aa r X i v : . [ h e p - t h ] D ec UUITP-49/20
Superconformal geometries and local twistors
P.S. Howe a, ∗ and U. Lindstr¨om b,c, † a Department of Mathematics, King’s College LondonThe Strand, London WC2R 2LS, UK b Department of Physics, Faculty of Arts and Sciences,Middle East Technical University, 06800, Ankara, Turkey c Department of Physics and Astronomy, Theoretical Physics, Uppsala UniversitySE-751 20 Uppsala, Sweden
Abstract
Superconformal geometries in spacetime dimensions D = 3 , , S -supersymmetry transformations become subsumed into super-Weyl transformations. Thenumber of component fields can be reduced to those of the minimal off-shell conformalsupergravity multiplets by imposing constraints which in most cases simply consists oftaking the even covariant torsion two-form to vanish. This must be supplemented byfurther dimension-one constraints for the maximal cases in D = 3 ,
4. The subject is alsodiscussed from a minimal point of view in which only the dimension-zero torsion is intro-duced. Finally, we introduce a new class of supermanifolds, local super Grassmannians,which provide an alternative setting for superconformal theories.
Dedicated to this year’s Nobel Laureate in Physics, Sir Roger Penrose,in appreciation of his many achievements, including the invention of twistor theory ∗ email: [email protected] † email: [email protected] ontents D = 3 D = 4 D = 6 D = 5
238 Minimal approach 259 Local Super Grassmannians 2610 Summary 27
Conformal symmetry has been extensively studied over the years because of its relevanceto various aspects of theoretical physics: two-dimensional conformal theory models andstatistical mechanics; four-dimensional N = 2 and 4 superconformal field theories; asan underlying symmetry that may be broken to Poincar´e symmetry in four-dimensionalspacetime, and as a tool to construct off-shell supergravity theories. It played an impor-tant role from early on in studies of quantum gravity [1]. Here we shall be interested in thegeometry of local (super) conformal theories as represented on bundles of supertwistorsover superspace.The first paper on D = 4 , N = 1 conformal supergravity (CSG) [2] used a formalismin which the entire superconformal group was gauged in a spacetime context. Althoughthis was not a fully geometrical set-up because supersymmetry does not act on spacetimeitself, but rather on the component fields, this paper nevertheless introduced the idea thatgauging conformal boosts and scale transformations could be very useful. After Poincar´esupergravity had been constructed in conventional (i.e. Salam-Strathdee [3]) superspace[4],[5], it was subsequently shown how scale transformations could be incorporated assuper-Weyl transformations [5, 6]. On the other hand, in the completely different approachto superspace supergravity of [7], super scale transformations were built in right from the1tart. However, it has turned out to be difficult to extend the latter approach to othercases, such as higher dimensions or higher N . The superspace geometry correspondingto all D = 4 off-shell CSG multiplets was given in [8], [9] using conventional superspacewith an SL (2 , C ) × U ( N ) group in the tangent spaces together with real super-Weyltransformations. The superspace geometries corresponding to most other off-shell CSGmultiplets have also been described, from the conventional point of view in D = 3 [10, 11,12] and from conformal superspace in D = 3 , D = 6(the (1 ,
0) theory) [18], [19], [20]. The D = 4 N = 2 theory [21] and the D = 6 (1 , D = 6 (1 ,
0) theory was formulated in in projectivesuperspace [23]. This theory as well as the D = 6 (2 ,
0) theory were recently discussed interms of local supertwistors in [24].In the non-supersymmetric case a standard approach is to work with conventionalRiemannian geometry augmented by Weyl transformations of the metric. The Riemanntensor splits in two parts, the conformal Weyl tensor, and the Schouten tensor which isa particular linear combination of the Ricci tensor and the curvature scalar. This objecttransforms in a connection-type way under Weyl transformations and can be used toconstruct a new connection known as the tractor connection [25, 26, 27]. This takes itsvalues in a parabolic subalgebra of the conformal algebra and acts naturally on a vectorbundle whose fibres are R ,n +1 (in the Euclidean case), thus generalising in some sense thestandard conformal embedding of flat n -dimensional Euclidean space. Although this ideadoes not carry over straightforwardly to the supersymmetric case, a similar construction,which does, can be made by replacing the ( n + 2)-dimensional fibre by the relevant twistorspace. This formalism is called the local twistor formalism and was introduced in D = 4in [29]. It has been discussed in the supersymmetric case for N = 1 , D = 4 byMerkulov, [30, 31, 32]. Such a formalism depends on the dimension of spacetime becausethe twistor spaces also change.In the following we shall take a slightly different approach to that of Merkulov in that westart from a connection taking its values in the full superconformal algebra. The conformalsuperspace formalism alluded to previously [13], [14] is a supersymmetric version of theCartan connection formalism [33], which was first mentioned in the superspace context in[34]. The formalism we advocate here can be thought of as an associated Cartan formalismin that the connection acts on a vector bundle rather than a principal one.In section 2 we briefly discuss bosonic conformal geometry starting from a local scale-symmetric formulation from which we recover the standard formalism; we also reviewthe rˆole of the Schouten tensor. We then give a brief outline of the local twistor pointof view in D = 3 , ,
6. A connection for the full superconformal group in the twistorrepresentation is given and the standard formalism is recovered by a suitable choice ofgauge with respect to local conformal boost transformations. This is then generalisedto the supersymmetric case starting with a quasi-universal discussion in section 3, andfollowed up by details of the D = 3 , D = 5 , N = 1 case, which is slightly different, is discussed. In all cases Global supertwistors and the relation to conformal supersymmetry were introduced in [28] D = 5 (N=1). In section 9 weintroduce a new class of supermanifolds which we call local super Grassmannians, whichcould be useful in constructing alternative approaches to the subject. In section 10, wemake some concluding remarks. From a mathematical point of view a conformal structure in n dimensions can be thoughtof as G -structure with G = CO ( n ), the orthogonal group augmented by scale transfor-mations, see, for example [26, 27]. This group does not preserve a particular tensor, butonly a metric up to a scale transformation, so that angles but not lengths are invariant.Similarly, a classical non-gravitational theory of fields which is locally scale invariant ina background gravitational field will be conformally invariant with respect to the stan-dard conformal group of flat spacetime when the metric is taken to be flat. So conformaltransformation are in this sense already present in a theory which is CO ( n ) invariant.Nevertheless, it can sometimes be of use to make conformal boosts manifest. This canbe done, for example, in the Cartan formalism which we shall describe shortly. We shallstart from a Weyl perspective in which the structure group is CO ( n ), so that, in additionto the usual curvature, there is also a scale curvature. It can easily be seen that thetorsion is invariant under shifts of the scale connection, and from there, one can eitheruse this shift symmetry to set the scale connection to zero, or introduce a new connectionin order to make the curvature invariant. The first method leads back to the conventionalapproach, whereas the second leads to the Cartan formalism