N = 1 supercurrents of eleven-dimensional supergravity

Journal of High Energy Physics, May 2018

Abstract Eleven-dimensional supergravity can be formulated in superspaces locally of the form X × Y where X is 4D N = 1 conformal superspace and Y is an arbitrary 7-manifold admitting a G2-structure. The eleven-dimensional 3-form and the stable 3-form on Y define the lowest component of a gauge superfield on X × Y that is chiral as a superfield on X. This chiral field is part of a tensor hierarchy giving rise to a superspace Chern-Simons action and its real field strength defines a lifting of the Hitchin functional on Y to the G2 superspace X × Y . These terms are those of lowest order in a superspace Noether expansion in seven N = 1 conformal gravitino superfields Ψ. In this paper, we compute the O(Ψ) action to all orders in the remaining fields. The eleven-dimensional origin of the resulting non-linear structures is parameterized by the choice of a complex spinor on Y encoding the off-shell 4D N = 1 subalgebra of the eleven-dimensional super-Poincaré algebra.

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N = 1 supercurrents of eleven-dimensional supergravity

Accepted: May = 1 supercurrents of eleven-dimensional Katrin Becker 0 1 2 Melanie Becker 0 1 2 Daniel Butter 0 1 2 William D. Linch III 0 1 2 Texas A 0 1 2 M University 0 1 2 0 Y that is chiral as a 1 College Station , TX 77843 , U.S.A 2 George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy Eleven-dimensional supergravity can be formulated in superspaces locally of the form X Y where X is 4D N = 1 conformal superspace and Y is an arbitrary 7-manifold admitting a G2-structure. The eleven-dimensional 3-form and the stable 3form on Y de ne the lowest component of a gauge super eld on X super eld on X. This chiral eld is part of a tensor hierarchy giving rise to a superspace Chern-Simons action and its real eld strength de nes a lifting of the Hitchin functional Extended Supersymmetry; M-Theory; Supergravity Models; Superspaces - HJEP05(218) on Y to the G2 superspace X Y . These terms are those of lowest order in a superspace Noether expansion in seven N = 1 conformal gravitino super elds . In this paper, we compute the O( ) action to all orders in the remaining elds. The eleven-dimensional origin of the resulting non-linear structures is parameterized by the choice of a complex spinor on Y encoding the o -shell 4D N = 1 subalgebra of the eleven-dimensional super Poincare algebra. Keywords: Extended Supersymmetry, M-Theory, Supergravity Models, Superspaces 1 Introduction 2 Review and summary Decomposition of the metric Decomposition of the 3-form Chern-Simons and Kahler actions Additional gravitino super elds 4 5 6 7 2.1 2.2 2.3 2.4 3.1 3.2 3.3 3 Extended supersymmetry and the gravitino supercurrent The main result The -transformations and the gauge-for-gauge symmetry of Determining F and Gij from transformations Connecting with 11D superspace The conformal supergravity supercurrent Toward higher-order terms and a more covariant formulation Outlook A 4D superspace and supergeometry B Useful variational expressions B.1 The origin of covariantized transformations B.2 Arbitrary variations of the Chern-Simons and Kahler actions realized in superspace [8] when the theory is truncated to four dimensions [9, 10]. As with all theories representing more than nine supercharges, the super-Poincare algebras close only up to the component eld equations of motion [11]. That is, these supersymmetries and the superspaces in which they are manifest are \on-shell". An alternative when only a part of the Poincare algebra is required to be manifest is to represent only the supersymmetries associated to the subalgebra linearly. That is, we may contemplate describing the eleven-dimensional supergravity theory as a bration { 1 { by simpler superspaces. The simplest such choice (or at least the most well-developed by far) is 4D N = 1 superspace. This superspace has many user-friendly features (e.g. o shell, nitely many auxiliary elds, chiral representations) not the least of which is being the most relevant phenomenologically. We therefore propose to study eleven-dimensional supergravity in superspaces that are locally of the form X Y where Y is any Riemannian 7-manifold and X is a curved 4D N = 1 superspace. This program was initiated in [12{16]. In this paper, we extend the construction of eleven-dimensional supergravity in superspace by deriving the supercurrents for the 4D N = 1 conformal supergravity and seven conformal gravitino prepotentials (the spin-2 and spin- 32 parts) to all orders in the remaining (spin 1) super elds. This is a necessary step in the determination of the action to all