Jhep10(2016)049

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JHEP10(2016)049 = 1 ,N Springer , = 3 October 4, 2016 August 29, 2016 October 11, 2016 : D : : Received Accepted 10.1007/JHEP10(2016)049 Published doi: Dedicated to the memory of Mario Tonin. Published for SISSA by a,b [email protected] , and P.A. Grassi a,c . 3 1607.05193 R. Catenacci The Authors. c a,b Supergravity Models, Superspaces, Diﬀerential and Algebraic Geometry

By using integral forms we derive the superspace action of , [email protected] [email protected] INFN — Sezione divia Torino, P. Giuria 1, 10125Gruppo Nazionale Torino, Italy di FisicaP.le Matematica, INdAM, Aldo Moro 5, 00185E-mail: Roma, Italy Dipartimento di Scienze eViale Innovazione T. Tecnologica, Universitàdel Michel, Piemonte 11, Orientale, 15121 Alessandria, Italy b c a Open Access Article funded by SCOAP ArXiv ePrint: spacetime surfaces embedded into theaction supermanifold. based We on show the how group thecomponent manifold group supergravity approach geometrical actions, interpolates thus between providing the another superspace proof and of the theirKeywords: equivalence. Abstract: supergravity as an integral onpicture a changing operators, supermanifold. here The playing the construction to rôleof Poincaréduals is the based lower-dimensional on target space L. Castellani, The integral form of supergravity JHEP10(2016)049 ]. That 14 (a.k.a. 1 rheonomic 6 ]). It also provides a manageable ]. 11 19 – 17 . The former denotes a flat bosonic spacetime 2 – 1 – 9 constraints consistent with the Bianchi identities, supermanifold = 3 supergravity, local supersymmetry being realized 13 D and ad hoc 12 the constraints on the curvatures. The action is written as forms. As is well known, differential forms on superspace 4 provides a superfield action which yields both the correct space- and 2 superspace integral 5 2 ], the same action can be written as the integral over the whole super- 1 12 ]. For recent developments see for ex. [ = 1 supergravity in the two frameworks ) approach 10 ]), supplemented by – N 2 10 = 3, ] and [ 1 In this paper we find a bridge between the two formalisms by a novel technique based At the moment, however, there is no explicit dictionary between the superfield ap- On the other hand, the construction of a 3d N=1 supergravity in the We distinguish between for reviews on the group manifold approach see for ex. [ 2.1 Superspace 2.2 Supergroup2.3 manifold Equivalence D 1 2 proach and the group manifold approach. on the integration of with additional fermionic coordinates, while the latter the full-fledged supermanifold according to [ a Lagrangian 3-form integrated over aAs bosonic discussed submanifold in of [ themanifold complete of supermanifold. an integral3-dimensional form, bosonic using submanifold. the Poincarédual that encodes the embedding of the provides an off-shell formulation of as a diffeomorphism in the fermionic directions. group manifold time equations of motion, general relativity that includesbeen fermions revisited and as a local workable supersymmetry. exampleple in [ For many this textbooks and reason researchmodel it papers of has (see superfield for exam- supergravity, withaction a (see superfield [ action integrated on superspace. 1 Introduction Three dimensional supergravity is one of the simplest models of a consistent extension of 4 The actions and their5 equivalence Outlook and perspectives A Properties of the susy PCO 3 Contents 1 Introduction 2 Superspace versus supergroup manifold JHEP10(2016)049 3 , 2 = 1 , ,N = 1 or PCO) a = 3 D ) with α , θ refer to the bosonic a x m and , which is closed (in general n L ] and the group-manifold formalism Picture Changing Operator (where 10 , ) 1 m | n ( ]). Multiplied by a suitable closed Poincaré 17 with a set of coordinates ( SM – 2 – 2) | (3 R ]. Another Poincarédual, differing from the first by 10 ]). These can be incorporated into a consistent differential , = 1 supergravity in two frameworks, namely in the group- 1 16 – N 12 = 3 . in the two frameworks. 