with the symmetric part ofthe conformal boost connection identified with the Schouten tensor, which will be definedbelow.Let e a denote a local basis for the tangent space with dual basis forms denoted by e a . So e a = e am ∂ m ; e a = dx m e ma , where x m denotes local coordinates and where the coordinateand preferred bases are related to each other by the vielbein e ma and its inverse e am .Infinitesimal local co ( n ) transformations act on the frames by δe a = − ˆ L ab e b : δe a = e b ˆ L ba , (2.1)where ˆ L ab = L ab + δ ab S . (2.2)The parameter L ab denotes a local o ( n ) transformation preserving the flat metric η ab , and S is a local scale transformation, although we note that η ab is not invariant under thelatter.The torsion 2-form is defined by T a = ˆ De a = de a + e b ω ba + e a ω := De a + e a ω , (2.3)3here ω ab is the o ( n ) connection and ω := e a ω a the scale connection. In components T abc = f abc + 2( ω [ a,b ] c − δ [ ac ω b ] ) , (2.4)where f abc = 2 e [ am e b ] n ∂ m e nc . (2.5)It is clear that the torsion is invariant under shifts of ω a by a parameter X a , say, providedthat the o ( n ) connection tranforms by ω a,bc → ω a,bc + δ ac X b − η ab X c . (2.6)It is also clear that we can use ω a,bc to set the torsion to zero while still maintaining thissymmetry, and from now on we shall take this to be the case.The curvature 2-form is given by ˆ R ab = R ab + δ ab R where R is the scale curvature.Making use of the first Bianchi identity we find R ab,cd = C ab,cd + 4 δ [ a [ c Q b ] d ] (2.7)where the trace-free part C ab,cd is the Weyl tensor and where Q ab = P ab + 12 ( R ) ab , (2.8)with the symmetric part P ab being the Schouten tensor (also known as the rho-tensor).It is given by P ab = 1 n − (cid:18) R ab − n − η ab R (cid:19) . (2.9)Here R ab is the usual symmetric Ricci tensor and R the curvature scalar. Under a finiteshift of ω a accompanied by a (2.6) transformation we find∆ Q ab = ˆ D a X b − X a X b + 12 η ab X , (2.10)where ∆ denotes a finite change.At this stage one option is to use X to set the scale connection to zero. This gaugewill be preserved by a combined X transformation and a scale transformation providedthat X a = Y a := S − D a S = e am ( S − ∂ m S ) = e am Y m , (2.11)where S here denotes a finite scale transformation. So at this point we have regained theconventional formalism: the local scale transformations are no longer regarded as partof the tangent space group but instead are simply rescalings of the metric without anadditional scale connection. The transformation of P ab is given by (2.10) but with X replaced by Y , accompanied by an overall factor of S − . This factor then disappears in acoordinate basis so that the usual formula is recovered.In the conventional formalism one can define a new connection, the tractor connection,which takes its values in the Lie algebra of the conformal algebra so ( n + 1 , R n +1 , . However,the tractor formalism cannot be adapted directly to the supersymmetric case becausethe superconformal groups are not simply given by super Lorentz groups in two higherdimensions, one of which is timelike. Instead, one should think about supertwistorsbecause they naturally carry the fundamental representations of superconformal algebras.It is therefore more relevant to study local twistor connections, introduced in [29] in thenon-supersymmetric case. Then one has to consider different twistor spaces according tothe dimensions of spacetime. In general we can write an element of the conformal Liealgebra, h , in the form h = (cid:18) − a αβ b αβ ′ c α ′ β d α ′ β ′ (cid:19) (2.12)where α, α ′ etc denote spinor indices which have two components for D = 3 , D = 6. The diagonal elements a, d are Lorentz and scale transformationswhile the off-diagonal ones are translations, b , and conformal boosts c .The (Lie algebra valued) connection is A = (cid:18) − ˆ ω ef ˆ ω ′ (cid:19) (2.13)where each entry is a one-form with the index structure given in the previous equation.The diagonal elements are connections for Lorentz and scale transformations, while theoff-diagonal entries are the vielbein form e and the conformal connection f . The trans-formation of A is A 7→ g − A g + dg − g , (2.14)and the curvature F is defined by F = d A + A . (2.15)It transforms covariantly under g without a derivative term. Its components are given by F = (cid:18) − ˆ R αβ T αβ ′ S α ′ β ˆ R α ′ β ′ (cid:19) , (2.16)where the diagonal terms are the covariant Lorentz and scale curvatures, T is the torsionand S the conformal curvature. In terms of the standard torsion and curvature one has F = (cid:18) − ˆ R αβ − e αγ ′ f γ ′ β T αβ ′ Df α ′ β ˆ R α ′ β ′ − f α ′ γ e γβ ′ (cid:19) (2.17)The objective now is to construct an element of the conformal group depending on thescale and conformal parameters, and a connection one-form with values in the conformalalgebra which will transform in the required way provided that the transformation of theSchouten tensor is as given above in (2.10). The group element g is given by g = (cid:18) S − ,S − C S (cid:19) , (2.18)5here the diagonal elements involve unit matrices and C is a covector, C α ′ β . Under aconformal transformation of this form one can straightforwardly compute the changes inthe components of A to be ˆ ω ˆ ω − eC − Y , (2.19)ˆ ω ′ ˆ ω ′ − Ce − Y , (2.20) e eS , (2.21) f S − ( f − ˆ DC + CeC ) , (2.22)where again the index structure on the various elements follows from the original defini-tions, and where Y = S − dS . In order to compare with the previous discussion we needto eliminate the scale curvature and express the conformal boost parameter in terms ofthe scale parameter S . In addition, we use the Lorentz connection to set the torsion tozero as usual. The conformal connection f is a covector-valued one-form, f b = e a f ab , andwe can use the anti-symmetric part of f ab to set ( R ) ab = 0. If we take the trace of thetransformation of ˆ ω , which is proportional to the transformation of ω , we can see thatthe conformal boost parameter can be used to set ω = 0, so that R = 0 as well. Thisgauge will be preserved if C a ∝ Y a . This gives us the desired result: the scale curvatures R and R are both zero as is the antisymmetric part of f so that we can identify theremaining symmetric part f ( ab ) with the Schouten tensor P ab .These transformations agree with the usual ones in the conventional formalism, ex-pressed in spinor notation and with respect to an orthonormal basis. For the connectionform ω ab this translates to ω ab e b Y a − e a Y b . (2.23)For P ab we recover the standard transformation for the tractor connection which in anorthonormal basis reads P ab S − ( D a Y b + Y a Y b − η ab Y ) . (2.24)Now let us return to the Weyl picture with non-zero scale connection but still with thetorsion taken to be zero. If we redefine the Lorentz and scale curvature two-forms by R ab R ′ ab = R ab + 2 e [ a ∧ Q b ] R R ′ = R − e a ∧ Q a , (2.25)where Q b := e a Q ab , then we observe that the primed quantities are invariant underinfinitesimal X gauge transformations, for which δQ a = ˆ DX a . (2.26)We can also define a new curvature two-form R ′ a := ˆ DQ a . (2.27)6e can interpret X as a local conformal boost parameter, Q a as the corresponding gaugefield and R ′ a as its curvature. We have thus arrived at the conformal gauging picture start-ing from the Weyl perspective. In fact, we can identify the combined primed curvaturestogether with the ˆ R curvatures, and Q with f , in (2.17). Of course, this is just the con-verse to deriving the conventional point of view starting from the conformal perspectiveas discussed, for example, in [14]To conclude this outline of non-supersymmetric conformal geometry we briefly reviewthe theory of Cartan connections of which conformal gauging is an example. Let H, G beLie groups, H ⊂ G , with respective Lie algebras