orders in the gravitino super eld. Additionally, it elucidates the eleven-dimensional origin of the conformal compensator elds: when supergravity is navely switched o , these are super uous component elds required for the superspace embedding of the 3-form hierarchy. However in the superspace splitting X Y , local superconformal symmetries emerge in addition to those inherited from eleven dimensions. The consistency of our super eld description hinges on the fact that precisely the seemingly-super uous components \compensate" for these fake symmetries. (They are their Stuckelberg elds.) We stress that the conformal symmetry is in the X factor and arises as an artifact of the splitting.1 In section 2, we review the results of our previous work [12{16] on the embedding of eleven-dimensional supergravity into o -shell superspaces X Y locally of the form R4j4 R7. The component elds of eleven-dimensional supergravity consist of an elf-bein, a gravitino, and a 3-form gauge eld. Decomposing these elds in a (4 + 7)-dimensional split leads to the embedding of the 3-form components into a collection of 4D N = 1 p-form super elds with p = 0; : : : ; 3 that are q-forms with q = 3 p in the additional seven directions. The original eleven-dimensional abelian gauge symmetry with 2-form gauge parameter decomposes into a set of gauge transformations in 4+7 dimensions that transform these super elds into each other. In addition, the p-forms in the tensor hierarchy are charged under the Kaluza-Klein vector gauging di eomorphisms along Y . This gives rise to a non-abelian gauging of the tensor hierarchy and a Chern-Simons-like action which we embedded in superspace [12, 13, 15]. One of the component elds in the tensor hierarchy is a [0; 3]-form, that is a 0-form in spacetime X and a 3-form in the internal space Y . It is naturally contained in the bottom component of a chiral multiplet; since this bottom component is required by supersymmetry to be complex, the pseudo-scalar [0; 3]-form is naturally paired with a scalar [0; 3]-form which is invariant under the abelian gauge symmetry. This object de nes a symmetric bilinear form that can be taken to be positive-de nite for generic 7-manifolds Y [21]. As such, it de nes a Riemannian volume on Y , and the super eld containing it de nes a Kahler potential on the superspace. Modi ed by a certain function of another of the tensor hierarchy super elds, this de nes a second N = 1 supersymmetric action which, together with the Chern-Simons action, describes the dynamics of elds with 4D spin 1Attempts have been made to de ne the superconformal symmetry directly in eleven-dimensional superspace [17, 18], and to use it to represent all 32 supersymmetries o shell [19, 20]. { 2 { HJEP05(218) A peculiar feature of this action is its invariance under global 4D N = 1 superconformal transformations, which include both scale (Weyl) and chiral U( 1 )R transformations. It was argued in [14] that this is naturally enhanced to a local superconformal symmetry when coupling to N = 1 supergravity. It was shown in [16] that the coupling to N = 1 supergravity and the additional seven gravitino multiplets could be achieved at the linearized level | that is, considering the action quadratic in uctuations about an on-shell 4D Minkowski background with an arbitrary internal manifold of G2 holonomy and vanishing 4-form ux. The explicit N = 1 superspace action was given and its component action was shown to match that obtained directly from 11D. Our goal in this paper is to proceed beyond the approximations of [14, 16] by constructing the coupling of both N = 1 conformal supergravity and additional seven gravitino super elds to the non-linear action of the elds of lower 4D spin. Speci cally, we will expand in super elds (denoted Ha and i below) containing component spins 32 but keeping the exact non-linear dependence on the remaining super elds at each order. To lowest order, this was done in reference [14] and checked to reproduce the non-linear scalar potential upon component projection. In section 3, we extend this result to the leading order in the seven gravitino super elds i by constructing the complete supercurrent J i (cf. (3.8)), and motivate its derivation with a careful analysis of the symmetries of the conformal gravitino super elds. The intricate compensator mechanism associated to this superconformal symmetry has a strikingly simple interpretation when it is derived from eleven dimensions. As we show in section 4, reducing the 11D N = 1 superspace frame to 4D N = 1 can be done in di erent ways parameterized by a complex scalar and a complex spinor of SO(7). By studying the gauge transformations of this parameterization, we discover