2) | D (3 SM action ]. Since the two Poincaréduals are in the same cohomology class, the two 10 , 2. The same set of coordinates will be also used to parametrize a local patch of 1 , = 1 α The paper is organised as follows. In section 2 we discuss the equivalence between Furthermore, the expression of the action written as the integral of a Lagrangian In particular there is a canonical Poincarédual that produces the standard spacetime The bridge between the superspace action of [ = 0 when auxiliary fields are present [ L 2.1 Superspace First, we parametrize the superspace and a supermanifold 2 Superspace versus supergroupWe manifold want to formulate manifold approach and thesupersymmetric superspace approach. Let us first clarify what we mean by basic ingredients for thesupergravity: superfield and the the constraints, group theprove manifold the Bianchi equivalence formulations between identities of the and groupspacetime manifold their action (rheonomic) and solutions. formulation, the the superspace component In action.work section In and section 4 in 5 we we the list appendices some perspectives we for give future some further details on the PCO. limit the number of independentconstraints component built fields. in directly Itthe into would closure be the of advantageous action. the to PCO have This implies the is exactly achieved those in constraints. superspace the and present group-manifold formulations formulation: in general terms. In section 3 we provide the action of [ actions are equal. three-form times a PCOformulation of clarifies supergravity an is additional redundant issue. in the sense As that recalled one above, needs the some superfield constraints to Choosing Poincaréduals in theLagrangian same is cohomology closed. class does not change the actionaction if with the auxiliary fieldsa of total [ derivative, leads to an expression for the action that coincides with the superfield calculus where top forms do exist, and can beis integrated provided on by the the supermanifold. d group-manifold three-form Lagrangian dual form (known init the becomes string an theory literature integral as top form, and therefore can be integrated on the supermanifold. and fermionic dimensions, respectively)of since differential forms. there is Indeedrespect no the to top fermionic the form 1-formsfermionic wedge in behave 1-forms. product like the Nonetheless, commuting one and usual can variables extend complex thereforelike with the forms space there (see of for forms is example by [ including no distribution- upper bound to the number of cannot be integrated on a supermanifold JHEP10(2016)049 ’s = θ α (2.3) (2.4) (2.1) Q ) = 0. where , their φ a ∂ x, θ ( a αβ F ). It is the α iγ Q 2.2 = 2 } β ) is a superfield, its ,Q x, θ ]), the Lagrangian is a α ( 1 Q F , { 2 ) of the theory. A superfield 2 θ ) (2.2) =0 x x, θ ) θ ( (

2 , ) is built out of superfields φ ) φ ) x, θ ( + x, θ x, θ ( x, θ F ( α ] ( θ F θ F ) F 2 2 ) are the ordinary derivatives with respect x α ( ), one can compute the Berezin integral by α D xd 2 Q ] 3 ,α α , ∂ 1 x d 2.2 , φ [ a 3 – 3 – φ ∂ ,α d [ 1 Z . ) = β ) + Z , φ and products thereof. x 0 ] = D and then selecting the highest term ( φ α x, θ α φ 0 [ ] = ( θ where ( = 3. The supersymmetry generator is defined as φ D D φ F [ a δ method, which minimises the amount of Feymann diagrams susy αβ D ∂ written in terms of the physical fields. The superderivative ) = ) are subject to constraints, and their variations have to be S α susy ) = S a is the functional 2.2 x, θ 2 ( ] refers to the integration variables. The integration over the ¯ θγ θ D φ 2 + ( ) follows from the fact that xd α supergraph 3 ∂ 2.4 d and the components are identified with the physical degrees of freedom. entering ( β = ), a local functional of the superfields , we obtain the superfield equations of motion. In the case of supergravity, θ φ φ α α θ and superderivative x, θ D a component action ( ∂ a αβ α ∂ F ) ). In addition, superspace action a α ≡ ¯ 2 θγ , θ ( θ a are the Dirac matrices for The property ( In the case of rigid supersymmetry, the action is invariantIn because the the case of variation local of supersymmetry, the oneThere needs to are impose the several vanishing of advantages in having a superspace action as in ( The supersymmetry of the action is easily checked: since The In the case of superspace (see for example the textbook [ being the supersymmetry generator satisfying the algebra x − α a α αβ most economical and compact wayof to freedom describe of the supergravity, complete ittechnique, action known encodes for as all all physical symmetries, degrees needed it for provides a a single powerfulrenormalization quantization scattering theorems amplitude. are mostly The manifest. supersymmetry cancellations and the non- derivative Lagrangian is a total derivative. Q γ ∂ which is the is defined as to ( supersymmetry variation is simply compatible with these constraints.expanding Given the ( action in powers of where the symbol [ is given by the Berezinthe superfields integral. Varying the actionthe under superfields an infinitesimal deformation of A generic superfield mightthat also there contain is some a auxiliary match fields between to off-shell complete bosonic the and spectrum fermionic so degrees of freedom. superfield can be expanded in its components with JHEP10(2016)049 , , A G are σ (2.6) ˜ . The G G , and other G -dimensional spacetime d -form lagrangians, whose . d α ˜ G . The coordinates of dθ ˜ α G ) x, θ ( = 0 (2.5) on the supergroup manifold A C σ A σ σ + ∧ a is expanded as a superspace 1-form as B σ dx A a σ ) BC – 4 – A C x, θ components. ( 1 2 A dθ σ + ] leads to the construction of A 19 is a 1-superform living in ) = ) is the lack of a fully geometrical interpretation, since it dσ – . For the fields to become dynamical, they must be allowed A , corresponding typically to the translation subgroup of A 17 x, θ 2.2 σ µ ( , corresponding to the fermionic generators of x A α σ θ -dimensional bosonic manifold reproduces the = 0). In other words one has to eliminate the extra degrees of freedom d x, θ dependence and to the ( θ A of the bosonic submanifold. The resulting equations yield the usual spacetime corresponds a one-form (vielbein) field σ ˜ A G . T θ , and include the supersymmetries as diffeomorphisms in the fermionic directions of The variational principle involves variations of the fields, and variations of the embed- Still one has a great redundancy, since Typically the fields one wants to retain as dynamical fields in this formulation are The supervielbein field A systematic procedure [ The fields of the theory are identified with the various components of the vielbein The main drawback of ( ˜ G and . In this respect supersymmetry transformations have a geometric interpretation similar ˜ due to the ding in field equations, together withfreedom the (“rheonomic constraints constraints”). needed to remove the redundant degrees of given by Grassmann coordinates coordinates corresponding to gaugeproduce directions. gauge Diffeomorphisms transformations, and in thebe these dependence removed last via of coordinates a the finite fieldsx gauge on transformation. these coordinates At can the end of the game all fields depend on on G to the one in the superfield approach. the spacetime coordinates equations must be nonvanishingof in the general. supergroup This manifold, is i.e. achieved a by “soft” considering supergroup deformations manifold restriction to a supergravity lagrangians. The local symmetries of the theory are the superdiffeomorphisms labelled by the adjoint index to develop a nonzero curvature, that is to say the right-hand side of the Cartan-Maurer 2.2 Supergroup manifold The logic of thiserator approach is algebraic:vielbein one satisfies starts the from Cartan-Maurer a equations: superalgebra, and to each gen- cannot be understood asfor an the integral superfield of action abut is differential it usually form is dictated not on by very arespect intuitive scaling manifold. the and properties group-manifold for The and approach constrained expression seems Lorentz superfields to covariance, it overcome does these not problems. always exist. In that JHEP10(2016)049 (2.8) (2.10) ) is in 2.9 ]) . 17 directions) yield is defined as the as follows dθ 2) | (3 (2.7) into the supermanifold is given by . Since the past litera- ) ], and involves Poincaré ˜ action which interpolates G (3) (3) SM 20 , L . M 12 (3) 2) | mother L ), generated by a Lie derivative. (0 x, θ, dx, dθ form on d ( rheonomic action Y = 0 (see the textbook [ ι ) and the integration is on the com- (3) ∧ top dθ L 2.7 ) + 0) = 1 | 2) | (3 (3) (3 N L = 0 (2.9) x, θ, dx, dθ L ( ι 2) | ( (3) (3) (3 d = 0 and ⊂SM – 5 – L of the supermanifold L = 3, θ d (3) = SM ι the supersymmetric action is given by D (3) M Z (3) Z 2) M | L ] = and on the embedding of = 0. It is less compact than the superfield formulation, (0 ` ] = φ Y [ σ = dθ (3) susy (3) M S L σ, δ [ = 0 and rheo has a direct correspondence with the component action, to which it θ is a total derivative and the action is invariant. Condition ( S (3) (3) L L is the rheonomic Lagrangian used in ( 0) | (3 . Changing the embedding corresponds to a diffeomorphism and it can be com- L 2) | (3 We have argued that the local symmetries of the group manifold action are the diffeo- The form of The supersymmetry of the action is expressed as a diffeomorphism in the fermionic In terms of these ingredients, the where plete supermanifold. The Poincarédual (alsoifold known to as the PCO) submanifold. localizes Integration the on full supermanifolds superman- is discussed in several papers (see The component action obtainedrelated by in field redefinitions. the Therefore twobetween there formulations must the exist must two a formulations. bea This the 3-form action same on is or, a the submanifoldsubmanifold, at rheonomic of and action. least, a denoting The bigger it way manifold by to is integrate by constructing a Poincarédual of that ture on group manifoldtheory, actions this for point supergravity has makes neededduals little some and clarification, reference integral reported to top in superintegration [ forms. 