h , g let P be a principal H -bundle overa base manifold M . A Cartan connection on P is a g -valued form ω equivariant withrespect to H , and such that ∀ X ∈ h ω ( X ) = X and ω gives an isomorphism from T p P to g , for any point p ∈ P .A simple example is given by an n -dimensional manifold M with G = SO ( n ) ⋉ R n , H = SO ( n ). Then g = g − ⊕ h , where g − corresponds to translations and h to rotations.The translational part of ω is identified with the soldering form, i.e. the vielbein, whilethe h -part corresponds to an so ( n ) connection. In the conformal case h = g ⊕ g where g = so ( n ) ⊕ R , and g = R n . So g o corresponds to rotations and scale transformationswhile g corresponds to conformal boosts. The grading of the Lie algebra then correspondsto the dilatational weights of the various components. The curvature of ω , R = dω + ω ,also has components corresponding to this grading, and it is straightforward to see thatthey correspond to the torsion, the curvature and scale curvature, and the conformalboost field strength respectively. In this section we shall present a quasi-universal formalism for superconformal geometriesin D = 3 , A on superspace taking its values in the appropriatesuperconformal algebra acting on the super vector bundle whose fibres are super-twistors.These have the form Z α = u α v α ′ λ i . (3.1)The spinor indices on ( u, v ) are two-component for D = 3 , D = 6. The primed spinor indices are dotted indices for D = 4 and are the same as theunprimed ones in all other cases. The odd part of a super-twistor is λ i , where i = 1 , . . . M ,with M = N , the number of supersymmetries for D = 3 , M = 2 N for D = 6.The connection is given by A = − ˆΩ αβ E αβ ′ E αj F α ′ β ˆΩ α ′ β ′ F α ′ j ˜ F iβ ˜ E iβ ′ Ω ij . (3.2)7or D = 3 the pair ( αβ ) on ( E, F ) are symmetric whereas in D = 6 they are antisymmet-ric. The connection components are as follows: ( E αβ ′ , E αj ) correspond to translations and Q -supersymmetry, ( F α ′ β , F α ′ j ) to conformal and S -supersymmetry, while the diagonal Ωscorrespond to Lorentz symmetry above the line and internal symmetry below the linewith the hats indicating that the scale connection is also included. ( ˜ E, ˜ F ) on the thirdline are appropriate transpositions of ( E, F ). We will occasionally refer to the internalconnection as Ω I .The curvature two-form is given by F = d A + A ; its components in matrix form are: F = − ˆ R αβ T αβ ′ T αj S α ′ β ˆ R α ′ β ′ S α ′ j ˜ S iβ ˜ T iβ ′ R ij , (3.3)where T αβ ′ = ˆ T αβ ′ + E αk ˜ E kβ ′ T αj = ˆ T αj + E αγ ′ F γ ′ j S α ′ β = ˆ DF α ′ β + F α ′ k ˜ F kβ S α ′ j = ˆ DF α ′ j + F α ′ γ E γj ˆ R αβ = ˆ R αβ + E αγ ′ F γ ′ β + E αk ˜ F kβ ˆ R α ′ β ′ = ˆ R α ′ β ′ + F α ′ γ E γβ ′ + F α ′ k ˜ E kβ ′ R ij = R ij + ˜ F iγ E γj + ˜ E γ ′ i F γ ′ j . (3.4)Here ˆ D denotes the exterior covariant derivative for the Lorentz, scale and internalparts of the algebra. The first terms on the right of the first two equations are thestandard even and odd superspace torsion tensors constructed in the usual way from theˆΩ connections, ˆ T αβ = ˆ DE αβ , ˆ T αj = ˆ DE αj , while the non-calligraphic curvature forms onthe right in the last three lines are the standard curvature tensors constructed in a similarfashion.In D = 3 , E αβ ′ , E αj ) can be identified, after converting pairs of spinorindices to vector indices and, if necessary, rescaling, with the basis one-forms of conven-tional superspace, E A = ( E a , E αi ), while in D = 4, ˜ E iα ′ , which becomes the complexconjugate of E αi in real superspace, is also required in order to complete the basis forms.The forms E A are associated with super-translations, and play the role of soldering formsin this context. This means that the translational part of the algebra can be subsumedinto super-diffeomorphisms.In a similar fashion, we can identify the pair ( F α ′ β , F α ′ j ) with a super-covector-valuedone-form F B = E A F AB . In D = 4 the tilded odd forms are essentially the complexconjugates of the untilded ones (in real spacetime) and similar identifications can bemade in other dimensions. 8e remark in passing that, although there is also a conformal supergravity theoryin D = 5 for N = 1 [35],[36], which can be described in conventional superspace [16]and in conformal superspace [17], it does not admit a straightforward description in thesupertwistor formalism. In this case the spinor indices are four-component, with nodistinction between primed and unprimed, vectors can be represented by skew-symmetricsymplectic-traceless spinors, e.g. E αβ = − E βα , η αβ E αβ = 0, where η αβ is the symplecticmatrix (charge conjugation matrix), and where the spacetime part of the curvature is γ a -traceless, ˆΩ αβ ( γ a ) βα = 0. However, the bilinear fermion terms in the bosonic curvaturesin (3.4) do not preserve these contraints. Nevertheless, we shall see in section 7, that theformalism can be amended to take the D = 5 case into account.The Bianchi identity is DF := d F + [ F , A ] = 0 . (3.5)Written out in components this is, for the torsions,ˆ D T αβ ′ − E αγ ′ ˆ R γ ′ β ′ − ˆ R αγ E γβ ′ − iE αk ˜ T kβ ′ − i T αk ˜ E kβ ′ = 0ˆ D T αj + ˆ R αγ E γj − E αk R kj + i T αγ ′ F γ ′ j − iE αγ ′ S γ ′ j = 0 , (3.6)for the Lorentz, scale and internal symmetry curvatures, D ˆ R αβ − T αγ ′ F γ ′ β − T αk ˜ F kβ + E αγ ′ S γ ′ β + E αk ˜ S kβ = 0 D R ij − T γ ( i F | γ | j ) + 2 E γ ( i S | γ | j ) = 0 , (3.7)and for the superconformal curvatures,ˆ D S α ′ β + ˆ R α ′ γ ′ F γ ′ β + F α ′ γ R γ β + i S α ′ k ˜ F kβ + iF α ′ k ˜ S kβ = 0ˆ D S α ′ j + ˆ R α ′ γ ′ F γ ′ j − F α ′ k R κj + i S α ′ γ E γj − iF α ′ γ T γj = 0 . (3.8)We now consider the supersymmetric counterpart of the group element (2.18) given by g = S − S − C S Γ S − ∆ 0 1 (3.9)with inverse given by g − = S S − ˜ C S − − S − Γ − ∆ 0 1 , (3.10)where C + ˜ C − Γ∆ = 0 . (3.11)For the moment we can think of g as an element of the complex supergroup SL ( M | M ),where ( M | M ) denote the even and odd dimensions of supertwistor space. In the realcases we will find that ∆ is related to Γ and that C will have symmetry properties9epending on the case in hand. The transformation of the connection A is given by(2.14). For the superspace basis forms we find E αβ ′ SE αβ ′ (3.12) E αj S ( E αj + E αγ ′ Γ γ ′ j )˜ E iβ ′ S ( ˜ E iβ ′ − ∆ iγ E γβ ′ ) , while for the connections we findˆΩ αβ ˆΩ αβ − E αγ ′ C γ ′ β − E αk ∆ kβ − δ αβ Y (3.13)ˆΩ α ′ β ′ ˆΩ α ′ β ′ + ˜ C α ′ γ E γβ ′ − Γ α ′ k ˜ E kβ ′ − δ α ′ β ′ Y Ω ij Ω ij − ∆ iγ E γj + ˜ E iγ ′ Γ γ ′ j − ∆ iγ E γγ ′ Γ γ ′ j , For the S - supersymmetry connections we have F α ′ j S − (cid:16) F α ′ j − ˆ D Γ α ′ j + ˜ C α ′ β ( E βj + E αγ ′ Γ γ ′ j ) − Γ α ′ k ˜ E kβ ′ Γ β ′ j (cid:17) (3.14)˜ F iβ S − (cid:16) ˜ F iβ − ˆ D ∆ iβ + ( ˜ E iγ ′ − ∆ iγ E γγ ′ ) C γ ′ β − ∆ iγ E γj ∆ jβ (cid:17) , The conformal connection, F α ′ β , transforms as F α ′ β S − ( F α ′ β − ˆ DC α ′ β + Γ α ′ k ˆ D ∆ kβ + F α ′ k ∆ kβ − Γ α ′ k ˜ F kβ (3.15)+ ˜ C α ′ γ E γδ ′ C δ ′ β + ˜ C α ′ γ E γk ∆ kβ − Γ α ′ k ˜ E kδ ′ C δ ′ β ) . The transformations of the field strengths can also be found straightforwardly. For thetorsions we have T αβ ′ S T αβ ′ (3.16) T αj S ( T αj + T αγ ′ Γ γ ′ j )˜ T iβ ′ S ( ˜ T iβ ′ − ∆ iγ T γβ ′ ) , while for the Lorentz, scale and internal curvatures we have:ˆ R αβ ˆ R αβ − T αγ ′ C γ ′ β − T αk ∆ kβ (3.17)ˆ R α ′ β ′ ˆ R α ′ β ′ + ˜ C α ′ γ T γβ ′ − Γ α ′ k ˜ T kb ′ R ij