that the bilinears of the spinors (dressed appropriately with the modulus and phase of the scalar) reproduce the transformation rules de ning the superconformal compensators. With the gravitino couplings thus understood, we turn in section 5 to their superconformal graviton analog. Similarly to the superconformal gravitino multiplet, our analysis has been perturbative in the Y -dependence of the 4D N = 1 superconformal graviton: in references [14, 15] the 4D N = 1 SG part was treated non-linearly as a Y -independent background. In reference [16], Y -dependent uctuations were studied but only to quadratic order in all elds. In section 5, we begin to address this point by giving the transformations of the gauged tensor hierarchy elds under the 4D N = 1 superconformal symmetry to the rst non-trivial order, now with Y -dependent gauge parameter. We then derive the complete 4D N = 1 conformal supercurrent by requiring invariance under these extended symmetries. This completes the superspace construction to lowest non-trivial order in the spin-2 and 32 components including dependence on all eleven dimensions but treating the spin 1 components exactly. To go beyond this order in spin elds requires understanding the non-linear terms in the gravitino expansion. In section 6 we take the rst step in this 3 2 direction by constructing modi cations of the hierarchy eld strengths that are invariant under part of the seven extended superconformal symmetries. { 3 { We begin by reviewing our previous results [12{16] on the description of eleven-dimensional . supergravity in terms of superspaces of the form X Y where X is a 4D N = 1 supermanifold and Y is a real 7-manifold. Locally X is of the form R4j4 with coordinates (xm; ; and indices m = 0; : : : ; 3 and ; = 1; 2 from the middle of the alphabets. Following the early/late convention, tangent indices are taken from the beginnings (e.g. a, , and ). Local coordinates on Y will be denoted by yi with i = 1; : : : ; 7; we will generally not need tangent indices for Y . The body of X (i.e. its bosonic part) will be denoted by X and eleven-dimensional indices on X Y will be denoted in bold so that, for example, and 3-form (section 2.2) and their superspace gauge transformations, eld strengths and Bianchi identities. With these ingredients, we build an invariant action consisting of a superspace volume term and a Chern-Simons-like term (section 2.3). We conclude our review by introducing the gravitino super elds and de ning the gravitino expansion (section 2.4). 2.1 Decomposition of the metric Let's rst discuss the elds that arise from decomposing the 11D metric, as these will play a role in de ning the covariant derivatives in 4D N = 1 superspace. We employ the standard Kaluza-Klein decomposition (2.1) (2.2) gmn = gmn + AmiAnj gij Amj gji : ! gij Anj gij A priori, the 4D metric gmn, the Kaluza-Klein gauge eld Ami, and the Y polarizations of the metric gij each depend on all eleven coordinates. The 4D metric gmn is encoded in a real super eld H _ = ( a) _ Ha with a linearized gauge transformation H _ = D _ L D L _ : This de nes it as an irreducible superspin- 32 representation: at the component level it contains the spin-2 polarizations of the frame ema, the N = 1 gravitino m , and an auxiliary vector eld dm. The gauge symmetry (2.2) actually encodes local N = 1 superconformal transformations, so that the frame is de ned up to an overall Weyl rescaling, the gravitino is de ned up to a shift in its spin-1/2 part corresponding to the action of the special superconformal S-supersymmetry, and dm is de ned up to chiral U( 1 )R transformations for which dm is the gauge eld. In [16], the super eld Ha appeared explicitly, but this quickly becomes unwieldy when going beyond quadratic order. It is usually much simpler to employ a curved Wess-Zuminotype superspace where the prepotential Ha is encoded in the super-vielbein EM A. The natural superspace to employ is 4D N = 1 conformal superspace [22], where the N = 1 superconformal symmetry is explicitly gauged and which describes precisely (in supergeometric language) the eld content of Ha. However, there is a caveat to employing this { 4 { superspace: We have to assume that the super-vielbein (and its prepotential Ha) do not depend on y, since in the language of [22], the super-vielbein is gauge invariant under any internal gauge symmetries. In the context of our split spacetime, these include the GL(7) di eomorphisms of Y . There is no technical obstruction to developing a superspace that relaxes this condition, but it does not yet exist, and we will revisit this point in section 6.2 The non-abelian Kaluza-Klein gauge elds Ami are described by a real unconstrained