2.3 Equivalence reduces after setting but more transparently related to the component action. morphisms on the supergroup manifold.manifold action This certainly resulting holds from true the if integration one of considers a a group If the Lagrangian satisfies the variation of fact equivalent to the rheonomic constraints mentioned above. component action is retrieved by setting directions of the supermanifold and therefore the variation of pensated by a changeTherefore of the the variational equations Lagrangian canembedding, be and obtained considering by theperspace. varying resulting the equations fields Projections as for of 2-form anthe these equations rheonomic arbitrary equations on constraints, the in necessary whole to the remove su- fermionic unwanted degrees directions of ( freedom. The correct integral over a bosonic submanifold and depends on theSM superforms JHEP10(2016)049 . 2 d / 2) | in into 1) (3 (2.11) (2.12) 2] − (3) d/ [ d SM ( M = 1 super d -derivatives 1)2 r (the transla- N form degree p − α d ). Inserting the are , Q β ) ) ψ r ψ ( ( ( is discussed and its δ δ ): the and 2 q δ | ∧ 2) is the derivative of the β p | ab )). Then, any variation ) ι ). The main goal of the (0 susy α α α dθ ,L ι ι Y ψ ( 0) yielding the component a 2.2 ( δ . αβ P ab δ ) (where − γ A α b αβ , dx, E V 0 dθ ) = ( ∧ ) x, = 0 (3.1) p ( . As recalled, the group-geometric a dθ ) = ( ( C 0) 0 ψ B V | , δ 2) ( σ | α (3 2 1 = δ ∧ dθδ L − 2) and they can be integrated on ( | 2) B , dθ | Ω a σ (0 susy d Y dx with 3 off-shell degrees of freedom ( BC = – 6 – , and can be integrated on this supermanifold. -exact terms, namely if they belong to the same A ) a d 2)-form. q C | 2) is closed (the action does not depend on the em- | | p V 1 2 ( (0 0) with 4 off-shell degrees of freedom (( | Y is an integral form, built as the product of the rheo- + is closed and not exact (it belongs to the cohomology α (3 δ A SM 0)-superform (constructed as discussed above), and the ψ L 2) | 2) | | dσ (0 (0 are those with (3 , Y Y ) 2) ∧ | dθ , which is a (0 (3 ], the equivalence of the different formulations of 0) ( where the latter counts the number of delta functions. In general | δ 2) ’s are related by | (3 2 21 Y θ )), and its variation under the change of the embedding of (0 L SM Y 2) | = , which is a (3 (3 2) we project the Lagrangian to 0) | = 1 supergravity in the two frameworks | (0 st ] for the definition of the Poincarédual on supermanifolds). Only the inte- (3 2) ) are the components of the supervielbein | SM is an integral form with negative form degree (derivatives of the delta func- Y L N α 12 (0 st -exact: d, 2) | d ( Y , ψ 1 a 2) − | is ( picture number q V = 3, (0 2) | H dimensions), and a gravitino (3 In a related work [ We propose the two different choices The Poincarédual/PCO D )-forms are integral forms on d q | p procedure to build supergravitythe actions superalgebra starts is from thetion a superPoincaréalgebra, generated generators, superalgebra. by the In Lorentzconstants the generators of case and the at superalgebra the are hand supersymmetry encoded charges). in the The Cartan-Maurer structure equations The theory contains ain vielbein 1-form dimensions for Majorana or Weyl).here The provided mismatch can by be a cured by bosonic an 2-form extra bosonic auxiliary d.o.f., field Chern-Simons theory hasproperties been are studied. described in that The paper. flat version3 of where ( delta function with respectfirst to its PCO argument and action. The secondpresent choice work leads is to to the prove superspace this equivalence. action in ( of the embedding isbedding). ineffective Also, if if two cohomology class, the corresponding actions are equivalent. SM where Ω tions act as negative degree forms: for example The integral forms of Thus the Lagrangian nomic action Poincarédual/PCO class gral forms can be integrated.the The pseudo-forms complex of which differential are forms polynomials onof a in the supermanifold contains delta function)and . the They are( characterized by two numbers ( for example [ JHEP10(2016)049 (3.2) (3.3) (3.4) (3.5) (3.6) (3.16) (3.11) (3.13) (3.14) α ) bc ρ a γ ψ B ( a . The algebra is further 2 ψ α λ abc ab ¯ ψγ in the superspace language ψ i 2 → ¯ 5 (3.7) (3.8) (3.9) (3.10) ψγ c − f = a = 0 = 0 = 0 = 0 1 c α ψ, B V a ρ ψ 2 1 ab Ξ a + λ R a c ab ¯ ≡ D ψγ ψ R γ D a → i ψ ab ψ a ψ V ψ V ¯ R − ψγ ]). a c , ψ 1 4 i b 2 ¯ α ψγ a ab 17 and the gravitino ) V i ¯ ≡ D 2 θ f γ c + + a (3.15) b γ – 7 – ab 2 λV ρ a a ψ + − Ξ c abc ab ω ¯ cb d R b D ψ ρ c ab . In the present case the cotangent basis is given → ψ V + ρV ω (¯ V V conventional