7→ R ij − ∆ iγ T γj + ˜ T iγ ′ Γ γ ′ j − ∆ iγ T γγ ′ Γ γ ′ j , The scale curvatures for D = 3 , R αβ given in (3.4), whilefor D = 4 we have to take the sum of the traces of ˆ R αβ and its complex conjugate; in allcases, we can write ( R ) AB = ( d Ω ) AB + kF [ AB ] , (3.18)10here k is a constant depending on the dimension of spacetime, and where F [ AB ] isthe graded anti-symmetric part of F AB , the latter having no symmetry. Since F A is aconnection one-form we are free to add a tensorial part to it and thereby set the scalecurvature R = 0, after which ( d Ω ) AB = − F [ AB ] . As in the bosonic case Ω transformsby a shift under conformal and S -supersymmetry transformations, as can be seen fromthe trace of the first line in (3.13), and can therefore also be set to zero. We are then leftwith the symmetric part of F AB which we can be identified as the super Schouten tensor.There is then a residual local super-Weyl invariance which we will present in more detailbelow for each case.To put more flesh on this general outline we shall now go through the various cases inturn. D = 3 The superconformal group in D = 3 is SpO (2 | N ). This is the same as the orthosymplecticgroup, but with the symplectic factor written first to indicate that it refers to the spacetimepart. It acts on the supertwistor space C | N (in the complex case) and consists of (4 | N ) × (4 | N ) matrices g which preserve the symplectic-orthogonal form J = − N . (4.1)The invariance condition is gJ g st = J (4.2)where the st superscript denotes the super-transpose of the matrix g . This is the same asthe ordinary transpose except for the odd component in the lower left corner which hasan additional minus sign. For a Lie superalgebra element h we have, correspondingly, hJ = − J h st (4.3)The reality condition needed to restrict to real spacetime (and superspace) is gKg ∗ = K (4.4)where K has a similar structure to J but with the minus sign on the second row replacedby a plus sign. The form of a real superalgebra element h is therefore h = a ib γic d δδ t − γ t e , (4.5)where d = − a t , b and c are symmetric, e is anti-symmetric, γ is real and δ is imaginary.Since the connection is a g -valued one-form it can be written as A = − ˆΩ αβ iE αβ E αj iF αβ ˆΩ αβ F αj ˜ F iβ ˜ E iβ Ω ij . (4.6)11ere E αβ and F αβ are real and symmetric, E αj is real and F αj is imaginary (for laterconvenience). On the bottom row ˜ E iβ = − E βi , while ˜ F iβ = F βi , where the o ( N ) indicesare raised or lowered by the flat Euclidean metric. The Lorentz and scale connections areˆΩ αβ = Ω αβ + 12 δ αβ Ω ; ˆΩ αβ = Ω αβ + 12 δ βα Ω , (4.7)with Ω αβ = Ω β α as the trace-free Lorentz connection and Ω the scale connection. Asusual two-component spinor indices are raised and lowered with the epsilon tensor, so Ω αβ is symmetric.It is straightforward to compute the components of the curvature two-form, but for themoment we shall focus on the scale curvature. It is given by R = R + E αβ F αβ − E αi F αi , (4.8)where R = d Ω . The last two terms can be rewritten as E αβ F αβ − E αi F αi = − E a F a − E αi F αi := − E A F A = − E B E A F AB , (4.9)Note that F AB is not necessarily graded-antisymmetric, although it is when contractedwith two sets of basis forms. (We have suppressed the wedge symbol in the two-formsabove). Here we have identified E αi as the standard odd basis forms of superspace and set E αβ = − E a ( γ a ) αβ , where E a are the standard even basis forms. (Note that this involvesa rescaling since there would normally be a factor of when going from bi-spinors tovectors.) We therefore have ( R ) AB = ( R ) AB − F [ AB ] . (4.10)As discussed briefly above, we can use the freedom to adjust a connection by a tensorialaddition to set ( R ) AB = 0, after which ( R ) AB = 2 F [ AB ] . As we shall see shortly below,superconformal gauge transformations (i.e. conformal boosts and S-supersymmetry) canbe used to set Ω = 0, after which F AB will become graded symmetric. We can thenidentify F AB as the super-Schouten tensor.We shall now exhibit a finite superconformal transformation which will allow us toset Ω = 0 and to identify residual superconformal transformations in terms of scaletransformations, or what one could call (finite) super Weyl transformations in this context.It is given by g = S − C S Γ S − Γ t . (4.11)where C = − iS − ( C + i t ) . (4.12)The parameters ( S, C,
Γ) are those for scale transformations, conformal boosts and specialsupersymmetry respectively. C is symmetric on its spinor indices (i.e. it is a vector) andreal, while Γ is taken to be imaginary. It is straightforward to check that this is indeed12n element of the superconformal group; one can find the effect of such a transformationon the connection by using the standard formula A → dg − g + g − A g . (4.13)For the moment we shall focus on the transformation of the scale connection Ω . It isgiven by Ω → Ω − S − dS − E αβ C αβ + E αi Γ αi = Ω − Y + E a C a + E αi Γ αi . (4.14)where the one-form Y := S − dS . We learn two things from this equation: first, theparameters C, Γ can be used to set the even and odd components of the one-form Ω tozero, and second, we can determine the residual symmetry transformations in terms of Y .In other words, having set Ω = 0 we have C a = Y a = S − D a S ; Γ αi = Υ αi = S − D αi S (4.15)where the derivatives are now the standard superspace covariant derivatives, and wherewe have denoted the odd component of Y by Υ for later use. The finite transformationsof the basis forms are: E a → SE a (4.16) E αi → S ( E αi − iE a ( γ a ) αβ Υ βi )The finite changes of the Lorentz and o ( N ) connections are:∆Ω αβ = E a ( γ a ) ( α | γ | ( Y γβ ) + i ) γβ ) ) + E ( αi Υ iβ ) (4.17)∆Ω ij = 2 E α [ i Υ αj ] + iE a Υ αi ( γ a ) αβ Υ βj The transformations of the superconformal connections are F αj S − (cid:0) F αj − D Γ αj + ( iC αβ + Γ ( αk Γ β ) k )( E βj + E αγ Γ γ j ) − Γ αk E βk Γ βj (cid:1) (4.18)and F αβ S − (cid:0) F αβ + iDC αβ − D Γ ( αk Γ β ) k + 2 F ( αk Γ kβ ) (4.19)+ C αγ E γδ C δβ + 2 iC ( αγ E γk Γ kβ ) + iC ( αγ E γδ (Γ ) δβ ) + Γ αγ E γk Γ β ) k + Γ αγ E γδ Γ δβ (cid:1) . Here Γ αβ := Γ αk Γ βk , is antisymmetric on α β , while the symmetrisations are on α, β only.The formalism given above applies quite generally regardless of whether any constraintshave been imposed or not. The basic constraint we shall choose for D = 3 (and in factin all cases) is T = 0, which is clearly superconformally invariant, from (3.16). We cannow use conventional constraints, including some for the conformal and superconformal13otentials, as well as the Bianchi identities, to show that we can always choose T = R = 0,remembering that we always choose the scale curvature to vanish. Thus in D = 3 theonly covariant field strengths that are non-zero are the internal curvature R I and thesuperconformal curvature S A = ( S a , S αj ). From the Bianchi identity (3.6) we can thensee that R αiβj,kl = ε αβ W ijkl , (4.20) R aβj,kl = ( γ a λ ) βjkl , R ab,kl = F abkl , for N ≥
2, where each field is totally antisymmetric on its SO ( N ) indices. In fact, for N ≥
2, the leading component in the Weyl multiplet, i.e. the conformal supergravity fieldstrength multiplet, is the leading non-zero component in R I .For N = 1 R I is identically zero, and only S A survives. From the Bianchi identitiesit is then easy to show that the only non-zero components are S ab,γ and S ab,c . Theformer is equivalent to a gamma-traceless vector-spinor, while the latter is equivalent toa symmetric traceless tensor. These are the Cottino and Cotton tensors respectively. Itis then straightforward to see that the component fields in the Weyl multiplet can bearranged diagrammatically as follows : [4 , ,
1] [3 , ,
2] [4 ,
0] [2 , ,
3] [1 ,
3] [0 , p, q ] entry denotes a field with p antisymmetrised internal indices and q sym-metrised spinor indices. For N ≤ N = 6 we note that there is an extra U (1) gauge field not included in the superconformalgroup. This field plays an important role in the BLG formalism for multiple membranes[39, 40], and was discussed in the superspace context in [12]. For N = 6, therefore, wehave an additional closed two-form field strength G with components G αiβj = ε αβ W ij (4.21) G aβj = ( γ a λ ) βj , This multiplet was discussed in the context of the supermembrane in [38].