prepotential Vi. Just as with the supergravity prepotential, it is more convenient to use a covariant description where the Kaluza-Klein multiplet is encoded in super-connections AM i which covariantize the N = 1 superspace derivatives r, so that the super-connections transform as Ai = r i with i a real super eld describing internal di eomorphisms. The Kaluza-Klein connections deform the algebra of the conformal superspace derivatives by introducing the curvature terms i is a chiral eld strength obeying the standard conditions r _ W i = 0 and i = r _ W _ i. Here Lv denotes the Lie derivative along the vector eld v 2 X(Y ). interior product (contraction) with the vector eld v. In contrast to the super-vielbein, the connection A may depend on y. The remaining Y polarizations gij of the metric are 28 real scalars from the point of view of X. We will address their embedding into super elds presently. 2.2 Decomposition of the 3-form The eleven-dimensional 3-form splits up into a \tensor hierarchy" of p-forms, Cmnp ! Cmnp ; Cmn k ; Cm jk ; Cijk which are embedded into a tower of p-form super elds. (The remaining 28 scalars from gij will also be embedded in these elds.) Being forms also in the seven directions, they are charged under the non-abelian gauge eld. The abelian part of the gauge transformation is parameterized by the super elds ij (chiral), Ui (real), and (chiral) encoding the components of an eleven-dimensional super-2-form. Their weights are summarized in table 2. The non-abelian part g = diff(Y ) acts by the Lie derivative with respect to the real scalar super eld i . 2Actually, this is not as constraining a scenario as it might seem. Because the N = 1 supergeometry describes the metric only up to Weyl rescalings, if the external metric gmn(x; y) factorizes as gmn ! e'(x;y)gmn(x), with y-dependence sequestered in a conformal factor, then the super-vielbein can be taken to just describe the y-independent piece. In realistic scenarios, this would describe the actual background con gurations of interest. Then an explicit y-dependent Ha could be introduced to describe general ydependent uctuations about that background. { 5 { (2.3) (2.4) X i Vij ijk V i Cmnp | | | | 2-forms vectors scalars auxiliaries Cmn i | | | | | | Cm ij | gmi G Hi | | Cijk, Fijk dX | dij fijk d i of the gauged Chern-Simons super eld hierarchy. The bosons G, Hi, and seven of the Fijk can all be removed by a choice of Wess-Zumino gauge (cf. section 2.4). The super elds of the hierarchy (not including the KK vector just described) are as follows (we mention only the embedding of bosons, summarized in table 1): Scalars. There are 35 chiral elds ijk containing the 35 pseudo-scalars Cijk from the 3-form and 28 metric scalars gij . (The remaining seven scalars will be shown to be pure gauge.) These transform under non-abelian internal di eomorphisms and under the abelian tensor hierarchy gauge transformations as Vectors. There are 21 real, unconstrained vector super elds Vij containing vector elds It has two eld strengths Eijkl (chiral) and Fijk (real) and F = (Vij is the vector multiplet prepotential we introduce next.) The lowest component of Fijk is related to a Riemannian metric gij by 1 144 pg gij := klmnpqrFiklFmnpFjqr : As we explain in detail in section 4, this metric is a generalization of the G2 structure metric and Fijk that of a G2 structure for the internal manifold Y . It is not in general closed or co-closed. Cm ij of the 3-form. These transform as Their chiral eld strengths W ij V = L V + 1 2 W should not be confused with those of the KK vectors. = L 2i 1 2i { 6 { (2.5) (2.6) (2.7) (2.8) (2.9) 2-forms. There are seven 2-form gauge elds Cmn i in the chiral spinor super elds i (2.10) (2.11) (2.13) (2.14) (2.15a) (2.15b) (2.15c) (2.15d) (2.15e) (2.15f) !h( ; v) := r v + _ r _ v + r v + r _ _ v : (2.12) It contains an excess of seven real scalars in the bottom component of Hi, but these will be shown to be pure gauge. 3-form. The 3-form Cmnp is encoded in a real scalar super eld X with gauge transformation and a \reduced" chiral eld strength X = L X + 1 2i r r _ _ !h(W ; U ) G = 41 r2X + W The bottom component of G is a complex scalar, which can be interpreted as a the chiral compensator super eld of (modi ed) old minimal supergravity [23, 24] (also known as 3-form supergravity [25])]. Its phase can be eliminated by a choice of U( 1 )R gauge and its modulus can be eliminated (or absorbed into the metric) by a choice of Weyl gauge. Being given explicitly in terms of the prepotential super elds, the eld strengths are the solutions to the Bianchi identities The eld strength Hi is real where we de ne the shorthand H = r _ _ + W : !h(W; V ) 1 2 1 2i r W 0 = 2i E E 1 2 4 r r F = 41 r2H = r _ G = 0: W W expressing the fact that these forms are closed in the extended de Rham complex [15]. These rules and de nitions are compatible with local superconformal symmetry on X. In particular, we can consistently assign scaling dimensions ( ) and U( 1 )R weights (w) in addition to engineering dimension (d) to all the gauge parameters, prepotentials, and eld strengths. These are summarized in table 2. { 7 { G(X) H( ) W (V ) F ( ) W (V) L . constructing F ^ F in superspace. This was the approach taken in reference [15] where it was shown that the composite superforms are given by F = E + E F W = EW 4i r2(F r F ) H = E + E H + !