constraints γ are a cotangent (vielbein) basis, dual to the tangent a + a 4 + V b c a G c c b b a a ab a a A in the group-manifold approach. We use the following V V ¯ ¯ ψγ ψγ ω V ω a σ R ω + ). i a ,V 2 1 4 i cd V D α − − f V ab ) ab 3.6 ] − − a ab (3.12) − a ω b ab ρ f V ∂ R 0 γ λ a dψ dB dω dV [ dH = 0 = = = = in order to match the degrees of freedom (and thus becomes a c ρ (FDA), see for ex. [ a → ρ = = = = (¯ ab B H df R a ρ 3 R ab ab H c R ω R = , the spin connection a c,α V ab ¯ θ is the Lorentz covariant derivative, and exterior products between forms are rheonomic parametrizations D As explained above, the redundancy introduced by promoting each physical field to The generalized Cartan-Maurer equations of the FDA yield the definitions of the with parametrizations invariant under the rescalings ( a superfield hasparametrizations. to They be are tamed knownand as by as imposing some algebraic constraints on the curvature Taking exterior derivatives of both sides yields the Bianchi identities: where understood. The Cartan-Maurer equations are invariant under rescalings Lorentz curvature, the torsion, the gravitino field strength and the 2-form field strength: vectors on the supergroupby manifold the vielbein extended with a 2-form Free Differential Algebra where the left-invariant one-forms JHEP10(2016)049 (3.20) (3.21) (3.23) (3.17) (3.19) (3.22) (3.18) ) by a term ). Similarly, is a superfield 3.3 3.11 R . α R, σ ψ , . α . , ) σ σ = r f σ i γ E E 3! α ( E , where i ∧ ∧ defined in ( − E ∧ R as fundamental fields, with ρ ρ , a ρ = a rs = 2 E E R B a 5 E A V ρσ ρσ rσ a α ω conventional constraints , α ab ρσ = T T = = 3 follows a different path, and i, c R CB with B a ! − ω + + ]. Note that, thanks to the presence , and is identified with the auxiliary rs + E E a D ab f ab σ σ ∧ ,T = 10 σ 1 ω T ∧ R E E − β 4 C E β α A B λ α ) A ∧ ∧ 0 ) = 0 ω ∧ differs from r r ω ab i, c r ab → a constraint on the spin connection − γ ρσ E E γ α − ( E T ( f 1 4 rσ rσ = 2 1 4 A a α – 8 – ab rσ AB = 0 and b 3 T T a R = 0 dE dω ω component. The solution for the other components + + , c + ρσ ,T ab = s s αβ 3 2 2

= s θ R soldering R E E A E scales as = correspond to the analogous terms in ( = = 0 , T AB ∧ ∧ ∧ 2 and the spin connection B ab r r H A R ab r ρσ rσ a are fixed by the Bianchi identities to the values: R ω E E A , c R E 5 of i rs rs E 2 ab rs a α 3 , c f 4 R T T and = ,T , c = = = 1 ]. The supertorsion 3 c ab rσ a ρσ a α ab , c 22 2 iγ T T , ,R R 1 2 , c ab rs 10 1 c R = component ρσ a T VVV . The index of the supervielbein now runs only on the superspace directions, and α Again there is a huge redundancy in that formulation, and one has to impose some The superspace formulation of supergravity in The contains the fields of the rheonomic approach as A The Bianchi identities thencontaining the imply scalar auxiliarycomponent, field and as the first Ricci component, scalarcan the as be gravitino found curvature in asbilinear [ in mixed fermions, and this reflects into the first constraint given above. To reduce the independent components, one imposes the which has the following expansion on the supervielbein basis has nonvanishing components The superfields one considers the supertorsion where the off-diagonal piecesrelated are to the set Lorentz to spin zero connection. and As a the consequence spinorial the part supercurvature of the connection is E constraints. First, one imposes the spacetime components of thethe curvatures. superspace To action, compare we have with to the clarify superspace the approach roleconsiders and of the the supervielbein superfield A=a, scalar superfield of theof superspace approach the of auxiliary ref [ field, the Bianchi identities do not imply equations of motion for the The coefficients JHEP10(2016)049 ) α m E 2.7 (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.1) ) and and 2.7 α µ = 0. This ), invariant E ˜ G (3) L d , i.e. abc c (3) . Indeed we observe that, L V b R V variations: ), the rheonomic action ( a . , V ε the most general Lorentz scalar δf c 2.2 2 . Comparing with the constraint ab f V f . b (3) 1 2 = 0 ¯ rs R ψγ and L V a f a − = . Using the gauge symmetries, one can ab if V = = 3 m f δω a fH abc H rs − dx = 0. This change in the spin connection a c f α ε εV m = α R a a ρσ E ρ = εV ab + – 9 – ¯ ¯ c ψγ ψγ + R = ε i i ¯ (4.8) ab µ ψρ a ¯ − D θ − i has to be identified with R dθ ,H f = = = = = 0 α µ + 2 = a E f ψ = 0 ab B ) such that ε ε ε ab V abc δ a δ = δ ε δω R δ c ), namely R α 3.12 is fixed by requiring the closure of V E ): ab α α 3.11 in ( R ψ a = in ( R reads (3) ab L R (3) L of the superfield approach one finds R rs = 6, and ensures the off-shell closure of the supersymmetry transformations given