14s well as G ab , where W ij is the SO (6) dual of W ijkl and λ i is the SO (6) dual of the 5-indexfermion on the left of the diagram above. For N = 8 an extra constraint is required inorder to avoid having two gravitons; this is achieved by imposing a self-duality constrainton W ijkl , and this in turn implies that the field content of the N = 7 and N = 8 Weylmultiplets are the same, so that the left-hand diagonal line can be terminated at N = 6.To conclude this section we translate the above results into conventional superspace.The main consequence is that the components of the superconformal potentials now ap-pear explicitly in the torsions and curvatures. These potentials are graded symmetric andmake up the components of the super-Schouten tensor as we remarked earlier. Makinguse of equation (3.4) which relates the conformally covariant tensors on the left to thestandard superspace ones on the right we find, for the even torsion two-form T a = − i E αi ( γ a ) αβ E βi , (4.22)or, in components, T αiβj c = − iδ ij ( γ a ) αβ ,T αibc = T abc = 0 , (4.23)which we recognise as the usual expressions [10]. For the odd torsion we find T αi = DE αi = ( γ a ) αγ E a ( E βj F βj,γi + E b F bγ i ) . (4.24)This implies for the components, T αi,βj γk = 0 ,T aβj γk = − ( γ a ) γδ F βj,δk ,T abγk = − γ γδ [ a F b ] δk , (4.25)where we have written the right-hand sides in terms of the super Schouten tensor. Forthe dimension-one component, since F αi,βj is antisymmetric under the interchange of pairsof indices, we have F αi,βj = ε αβ K ij + ( γ a ) αβ L aij , where K is symmetric on the internalindices and L antisymmetric, so that T aβj γk = ( γ a ) β γ K jk + ( γ b γ a ) βγ L bj k . (4.26)This differs slightly from the expression for the dimension one torsion given in [10, 12],but can be brought into agreement by a further redefinition of the dimension-one o ( N )connection. (On the other hand (4.26) does agree with the form given in [11].) Thedimension three-halves torsion is the gravitino field strength, which in three dimensionscan be dualised to a vector-spinor Ψ aγk . It is given in terms of the super Schouten tensorby T abγk = − γ [ a ) γδ F b ] δk , (4.27)where Ψ aαi := 12 ε abc Ψ bcαi = ε abc ( γ b ) αβ F cβi . (4.28)The components of the standard superspace curvature tensors can also be easily com-puted from equation (3.4), using the fact that the covariant Lorentz and scale curvaturesare zero. 15 D = 4 The superconformal groups for D = 4 N -extended supersymmetry are SU (2 , | N ), for N = 1 , , P SU (2 , |
4) for N = 4. Elements of this group are matrices g with unitsuperdeterminant which obey g ∗ J g = J (5.1)where J = N , (5.2)and where the numerical subscripts denote the dimenions of the unit matrices. For anelement h of the corresponding Lie superalgebra su (2 , | N ) we have h ∗ J = − J h (5.3)Such an element can be written h = − ˆ a − ib − iε − ic ˆ a ∗ ϕ − ϕ ∗ − iε ∗ e , (5.4)where b and c are hermitian, e is anti-hermitian, and the factors of i are put in forconvenience (we follow the conventions of [42] here). The supertrace condition then impliesthat − a + ¯ a = e kk , (5.5)where ˆ a αβ = a αβ + 12 δ αβ a . (5.6)The connection in D = 4 can be written A = − ˆΩ αβ − iE α ˙ β − iE αj − iF ˙ αβ ˆ¯Ω ˙ α ˙ β ¯ F ˙ αj − F tiβ − iE ∗ i ˙ β Ω ij . (5.7)and the curvature is F = − ˆ R αβ − i T α ˙ β − i T αj − i S ˙ αβ ˆ¯ R ˙ α ˙ β ¯ S ˙ αj − ( S t ) iβ − i ( T ∗ ) i ˙ β R ij , (5.8)where the expressions for the various components are given by (3.4) with appropriatefactors of i . We set ˆΩ αβ = Ω αβ + 12 δ αβ (Ω + i Ω ) , (5.9)where Ω and Ω are the scale and U (1) connections respectively and where Ω αβ is trace-free. 16 group element depending on scale and superconformal transformations is easily con-structed in a similar fashion to D = 3. It is given by g = S − C S ¯Γ − S − Γ t . (5.10)where C = iS − ( C + i t ) (5.11)with C being hermitian. The inverse is given by g − = S C S − − S − ¯ΓΓ t . (5.12)where ˆ C = − iS − ( C − i t ) . (5.13)As in D = 3 we can use the antisymmetric part of the superconformal connectiontogether with a superconformal transformation to set R = R = 0 and leave residualsuperconformal transformations in terms of the scale parameter S . The transformationof the scale connection Ω under g is given by∆Ω = − Y + E α ˙ α C ˙ αα − i E αk Γ αk − i E ˙ αk ¯Γ ˙ αk . (5.14)The right-hand side can be rewritten as − Y + C , where C = E A C A = E a C a + E αi Γ αi − ¯ E ˙ αi Γ ˙ αi (after rescaling C α ˙ α → C α ˙ α and Γ αi → i Γ αi ). We can then use C to set Ω = 0,leaving residual superconformal transformations with Y = S − dS = C .The scale curvature R can be written as R = R − E A f A , (5.15)where R = d Ω and where f a := 2 F a ; f αi := − i F αi ; ¯ f ˙ αi := i F ˙ αi . (5.16)We therefore have ( R ) AB = ( R ) AB − f [ AB ] . (5.17)Finally, we can use C A to set (Ω ) A = 0 and the graded antisymmetric part of the super-conformal connection f [ AB ] to set R = 0. After this, we have residual superconformaltransformations determined by S , as above, while the remaining part of f AB is gradedsymmetric and can be identified as the super-Schouten tensor for D = 4.17he super-Weyl transformations of the basis forms and the connections are given by: E a SE a ,E αi S E αi , Ω αβ Ω αβ − E α ˙ γ ( C ˙ γβ + 4 i ¯Υ ˙ γk Υ βk ) + 2 E αk Υ βk , Ω ij Ω ij − E γj Υ γi + ¯ E ˙ γi ¯Υ j ˙ γ + 2 i Υ γi E γ ˙ γ ¯Υ ˙ γj ) , (5.18)where it is understood that the trace over α and β is to be projected out in the thirdline. The components of the super-Schouten tensor at dimension one are f αi,βj , f jαi, ˙ β pluscomplex conjugates. Graded symmetry then implies that f αi,βj = ε αβ f ( ij ) + f ( αβ )[ ij ] , (5.19)while f jαi, ˙ β is skew-hermitian. At dimension three-halves there is just a complex vector-spinor f aβj which can be identified with that part of the gravitino field-strength tensorwhich is not part of the Weyl supermultiplet.The basic constraint for D = 4 is T a = 0, as for the other cases. Using the Bianchiidentities and choices for the connections, including the superconformal ones, one findsthat the non-zero components of the covariant torsion T γk are T i j γk ˙ α ˙ β = ε ˙ α ˙ β χ γijk , T j kα ˙ α,β, ˙ γ = i ¯ M jkαβ , T jα ˙ α,β,γk = − i