(W; F ) G = EG + W W 1 2 4i r2(F H) : ir F W F + ir _ F W _ F (2.16a) (2.16b) (2.16c) (2.16d) These complicated composite elds satisfy the same Bianchi identities (2.15) as the eld strengths after which they are named. (This apparently highly non-trivial fact is simply the fact that the wedge product of closed forms is closed [15].) Because of these closure relations, the invariant action can be written SCS = 12LCS = i 1 Z 2 Z d 2 Here E and E are the full superspace and chiral superspace volume densities, respectively.3 The superconformal weights of these measures can be found in table 3. 3The volume densities are further discussed in appendix A. They are y-independent as they are built invariance under internal di eomorphisms. purely from the curvatures themselves. The most general possibility is given by [16] SK = 3 Z 2 d x The ingredients are as follows (cf. section 2.2): g(F ) = det(gij (F )) is the determinant of the Riemannian metric on Y obtained from Fijk using (2.7). This factor is needed for the integrand to be a scalar density under internal di eomorphisms [14, 21, 26]. G is the superconformal primary chiral super eld strength carrying the 4-form eld strength Fmnpq along the four-dimensional spacetime. It has weights (3; 2; 1) (cf. supergravity. Its presence ensures the proper conformal weight of the integrand. F is (for now) an arbitrary analytic function of the Weyl-invariant combination x := jHj2 = (GG) 2=3 gij HiHj (2.19) of the 3-form eld strengths Fmnp i. Its precise de nition will be given later. We will refer to SK as the Kahler action.4 One of our goals of this paper is to determine F from invariance under the extended (non-manifest) supersymmetry (cf. section 3.1). 2.4 Additional gravitino super elds In addition to the above supergravity-coupled action, we must introduce gravitino superelds that capture the dynamics of the seven additional spin-3/2 elds and which make the full action invariant under extended supersymmetry. These should be contained within an unconstrained spinor super eld i subject to a large gauge transformation that eliminates most of its component elds. The goal is to determine that gauge transformation, both for the gravitini and for the matter elds, and determine what constraints it places on the function F (x). In [16], we constructed the second-order action for uctuations about a at Minkowski background times a G2-holonomy manifold Y with closed and co-closed 3-form 'ijk(y) and its corresponding G2-holonomy metric gij (y). Two prepotentials took background values: h i = i' and hXi = 2. The uctuation elds and the gravitino super eld were subject to 4Strictly speaking, this is a misnomer since a Kahler potential depends on chiral multiplets and vector multiplets gauging their isometries. The deformation considered here also contains the tensor multiplets (chiral spinors) the linearized gravitino gauge transformations 0 1 1 ijk = 1Vij = i = 1Vi = i = i + gij D 1 1 2i '~ijklD2 l 2i 'ijk( k j k ) 1 2 i ( i + i) : (2.20a) (2.20b) (2.20c) (2.20d) (2.20e) i is a chiral spinor and i is an unconstrained complex super eld. Here we have assigned a gravitino weight 1 to , , and , and denoted the above transformations by 0 and 1 corresponding to how they change the gravitino weight of the corresponding eld. One expects on general grounds that the full non-linear action takes the form of a power series expansion in the gravitino super eld i. Schematically, S11D = S0 + S1 + S2 + O( 3) ; where S0 consists of the zeroth-order Kahler (2.18) and Chern-Simons actions (2.17). The extended supersymmetry transformations should act on the super elds schematically as = 0 + 1 + 2 + : : : where 0 is non-vanishing only for the gravitino and corresponds to the non-linear generalization of (2.20a). Then solving the equations S11D = 0 order-by-order, that is, 1S0 + 0S1 = 0 ; 2S0 + 1S1 + 0S2 = 0 ; etc. we can determine the higher-order modi cations of the action as well as higher-order modi cations to the supersymmetry transformations. Note that these transformations are in nitesimal (thus rst-order) in and , and so n will be O( n ) while for any other eld n will be O( n 1). In this paper, we will be concerned with the rst step of the Noether procedure, that is, determining the complete rst-order modi cation S1 to the action and the full contributions to 1 for the matter elds and 0 for the gravitino super eld. The complete expressions for these quantities are given in eqs. (3.4), (3.6), and (3.7) of the next section. 