli, metries, edited by J. Wess and E. A. Ivanov (Springer, J. High Energy Phys. 02 (2013) 121. Berlin, 1999), p. 46; Nucl. Phys. B541, 615 (1999). [57] D. Butter, S. M. Kuzenko, J. Novak, and G. Tartaglino- [27] S. J. Gates, Jr., M. T. Grisaru, M. E. Knutt-Wehlau, and W. Mazzucchelli, J. High Energy Phys. 09 (2013) 072. Siegel, Phys. Lett. B 421, 203 (1998). [58] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, J. High [28] M. F. Hasler, Eur. Phys. J. C 1, 729 (1998). Energy Phys. 04 (2009) 007. [29] B. Zumino, in Recent Developments in Gravitation [59] P. Binetruy, G. Girardi, and R. Grimm, Phys. Rep. 343, 255 (Cargèse 1978), edited by M. Lévy and S. Deser (Plenum (2001). Press, New York, 1979), p. 405. [60] T. Kugo and S. Uehara, Prog. Theor. Phys. 73, 235 (1985). [30] J. Wess and B. Zumino, Phys. Lett. 79B, 394 (1978). [61] S. M. Kuzenko and S. A. McCarthy, J. High Energy Phys. [31] M. Roček and U. Lindström, Phys. Lett. 79B, 217 (1978); 05 (2005) 012. 83B179 (1979); U. Lindström, A. Karlhede, and M. Roček, [62] W. Siegel, Phys. Lett. 85B, 333 (1979). Nucl. Phys. B191, 549 (1981). [63] M. Muller, Nucl. Phys. B264, 292 (1986). [32] S. Deser and J. H. Kay, Phys. Lett. 120B, 97 (1983). [64] D. Butter, Ann. Phys. (Amsterdam) 325, 1026 (2010). [33] S. Deser, in Quantum Theory Of Gravity, edited by S. M. [65] S. Ferrara, L. Girardello, T. Kugo, and A. Van Proeyen, Christensen (Adam Hilger, Bristol, 1984), pp. 374–381. Nucl. Phys. B223, 191 (1983). [34] C. Closset, T. T. Dumitrescu, G. Festuccia, and Z. [66] D. Butter, S. M. Kuzenko, J. Novak, and G. Tartaglino- Komargodski, J. High Energy Phys. 05 (2013) 017. Mazzucchelli, J. High Energy Phys. 10 (2013) 073. [35] K. Hristov, A. Tomasiello, and A. Zaffaroni, J. High Energy [67] M. T. Grisaru and W. Siegel, Nucl. Phys. B187, 149 (1981). Phys. 05 (2013) 057. [68] M. T. Grisaru and W. Siegel, Nucl. Phys. B201, 292 (1982); [36] G. Festuccia and N. Seiberg, J. High Energy Phys. 06 (2011) B206496(E) (1982). 114. [69] W. Siegel, Nucl. Phys. B142, 301 (1978). [37] B. Jia and E. Sharpe, J. High Energy Phys. 04 (2012) [70] S. Deser, R. Jackiw, and S. Templeton, Phys. Rev. Lett. 48, 139. 975 (1982); Ann. Phys. (N.Y.) 140, 372 (1982); 185406(E) [38] H. Samtleben and D. Tsimpis, J. High Energy Phys. 05 (1988). (2012) 132. [71] B. S. DeWitt, in Relativity, Groups and Topology II, edited [39] C. Klare, A. Tomasiello, and A. Zaffaroni, J. High Energy by B. S. DeWitt and R. Stora (Elsevier, Amsterdam, 1984), Phys. 08 (2012) 061. pp. 381–738. [40] T. T. Dumitrescu, G. Festuccia, and N. Seiberg, J. High [72] D. D. K. Chow, C. N. Pope, and E. Sezgin, Classical Energy Phys. 08 (2012) 141. Quantum Gravity 27, 105001 (2010). [41] C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski, [73] L. P. Eisenhart, Riemannian Geometry (Princeton Univer- and N. Seiberg, J. High Energy Phys. 10 (2012) 053. sity Press, Princeton, 1926). [42] C. Closset, T. T. Dumitrescu, G. Festuccia, Z. Komargodski, [74] G. W. Gibbons and P. Rychenkova, Phys. Lett. B 443, 138 and N. Seiberg, J. High Energy Phys. 09 (2012) 091. (1998). [43] D. Cassani, C. Klare, D. Martelli, A. Tomasiello, and A. [75] P. van Nieuwenhuizen, Phys. Rev. D 32, 872 (1985). Zaffaroni, arXiv:1207.2181. [76] U. Lindström and M. Roček, Phys. Rev. Lett. 62, 2905 [44] J. T. Liu, L. A. Pando Zayas, and D. Reichmann, J. High (1989). Energy Phys. 10 (2012) 034. [77] M. Nishimura and Y. Tanii, J. High Energy Phys. 10 (2013) [45] T. T. Dumitrescu and G. Festuccia, J. High Energy Phys. 01 123. (2013) 072. [78] S. M. Kuzenko, J. Novak, and G. Tartaglino-Mazzucchelli, [46] A. Kehagias and J. G. Russo, Nucl. Phys. B873, 116 (2013). J. High Energy Phys. 01 (2014) 121. [47] P. de Medeiros and S. Hollands, Classical Quantum Gravity [79] B. M. Zupnik and D. G. Pak, Classical Quantum Gravity 6, 30, 175016 (2013). 723 (1989). [48] P. de Medeiros and S. Hollands, Classical Quantum Gravity [80] S. M. Kuzenko and G. Tartaglino-Mazzucchelli, J. High 30, 175015 (2013). Energy Phys. 03 (2013) 113.

085028-40