a α = With the usual group-geometrical methods, the action is determined as in ( Comparing the analysis in the superspace and the analysis in the rheonomic approach, Using these constraints, one finds that the only independent off-shell degrees of freedom rs a and close on all the fields without need of imposing the field equations. easily computed by taking(tangent the vectors dual Lie to derivative of the fields along the fermionic directions The remaining parameter yields below. The action is invariant under off-shell supersymmetry transformations which are under the rescalings discussedadmit the above, vanishing and curvatures solution then requiring that the variational equations and also imply the constraints, arising from the This action is obtained by3-form, taking given in for terms the of Lagrangian the curvatures and 1-form fields (cotangent basis of To uncover the relationand between the the component superspace action, action we have ( to discuss themthe in Lagrangian the corresponding frameworks. by a change of the spinof connection, the one can curvature setproduces to zero a the change last of termT in the parametrization 4 The actions and their equivalence (vielbein, gravitino and scalar auxiliaryof field) the are superform contained in expansion theidentify components the physical and auxiliary fields. we find that the auxiliary superfield JHEP10(2016)049 ) ). 2) | x, θ ( (3 (4.10) (4.13) (4.15) (4.11) F SM ]. We refer d, 0)-form and | ( 26 2) – | (0 23 H . α ). It is a (3 , ψ a . The expression in 4.1 The choice of the Poincaré V A ) (4.16) E δ 3 ψ ). dθ 2) ab | which contains the auxiliary field, ( δ (0 (4.14) ) ¯ R 3.15 integration theory [ ψγ ) γ Y E f ∧ )–( dθ i ( 2 3 0) δ abc | γδ 3.11 (3 + c Sdet( b a L V R b β V 2) θ | a V – 10 – α (3 ectoplasmic a ψ V V θ ) = a 2 γ f (4.9) (4.12) αβ SM 3 2 f V Z x, θ ( = = 9 = = 0 = = 0 = leads to the equations of motion: F ]), by varying the action keeping the submanifold fixed. theory interpolating between the rheonomic action, the a ρ 2) | f ab H df 17 SG R (0 st R S is the rheonomic action given in ( Y ’s is performed by integrating on the Dirac delta functions, and 0) dθ | mother B (3 , L ) is the use of the ψ , a 4.14 allows us to interpolate between the component action and the super- V , 2) | ab ) is the superdeterminant of the supervielbein (0 ω 2)-form: E | Y ] for a complete discussion and for the equations of motion in superspace. ’s, one can retrieve the component action. However, the computation is rather θ 10 , To retrieve the usual spacetime action one chooses for the Poincarédual/PCO the Finally, we are ready to discuss the relation between the two actions. As explained Let us move to the superspace action. As we have seen in the previous section, after Notice that the equations of motion are obtained from the rheonomic action principle Varying The dependence of the fields on the gauge (Lorentz) coordinates factorizes, and reduces to a multiplica- 1 3 It is closedThe and integration not over exact, the and it istive an factor in element front of of the the integral over cohomology the superspace. it is closed becausedual/PCO of thespace parametrizations action. ( following (0 superspace action and the component action is described by the superintegral: where the Lagrangian der in cumbersome already in the presentnent simplified action context. from A ( to better [ way to derive the compo- in the introduction, the the expression where Sdet( is a superfield and transforms as discussed in section 2. By expanding at the second or- (as explained in theThey textbook are [ 2-form equations and can be expanded onimposing the the basis constraints we arethe left Ricci with scalar a and superfield the Rarita-Schwinger term. To build the action we therefore consider JHEP10(2016)049 0) | 0)- | (3 ) by = 0 L (4.22) (4.23) 2) | 4.16 is closed (0 st 2) Y | ). Then the = 0 if (0 susy submanifold. ∧ 2) | Y 3 SG 0) (0 st ) (4.24) | M Y (3 δS E given by an exact i L ( 2) d | 2) d | one finds: . Notice that the su- (0 susy (0 st Sdet( 2) Y ρ | Y f ] (0 st ∧ θ 2 Y yields 0) ) since and ) (4.21) | δ ) (4.18) xd 2) 2) (3 | | 3 ψ ab 2) | L ( d (0 st (0 st ) (4.20) [ R ’s in δ = 0) (4.17) . Varying the submanifold via a (0 st Y 2) Y ) ψ θ | Z 2) Y ( γ | (3 ∧ 2 i i ψ (0 st δ , dθ ( ( ] we know a priori that β 0) δ Y d | ) yields an ordinary spacetime action, SM i 27 Z α (3 γδ ∧ = 0 i ) = 6 L a θ = 0 (4.19) = ψ 0) 4.15 d | αβ ( ab ( ≡ ( 2 γ 0) 2) (3 0) applied on | ) | ψV b | i δ L c a (3 ψ (0 st (3 V 2 ( – 11 – , we see that only the first two terms of | V a L Y L di ¯ 2 3 b ψγ d V 2 ∧ | is also dictated by Hodge duality: indeed it is the + V (3) i 0 susy a is connected to the Poincarédual/PCO