3 Extended supersymmetry and the gravitino supercurrent We begin this section by summarizing our main result: the non-linear extensions to the linearized transformation rules (2.20) and the correction to the action to rst order in gravitino weight. Subsequently, we will motivate these results, discuss their underlying physics, and sketch some of the derivations. where r 0 r . Then the variation of constituents 0 and V, so that As we are employing a covariant framework for the Kaluza-Klein gauge prepotential Vi, it appears implicitly in almost all of our formulae. For example, the spinor covariant derivatives r and r _ are given as r = eiLV r0 e iLV = r 0 LA ; r _ = e iLV r0_ eiLV = r _ 0 LA _ ; 0 does not possess the KK connection. This means that covariantly chiral superelds such as must be understood as = e iLV 0 where 0 is chiral with respect to has two parts arising from the variation of the independent ; on top of this, it is convenient to introduce additional pieces for its covariantized transformation. Similarly, additional terms are naturally included in the covariantized transformations of the other super elds. We nd (3.1) (3.2) (3.3a) (3.3b) (3.3c) (3.3d) are both (3.4a) (3.4b) (3.4c) (3.4d) (3.4e) V := V + := := + iL V ; 1 for the prepotentials of the tensor hierarchy. We emphasize that and chiral with respect to r. (Additional comments can be found in appendix B.1.) In terms of these covariantized transformations, the rst-order extended SUSY transformations are 9F 0 HmF npqHq(GG) 2=3 (GG)1=3 : (3.5) F ; 1 4 G i 2 i 2 1 2 G G G F + ; 1 2 i 2 i 2 G : H + G H ; In the transformation rule for , we have introduced the composite 4-form pg F (GG)1=3 = 3! 1 pg ijklmnp F mnp(F + jHj2F 0) The antisymmetric symbol ijklmnp is a tensor density with entries 1, and upper indices in the second line of (3.5) have been raised with the metric gij (F ). As we will explain presently, the zeroth-order transformation of the gravitino super eld turns out to be 0 i = i + G Gij r j + i W ij j ; where i is covariantly chiral and i is an unconstrained complex super eld, as in the linearized theory. In the full theory, we see that the second term is dressed with a factor of G as well as a complex rank-2 tensor Gij (see eq. (3.12) for its de nition) and a third term involving the 2-form contribution to the action is eld strength W ij has appeared. The rst-order gravitino gauges the seven supersymmetries not manifest in N = 1 superspace, J i can be interpreted as the Noether supercurrent for these supersymmetries. One can easily check that the rst line counters the transformation of the Kahler term whereas the second line counters the transformation of the Chern-Simons term. Further requiring -invariance generates no new contributions to the supercurrent, but instead determines the function F (x) (recall eq. (2.19)) in terms of as the solution to the quartic polynomial S1 = 1 Z 2 where the complex super eld J i is given by F^ := F 2xF 0 4 F x ^4 + F^3 1 = 0 : Together these imply, for example, that F 0(x) = 112 F^2 and F (x) = 13 F^ + 23 F^ 2, so that both F and its rst derivative can be expressed in terms of F^. Both F (x) and F^(x) possess in nite series expansions: F (x) = 1 + x 12 x 2 144 + ; F^(x) = 1 x 12 x 2 48 + + Both functions are monotonic. Whereas F (x) slowly increases without bound, F^(x) slowly tends to zero. Finally, the function Gij in the gravitino transformation (3.6) is given by Gij = (GG) 1=3 ^ 1gij + F 2i F^Fijk gkl Hl(GG) 1=3 : It has ( ; w; d) = ( 2; 0; 0) and is a Hermitian matrix: the rst term is real and symmetric, and the second term is imaginary and antisymmetric. These results appear rather complicated, but there proves to be a great deal of structure that constrains them. In the remainder of this section, we will motivate these results and sketch their derivations. -transformations and the gauge-for-gauge symmetry of Let us begin by justifying the transformations. We have chosen to x the gravitino's conformal and U( 1 )R weights as in table 2, and to identify i with no additional factors as in (3.6). Taking into account the engineering and conformal dimensions of the various elds, it is easy to see that we cannot assign a transformation to Vij , X, or Vi, provided we expect only eld strengths