12 ( N − δ jk ¯ χ lmn ˙ α χ βlmn , T kα ˙ α,β ˙ β, ˙ γ = ε αβ ¯Ψ k ˙ α ˙ β ˙ γ . (5.20)Here, vector indices have been converted to pairs of spinor indices using the sigma-matricesin the usual way: v a → v α ˙ α = ( σ a ) α ˙ α v a . (5.21)In (5.20) multiple internal indices are antisymmetrised, while the spinor indices on M and¯Ψ are symmetrised. The leading components of the Weyl multiplets are Ψ , M and χ for N = 1 , , N = 4 there is also a complex dimension-zero scalar field thatdoes not appear directly in the components of the torsion tensor. Instead it parametrisesthe coset space SU (1 , /U (1). The U (1) factor here is part of the superspace internalconnection, but for N = 4 not part of the gauge group of the superconformal group whichis only SU (4). An additional constraint is required at dimension one to enforce this. It is D αi χ α ijk = 0 . (5.22)The components of the covariant curvature can easily be found; for example, at dimen-sion one, they are [9]: 18 αiβj,γδ = 0 , R αiβj, ˙ γ ˙ δ = − ε αβ ¯ M ˙ γ ˙ δij , R αiβ,kjl = − ε αβ ¯ B ijkl , R jαi ˙ β,kl = − ¯ χ ˙ βikm χ jlmα , (5.23)where B ijkl = 12 D αl χ αijk . (5.24)Further details of the torsions and curvatures, including the Schouten terms, can befound in [8, 9]. D = 6 The (1 ,
0) theory in D = 6 has been discussed in SU (2) superspace in [23] and inconformal superspace [18], [19], [20]. The theory was also discussed earlier in harmonicsuperspace in [22] and, some years ago, in projective superspace [23]. In components, the(2 ,
0) supergravity dates back twenty years to [43]. Below we summarise the local twistorformulation of (1 ,
0) and (2 ,
0) 6 D conformal supergravity recently presented in [24].The complex conjugate of a four-component D = 6 spinor u α is denoted ¯ u ˙ α but thisrepresentation is equivalent to the undotted one as there is a matrix B α ˙ α relating the two,¯ u ˙ α = ¯ u α B α ˙ α . B is unitary, B ∗ B = 1, and satisfies B ¯ B = − Similar remarks hold forthe inequivalent spinor representation denoted by a lower index, v α say. So a twistor z consists of a pair of 4-component spinors and can be written z = (cid:18) u α v α (cid:19) . (6.1)A supertwistor in D = 6 can therefore be written in the form Z = u α v α λ i . (6.2)where i = 1 , . . . N for ( N,
0) supersymmetry, N = 1 ,
2. Here ( u, v ) are commutingobjects while λ is odd. The superconformal group is OSp (8 | N ) in complex superspaceand preserves the orthosymplectic metric K , so for an element g of the group we have gKg st = K = J . (6.3) We use the six-dimensional conventions of [44]. st denotes the supertranspose, which is the same as the ordinary transpose exceptfor an additional minus sign for each element in the bottom left (odd) sector. The 2 N × N matrix J is the Sp ( N ) symplectic invariant. In real spacetime we need to impose thereality constraint gRg ∗ = R = B − B N . (6.4)An element of the Lie superalgebra, h , has the form h = a αβ b αβ ε αj c αβ d αβ ϕ αj λ iβ ρ iβ e ij . (6.5)The orthosymplectic constraint implies that b and c are skew-symmetric and d = − a t , asbefore, while eJ = − J e t . (6.6)In indices, setting ( J ) ij = η ij , this implies e ij := e ik η kj = e ji . (6.7)For the odd components we have ρ = J ε t λ = J ϕ t . (6.8)or, in indices, ρ iβ = η ij ε βj ⇒ ρ iβ = − ε βi ,λ iβ = η ij ϕ βj ⇒ λ iβ = − ϕ βi . (6.9)Next we need to impose reality in order to move to real superspace. This is done withequation (3.3) but this time with R extended by the unit matrix in the odd-odd sector,as in (6.4). The result of imposing gRg ∗ = g , at the Lie algebra level is that a, b, c and d obey the same conditions as in the bosonic case while e satisfies e = − e ∗ . (6.10)For the independent odd components of h we have:¯ ε ˙ αi = − η ij ε βj B β ˙ α , ¯ ϕ i ˙ α = ( B − ) ˙ αβ η ij ϕ βj . (6.11)These constraints simply mean that ε αi and ϕ αi are symplectic Majorana-Weyl spinorsas one would expect. They are respectively the parameters for Q and S supersymmetrytransformations. Indices are raised and lowered according to the rule: X i = η ij X j ⇔ X i = X j η ji with η ik η jk = δ ji . A is A = ˆΩ αβ iE αβ E αj iF αβ ˆΩ αβ F αj − F iβ − E iβ Ω ij , (6.12)where E A = ( E a , E αi ), with E a = ( γ a ) αβ E αβ , will be identified with the even and oddsuper-vielbein one-forms of the underlying superspace, F A = ( F a , F αi ) is the connectionfor superconformal transformations, i.e. S -supersymmetry and standard conformal trans-formations, ˆΩ αβ ( ˆΩ αβ ) is the Lorentz plus scale connection and Ω ij the internal sp ( N )connection. On the bottom line, F iβ and E iβ are transposes of F αj and E αj with the inter-nal index lowered by ( J ) ij = η ij . The curvature two-form, F = d A + A , has componentsgiven in matrix form by: F = ˆ R αβ i T αβ T αj i S αβ ˆ R αβ S αj −S iβ −T iβ R ij , (6.13)where, from (3.4), with α ′ α , and with appropriate factors of i , T αβ = ˆ DE αβ + iE αk E kβ , T αj = ˆ DE αj + iE αγ F γj , ˆ R αβ = ˆ R αβ − E αγ F γβ − E αk F kβ , R ij = R ij − F iγ E γj − E γi F γ j , S αβ = ˆ DF αβ + iF αk F kβ , S αj = ˆ DF αj + iF αγ E γj . (6.14)Here, ˆ D is the superspace covariant exterior derivative with respect to scale, Lorentz andinternal symmetries, while the leading terms on the right, for the top four lines, are thestandard superspace and torsion and curvature tensors for the corresponding connections(extended by the scale connection).The detailed form of the Bianchi identity DF = 0 is given, mutatis mutandis , by (3.6)to (3.8). We shall now repeat the steps carried out in the non-supersymmetric case toreduce the conformal and superconformal boost parameters to derivatives of the scaleparameter. We introduce a group element g ( S, C,
Γ) where S is a scale parameter andΓ αi is an S -supersymmetry parameter. It is given by g = S − iS − ˜ C S Γ S − J Γ t , (6.15)where the index structure is as above, in (6.12) for example, where J is the Sp ( N )invariant discussed previously, and where˜ C + ˜ C t + i Γ J Γ t = 0 . (6.16)21f we write ˜ C = C − i J Γ t , (6.17)then, from (6.16), C is antisymmetric since Γ J Γ t is symmetric. Reality implies that C = BC ∗ B , ¯Γ = − B − Γ J . (6.18)Note also that the latter equation implies that Γ J Γ = B (Γ J Γ) ∗ B .Under such a transformation the components of A transform as follows: E αβ SE αβ ,E αj S ( E αj + iE αβ Γ βj ) ,F αβ S − (cid:16) F αβ − ˆ DC αβ − i ( ˆ D Γ [ αk )Γ β ] k + 2 iF [ αk Γ β ] k + i ˜ C γα E γδ ˜ C δβ − C γ [ α E γk Γ β ] k (cid:17) ,F αj S − (cid:16) F αj − ˆ D Γ αj + i ( E βj + iE βγ E γj ) ˜ C βα − E βk Γ αk Γ βj (cid:17) , ˆΩ αβ ˆΩ αβ − E αγ ˜ C γβ − E αk Γ βk + 12 δ αβ Y , Ω ij Ω ij + Γ αi E αj + Γ αj E αi + i Γ αi E αβ Γ βj . (6.19)The curvature transformations are obtained from those for the potentials by replacingthe latter by the former in the equations above. In addition, for the superconformalcurvatures, the derivative terms in (6.19) must be replaced by curvature terms as follows:ˆ DC αβ R [ αγ C | γ | β ] , ˆ D Γ αj R αβ Γ βj + Γ αk R kj . (6.20)If we take the trace of the third equation in (6.14) we find that2 R = 2 R − E A F A , (6.21)where we have defined the super-vector-valued one-form F A = ( F a , F αi ). By adjustingthis potential we can choose R = 0 so that the (graded) antisymmetric