in ( SM = d 0) , δ | Y 2) M Z | V because of the two i Z 2) α (3 2) | (0 susy | δ = L abc x (0 susy = is not zero, but is exact, and therefore the integrand is = (0 susy Y δψ 2) Y Y f | SG 2) L ≡ | SG (3 2) | S δS (0 st α (3 i 2)-cohomology classes [ Y SM | M Z directions corresponds to a variation of Z θ i = 0)-form: ) depends in general on the choice of the bosonic | SG = 6 : S = 0. Berezin integration in ( 4.15 (3) SG S M dθ = 0. The choice of 0) | (3 L d Computing now the term with Another Poincarédual can be chosen as follows The action ( -diffeomorphism. Therefore their difference is exact (since a Lie derivative acting on a θ and not exact, and fulfills the requirements for acontribute, Poincarédual. and using the curvature parametrizations for since Hodge dual of the (3 which is closed (by thecohomology 3d classes into Fierz (0 identity) and not exact. Since Hodge duality maps (3 a closed form gives an exact form), and we find the equivalence: with We prove in the appendix that Integrating by parts and(because it noting exceeds that the maximal 0 rank = of an integral form), we find that supersymmetric only up to a total derivative. This choice is encodeddiffeomorphism in in the the Poincarédual/PCO form, since the Lievariation derivative of the action due to the variation of the submanifold is: integrated on where all forms dependpersymmetry only variation on of that imposes JHEP10(2016)049 (5.1) inter- (4.26) = 3 case, = 3 super- PCO D ,D = 1 Λ or equivalently, (4.25) = 1 ) √ N = 2 supergravities) N + dθ ( , integrated on super- N f 2 0) 2) integral form | | , xδ → (3 is integrable on the super- 3 is closed, and this is a con- L . It is interesting to notice f d c = 10 ) 2) | 2) 0) E | | E ). b (0 D . The result is that the action (3 (3 we finally conclude that the two Y E R L a = 1 supergravity action, and its ∧ 3.15 R E 0) N | 2) )–( | abc (3 (3 ) = Sdet( L Λ L = 4 supergravity and higher dimensional i ψ = 11 and 3.11 ( 6 2) 2 | δ D − (3 a ∧ = 3 and ]. by a constant term c SM – 12 – 28 ,...D f Z D V 4 EE , a ∧ = 2 , b ¯ Eγ . Indeed, it can be shown that in V ) SG l Λ S = 1 − ∧ √ 2 | a N (0 V 4 = Y abc ∧ 2) | ) l (3 | = Ω (3 2) | L (3 ) is the supervielbein in superspace and we have used =0 α 2 l , but is definitely not the only one. One can wonder whether it would be is identified with the scalar superfield Vol 2) P | , ψ f a (3 = V 2) | SM = ( (3 = 0. L E 0) | As a final comment we observe that the form In conclusion, the group-manifold rheonomic Lagrangian The present formulation permits also the introduction of a cosmological constant term. The relation between integral forms and superspace formulation has also been used to formulate massive (3 4 L where there exists such a possibility and it will besupergravity discussed in the separately. multivielbein formulation in [ the present analysis on such models. manifold possible to construct a supergravity action as a non-factorized (3 sequence of the existence ofan the off-shell auxiliary formulation fields of for thetence the theory. of model This an at agrees action hand, withof principle i.e. the supergravity the in common models existence superspace. belief of (such about Noteis as the available however for even exis- that in example the absence rheonomic of formulation auxiliary fields and it would certainly be interesting to test gravity for simplicity. Nonetheless, theity present model formulation is and applicable in tomodels. particular any to supergrav- The mathematical frameworkpolating permits different superspace to formulations. explore Anthe different important different choices remark: formulations of the holds equivalence because between the Lagrangian d 5 Outlook and perspectives With the present work,different we superspace have established formulations of a supergravity. precise We mathematical have relation used between the two is closed using the rheonomic parametrizations ( space, yields bothsuperspace the usual version. spacetime Theby a essential total ingredients derivative, of and the the rheonomic proof constraints are with Poincaréduals the differing auxiliary field that ensure in the superspace framework,acquires by a shifting new the termthat superfield proportional this to new term the volume form Vol Recalling that actions are indeed equivalent. This is achieved by shifting the superfield where JHEP10(2016)049 (A.2) (A.3) i )) ) σ σ E )) E ( ( σ δ δ ) E ρ ( ∧ and we obtain for )) δ E ) σ i ( = 0 C ) ρ δ δ E (A.5) σ 3 ( E E E ρσ ( δ E E (A.4) δ ( ∧ ∧ ( i ∧ δ c ) 3 β ρσ B ) = 0 γ ι σ ∧ E σ E ( E α T δ E γ ι (A.1) cδ i E β )) = 0 ( )) ) ) ( ι ∧ T E σ δ σ BC αβ σ δ A α E ) ) ) ι ( E ∧ E + ∧ ρ E T ρ ( 2 ( ab ( ) d δ δ δ E δ αβ ρ δ γ E ) ( ) ) = β ( ( E E ρ )) ι ρ δ E b ). ∧ δ ab σ ( γ α A ∧ ∧ σ E E δ ι γ ι E δ ( ρσ ( E ( c ) T γ E δ b E ( δ , and the term vanishes because of the ρ ρσ ∧ ι ( αβ b E δ h ) δ ) E ab ρσ ) E ∧ ρσ ( γ E ) cd γ ( ) can be written as β ρ η ab ) ρ β ι ( ∧ ( δ T ρ γ ∧ ι E E α γ ) )) E β ( β . We check that, by using the conventional ( α ι ι = γ ι b ( E ι γ A.3 σ ι ∧ δ α h ( δ α – 13 – ∧ α E ρσ E δ E ι δ E ι ψ β ) αβ ρσ β ( γ ι ρσ ) cdb ρ ι E ∧ ι ∧ δ αβ ) α αβ ( αβ ) h ι ) = ab ) a δ E a δγ ∧ ) ρ β cd ( γ ≡ β ι ab a ab δ E E ρσ ι ( δ ab iγ α A.2 E ) α γ b γ γ α ( γ a ι E ( δγ ( ι ι = may be inferred by Hodge duality with the cohomology ( , δ E 3 b E α αβ T ,E c by definition. ( ι + 2 γ αβ ) cδ ρσ ρσ a 2 E ρσ E 2) ) ∧ 2) E | β d | T + δ ab αβ α h V c a ( ab ∧ (0 ) (0 susy d γ cdb E δ ∧ αβ β γ in the vielbein basis: β ( E a E ) Y ι Y ( ab ι E b δ ∧ = b = γ α T cd γ α ab A ∧ c ) ι a E ι ( ∧ γ E β c a b T ( c E +2 γ a δγ αβ ∧ αβ E E E ∧ ( γ R ) cδ ) E cd α a = 2 iγ a a ι T ∧ . The second term in ( ab ab cd 2 abc a E E γ a γ ab ( abc ( R T b γ E R b abc d E E = = = = ( = ( 1 3! . In this appendix we prove it directly. We use here the superspace notations ∧ ∧ a a The closure of a = E T ψV 3 a E ¯ ψγ Let study the second piece in ( It is invariant under Lorentz symmetry since all tangent indices are contracted with where we have used recast the first term as follows where antisymmetry of where we have used the parametrization of the torsion. Due to antisymmetrization, we can We expand the torsion the first term: is closed and not exact where Lorentz invariant tensors. It is also closed. To prove it, we observe Closure. class for the supervielbein supergravity constraints, the PCO We would like to thank C.useful Maccaferri, discussions D. and Francia, remarks. F. Del Monte, P. Fréand M. PorratiA for Properties of the susy PCO Acknowledgments JHEP10(2016)049 flat / = 0 2) (A.9) (A.6) | (A.11) (A.10) (in the (0 susy dψ ξ Y , dθ -exact. The . Therefore, a d 2) | ¯ , θγ (3 d 2) i 2 | (3 SM = 2) form as follows ). Expanding the flat | a ψ c into ) A.9 dV ) ) = Vol ψ ¯ ψγ ( β (3) (A.7) c ψ 2 ) ( (A.8) ψ δ V 2 ( ψ M δ ( δ ∧ αβ ) 2 β 2) ) ι | 2) δ ) α is totally symmetric with respect | α . In this sense, the PCO ) = 0 β ab (0 st ψ ψ ι ι (0 γ ( γ ψ ( 2) ι | Y α ( ( 2 Y δ ι αβ β b 2 δ , ξ = 3. (3 ) ι ι + δ β a αβ V α αβ b ι ab i ι ) D α c γ d ∧ V ι 2) ( SM ab | ψV ) V b 1 γ = a αβ αβ ψ ( ) − V ∧ ) b ( a 2) ¯ b ψγ | ab ∧ ab Ω V There are two ways to compute the difference γ ¯ (0 γ h V – 14 – ψγ . ( = ( ∧ ( ψ d α Y b ∧ a β . ξ a a αβ 0) ι V | a = dθ L 2) α V ¯ | iγ ψγ (3 V ∧ i iι = = (0 susy ω = flat a , the fact that / abc α Y δ = = = 2 2) V γ 2) | | ψ ) flat -exact c (0 / (0 susy d = = γ 2) Y | Y ( 2) and | δ and 0) (0 susy | R (0 ss/flat Y (3 2) Y | dθ = ω . The first uses the fact that they are, from the mathematical point d a (0 st 2) γ ∧ cδ | = 1 we have the following Chevalley-Eilenberg cohomology class ¯ Y θγ and using the derivative on the Dirac delta functions, we can rewrite T (0 susy i with the flat one a 1 2 Y N 2) 2) V | | + (0 ss/flat (0 susy α and Y Y dx 2) | (0 st = as follows Y a = 0 by using the Fierz identities. Now we can construct a (3 V = 3, with flat / 0) | The flat Cartan-Maurer equations immediately imply that 2) D | (3 (0 susy is the Hodge dualbosonic to vielbeins the Chevalley-Eilenberg cohomologyY class ( which is the volume form of the supermanifold representative which is supersymmetric (it isdω written in terms of supersymmetric variables) and is closed: It is manifestlyIn invariant under supersymmetry, and satisfies an interesting equation. where and therefore Therefore, we can relate twowe PCO’s can by infinitesimal relate changes of the background. With that Poincaréduals belongs to the same cohomologysecond class. way to Thus, verify the this differencesupermanifold) is is of to the observe PCO that is the variation under a diffeomorphism to the spinorial indices, and the FierzRelation identity in between between of view, the Poincaréduals of embeddingsif of the two a embeddings submanifold gives two submanifolds in the same homology class the corresponding where we have used JHEP10(2016)049 , D , ]. Nucl. , = 1 (1988) 94. N -dimensions ]. Phys. Rev. SPIRE , IN 22 -exact (since it ][ d Teor. Mat. Fiz. [ SPIRE (1 + 1) supergravity IN , in the proceedings ]. ][ = 1 N SPIRE , indeed belong to the same ]. (1988) 1070 IN D ]. -Dimensions. 1. Supergravity -dimensions. 2. An action for a 3 [ 2) ]. ]. | Balanced superprojective varieties 77 (0 susy ]. Funct. Anal. Appl. arXiv:1503.07886 , Y SPIRE Superspace or one thousand and one [ SPIRE IN (2 + 1) (2 + 1) SPIRE SPIRE hep-th/0108200 IN [ IN SPIRE IN ]. and ][ [ [ IN ]. 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