part of F AB isnow proportional to R AB . Taking the trace of the transformation of ˆΩ αβ we find2Ω − E a C a − E αi Γ αi + 2 Y , (6.22)so that we can use the parameters C a and Γ αi to set Ω = 0. This leaves residualtransformations determined by the scale parameter S , C A = 2 Y A = 2 S − D A S , (6.23)22here C A = ( C a , Γ αi ). We shall take the components of Y to be given by Y A = ( Y a , Υ αi )in order to clearly distinguish the even and odd components where necessary.The basic constraint that we shall choose is to set the even torsion two-form to zero, T a = 0 , (6.24)which is clearly covariant. Using this, conventional constraints corresponding to connec-tion choices (including superconformal ones) and the Bianchi identities, one finds that thecovariant torsions ( i.e. the torsion components of F ) are given by T αi βjγk = 0 , T aβj γk = ( γ a ) βδ G γδj k := ( γ bc ) βγ G abcj k , T abγk = Ψ abγk , (6.25)where G abcjk is anti-self-dual on abc (by its definition), anti-symmetric on jk and symplectic-traceless on jk for N = 2, and where Ψ abγk is the gamma-traceless gravitino field strength.For the curvature tensor components we find R αiβj,kl = 0 , R αiβj,cd = 4 i ( γ a ) αβ G acdij , R aβj,cd = − i γ a Ψ bc − γ c Ψ ab − γ b Ψ ca ) βj , R aβj,kl = − γ a χ ) β ( k,l ) j , R ab,cd = C ab,cd , R ab,kl = F ab,kl . (6.26)The dimension three-halves field χ αi,jk is antisymmetric on jk : it is a doublet for N = 1while for N = 2 it is in the of sp (2), i.e. it is symplectic-traceless on any pair ofindices. The graviton field-strength C ab,cd has the symmetries of the Weyl tensor, while F ab,kl is the sp ( N ) field-strength tensor. We have thus located all of the components ofthe conformal supergravity field strength supermultiplets except for the dimension-twoscalars which are given by C ij,kl = D αi χ αj,kl . (6.27)for the (1 ,
0) case this reduces to a singlet, while for (2 , C ij,kl is in the 14-dimensionalrepresentation of sp (2). Fuller details of this multiplet can be found in [24].The standard superspace torsion tensors differs from the covariant ones by componentsof the Schouten tensor F AB . D = 5 The D = 5 superconformal theory, which exists only for N = 1, where the algebra is f (4)[37], has been described in (conformal) superspace in [16], [17]. However,it does not fit23nto the supertwistor formalism as well as the other cases due to the constraints that onehas to impose on the top left quadrant of A (starting from the D = 6 , (1 ,
0) case). Theseare ( γ a ) αβ ˆΩ βα = 0 ,η αβ E αβ = 0 ,η αβ F αβ = 0 , ( γ a ) αβ ˆΩ βα = 0 , (7.1)where α = 1 , . . . i = 1 , F , but the fermion bilinears in the A term donot preserve them. Thus with the same definition of F as before, F = d A + A , thecurvatures in the top left quadrant should be defined byˆ R αβ = F αβ −
14 ( γ a ) αβ ( γ a ) γδ F δγ , T αβ = F αβ − η αβ η γδ F γδ , S αβ = F αβ − η αβ η γδ F γδ , ˆ R αβ = F αβ −
14 ( γ a ) αβ ( γ a ) γδ F δγ , (7.2)where the F s on the right-hand side are the top left quadrant entries in F , and where η αβ is the 4 × D = 5); the oddtorsion, S-supersymmetry curvature and internal curvature are defined as before. Therelations between the covariant curvatures and the standard superspace ones is the sameas in (6.14) with the difference that the unwanted terms have to be projected out as inthe preceding equation. For example, we have T αβ = T αβ − i h E αk E βk i , ˆ R αβ = ˆ R αβ − E αγ F γβ − h E αk F kβ i , (7.3)where the angle brackets indicate that the symplectic trace terms on the first line andthe ( γ a )-trace terms on the second line have been projected out. Note that the middleterm on the right in the second equation is automatically ( γ a )-traceless because E αγ and F γβ are now five-dimensional vectors, so that multiplying by γ a and taking the trace isidentically zero.We now briefly describe the constraints and the geometry that follows from them. Asin the other cases we set T αβ = 0 from which the even torsion two-form T αβ takes itsusual γ -matrix form, as can be seen from the first line in (7.3) on setting the left-hand-sideto zero. The only non-zero torsion components are at dimension one and three-halves.The former is T aβj γk = δ jk (cid:0) ( γ bc ) βγ G abc + 2( γ b ) βγ G ab (cid:1) , (7.4)24here G abc is the dual of G ab , G abc = 12 ε abcde G de . (7.5)The field G ab can be identified as the leading component of the D = 5 , N = 1 Weylmultiplet. The covariant dimension three-halves torsion is the gamma-traceless gravitinofield strength. Note that (7.4) can be obtained by dimensional truncation from the D = 6 , N = (1 ,
0) expression. The remaining components are a dimension three-halvesspinor, χ αi , and, at dimension two, the Weyl tensor C ab,cd , the sp (1) field strength F ab,kl and a scalar field C . The curvature components are formally the same as the D = 6 , (1 , R αiβj,cd = 4 iε ij ( η αβ G cd + ( γ a ) αβ G acd ) , (7.6)while the dimension three-halves spinor χ αi,jk → ε i ( j χ αk ) . Finally, the dimension twoscalar can be defined to be C = D αi χ αi . (7.7)The standard superspace torsions and curvatures can be obtained from the covariantones in the same way as in D = 6 , (1 , γ a tracesprojected out. The Schouten terms come from the equation for the odd torsion in (6.14). We shall now describe the superspace geometry corresponding to these conformal su-pergravity multiplets from a minimal perspective. Since the details coincide with thosederived previously we will be brief. We define a superconformal structure on a super-manifold with (even | odd) dimension ( D | D ′ ) to be a choice of odd tangent bundle T (ofdimension (0 | D ′ ) which is maximally non-integrable, so that the even tangent bundle T is generated by commutators of sections of T , and such that the Frobenius tensor, F ,defined below, is invariant under R ⊕ spin (1 , D − ⊕ g , where g is the internal symmetryalgebra for the case in hand: g = so ( N ) for D = 3, u ( N ) for D = 4, sp ( N ) for D = 6,with N = 1 ,
2, and for D = 5, with N = 1. The components of F with respect to localbases E αi for T and E a for T ∗ are given by F αiβj c = h [ E αi , E βj ] , E c i = − ik ij ( γ c ) αβ , (8.1)where k ij = δ ij for D = 3 and η ij for D = 5 , h , i denotes thepairing between vectors and forms. The R factor denotes an infinitesimal scale trans-formation, δE a = SE a , δE αi = − SE αi , while the spin and symplectic algebras act inthe natural way on the spacetime and internal indices. For D = 4, T is the sum of twocomplex conjugate bundles of dimension 2 N , T = T ⊕ ¯ T , and we have F αiβjc = h [ E αi , E βj ] , E c i = 0 , (8.2) F i j c ˙ α ˙ β = h [ ¯ E i ˙ α , ¯ E j ˙ β ] , E c i = 0 , F j cαi ˙ β = h [ E αi , ¯ E j ˙ β ] , E c i = − iδ ij ( σ c ) α ˙ β .
25e now introduce connections for sp ( N ) and spin (1 , D −
1) and define the torsion andcurvatures in the usual way. Note that this procedure involves the complementary basis E αi for T ∗ which is only determined modulo T ∗ , i.e. shifts of the form E αi E αi + L αib E b . (8.3)We could include this in the structure group, along with a corresponding connection,but we shall instead follow the standard procedure of using this freedom to impose someadditional constraints at dimension one-half. In addition we shall not include a scaleconnection so that we have the standard superspace geometrical set-up.It is clear that the Frobenius tensor is invariant under scale transformations of theform E a SE a , E αi S − E αi (with the same transformation for ¯ E i ˙ α in D = 4). If weimpose constraints to determine E a and E αi at dimension one-half we can then determinetheir transformations as well. A convenient one to consider, as it does not involve anyconnection terms, is ( h [ E αi , E b ] , E c i ) ( bc ) = 0 , (8.4)where the symmetrisation is understood to include lowering the c index on the left-handside. Making finite super-Weyl (scale) transformations, we find that this constraint willbe preserved if E a S − (cid:0) E a + i ( γ a ) γδ Υ γ k E δk (cid:1) ,E ai S (cid:0) E αi − iE a ( γ c ) αβ Υ βi (cid:1) , (8.5)where Υ αi = S − D αi S . The super-Weyl transformations for the Lorentz and internalconnections are given in (4.17) for D = 3, (5.18) for D = 4 and (6.19) for D = 6. The D = 5 case can be derived from D = 6 by dimensional truncation.Identifying F αiβj c with the dimension-zero torsion T αiβj c , imposing suitable constraintson various components of the torsion corresponding to fixing the odd basis E αi using(8.3) and making appropriate choices for the spin (1 , D −
1) and g connections, one canshow, with the aid of the usual superspace Bianchi identities and some algebra, that thecomponents of the torsion and curvature tensors can be chosen to agree with those derivedpreviously. The finite super-Weyl transformations in the general case were given in (5),from which it is straightforward to obtain particular cases.In addition to the fields of the conformal supergravity multiplet, this geometry will alsocontain the components of the super Schouten tensor F AB , whose transformations can befound in (6.19). We can recover the covariant forms for the torsions and curvatures byreversing the steps made earlier. In the previous sections we have considered superconformal geometries starting from localsupertwistors in various dimensions. In all cases we have used the standard split of su-pertwistor spaces into even twistors together with additional odd components. However,26e can choose different ( even | odd ) splittings which naturally give rise to formulations ofsuperconformal geometry on different supermanifolds. These include chiral, projective[46], [47],[49] and harmonic superspaces [48],and can be thought of from the viewpoint ofsuper flag manifolds as discussed from a pure mathematical perspective in [50],[51] andpresented in a more accessible form in [42]. More recently the present authors have usedthem to discuss super-Laplacians and their symmetries in the context of rigid supersym-metry [52, 53, 54]. In this section we shall give a brief discussion of some cases relevant tofour-dimensional complex spacetime, deferring a fuller discussion to a later publication.Grassmannians are spaces of p -dimensional planes in C n , with p < n , and super Grass-mannians are spaces of ( p | q ) planes in C m | n , with p ≤ m, q ≤ n (but with p + q < m + n ).For example, complex Minkowski space in D = 4 can be thought of as an open set in theGrassmannian of 2-planes in C ,. In the super case, consider D = 4-dimensional space-time with N = 2 supersymmetries. The corresponding supertwistor space is then C | ,and the Grassmannian of (2 |
1) planes in C | is known as analytic superspace. A super-twistor, i.e. an element of C | can be split into two halves u A and v A ′ , where A = ( α, A ′ = ( α ′ , ′ ), where we have used a prime to denote a dotted two-component spinorindex. This allows us to rewrite a supertwistor as Z A = (cid:18) u A v A ′ . (cid:19) . (9.1)We can then define a superconformal connection A acting on supertwistors in this basisin the form A = (cid:18) − ˆΩ AB E AB ′ F A ′ B ˆΩ A ′ B ′ (cid:19) , (9.2)with corresponding curvature F = d A + A = (cid:18) − ˆ R AB T AB ′ S A ′ B ˆ R A ′ B ′ (cid:19) . (9.3)The action of the superconformal group on A and F is formally the same as in thebosonic case, equation (2.14), but note that the diagonal transformations are not simplyscale and Lorentz transformations, because the diagonal subgroups are also supergroups.This is therefore a somewhat different way of looking at superconformal transformationswhich we propose to investigate further in a forthcoming paper.Finally, we note that such geometries can also be thought of super versions of para-conformal geometries [55].
10 Summary
In this article we have discussed the supergeometries describing off-shell conformal super-gravity multiplets in D = 3 for N = 1 to 8, in D = 4 for N = 1 to 4, and in D = 6for ( N,
0) supersymmetries with N = 1 ,
2, from the perspective of local supertwistors . We note that there is a related construction in four dimensions described in [56].
27n this formalism one introduces connections taking their values in the superconformalalgebras in the twistor representation, which can be thought of as an associated versionof the Cartan connection formalism. Similarly, the conformal superspace construction[13] is an adaption of the Cartan formalism to superspace, and is expected to be equiv-alent to our local supertwistor approach.. From this starting point one can then derivethe standard superspace formalism in a systematic fashion. In order to specialise to theminimal off-shell conformal supergravity multiplets one then has to impose constraints.In addition, we discussed the D = 5 , N = 1 case for which a slight amendment of theformalism is necessary. We also showed that the same results can be obtained fromthe minimal formalism in which only the dimension-zero torsion, or Frobenius tensor, isspecified. This minimal formalism was previously applied to D = 3 [10, 12], where anadditional self-duality constraint at dimension one is required, while it was also previouslyshown in the D = 4 case that the supergeometries also follow from the dimension-zerotorsion constraint [8]. In this case an additional dimension one constraint is required toensure the correct number of component fields. In section 10 we briefly introduced theidea of local super Grassmannians. Such superspaces are best viewed in terms of different( even | odd ) splittings of supertwistor space. Acknowledgement:
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