to appear on the right-hand-side of the covariantized transformations. Now keeping in mind the linearized transformation rule (2.20d) for i, the only nonlinear modi cation consistent with chirality involves a factor of G correcting the conformal and U( 1 )R weights. Incidentally, this is also why it is necessary to choose i to have a lower GL(7) index. If the index were raised, we would need to lower it with a chiral metric, but no chiral metric exists. appears only as r _ i and engineering dimension arguments. Considerations of various weights and chirality similarly restrict the transformation of the remaining prepotential ijk to at most be of the form (3.4a). The coe cient can be determined by requiring -invariance of the terms in the Chern-Simons action that are purely chiral | that is, terms that cannot be lifted to full superspace by eliminating an overall r 2 factor. Because the supercurrent term (3.7) is a full superspace integral, the purely chiral terms must cancel on their own, and this is only possible if transforms as in (3.4a). (To verify this, it helps to use the expression (B.5) for the general variation of the Chern-Simons action.) The transformations now uniquely determine the rst-order gravitino action (3.7). Because these transformations can be written purely in the language of di erential forms, knowing nothing of the metric gij or the function F , the Kahler and Chern-Simons actions must be canceled separately by their respective gravitino terms. The rst line of (3.8) exactly cancels the transformations of ijk and i in the Kahler term, whereas the second line of (3.8) can be seen to cancel the Chern-Simons term. The only terms in S1 we might not determine in this way are those that are -invariant | that is, terms where the gravitino . But no such terms can be written down by virtue of conformal Similar arguments can be made to motivate the transformations. There is no transformation one can postulate for i that is consistent both with chirality and its various weights. The other prepotentials are more subtle. Let us focus rst on Vi. The transformation given in (3.4e) matches in the linearized approximation and is the only nonlinear possibility up to multiplication by an overall function of the weightless combination jHj2 (2.19). To eliminate this possibility requires some physical insight. At the linearized level, the gravitino super eld transformation (2.20a) is subject to a gauge-for-gauge symmetry whereby can be shifted by an antichiral super eld. (This is necessary so that the counting of the degrees of freedom is correct: an without such a shift symmetry would introduce an additional gauge invariance absent in the linearized theory.) The transformation (3.4e) respects this gauge-for-gauge symmetry, but any such jHj2 modi cations would break it. That is, if i is an antichiral super eld i, the transformation (3.4e) leads to 1Vi = 12 (G i + h.c.), which is the form of a chiral gauge transformation of V, i = i + i, see section 3.1 of [16]. A compensating chiral gauge transformation with i = 12 G i leaves V invariant. A similar argument xes the transformations of Vij and X. To x the -transformation for ijk and to uncover the full expression for the transformation of the gravitino is more subtle. A practical approach is to rst require invariance around a eld con guration with Hi = 0. An obvious ansatz here is to generalize the linearized results to (introducing factors of G for weight) (3.13a) (3.13b) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) 1 2 ijk = term does not actually violate the gauge-for-gauge symmetry, as one can counter the chiral shift in by a shift of i involving W ij . In order for this chiral shift symmetry to hold at higher order in Hi, the i W ij j term that we have added cannot be dressed by any function of jHj2. So the most general expectation for the gravitino is the expression (3.6) for some complex rank-2 tensor Gij agreeing with (GG) 1=3gij when Hi vanishes. (The exact expression (3.12) cannot be so simply determined.) Let us record here the explicit form for the gauge-for-gauge symmetry: i = i ; i = G i ; i = iW ij j ; r _ i = 0 : We now have su cient information to completely determine the ijk transformation to all orders in Hi. Let's assume that the full transformation of is 2 r i 2 Zijkl l + 3 [iW jk] for some unknown covariant tensor Zijkl. This must reduce to (3.13b) in the limit where H vanishes. Now under the gauge-for-gauge symmetry (including the appropriate non-abelian transformation), we nd i = gij G(GG) 1=3 r j + ; ijk = 2 r i 2 (GG)1=3F~ijkl l + i = gij G(GG) 1=3 r j + i W ij j + O(H) : canceled if Thus, This transformation in the W W part of the Chern-Simons action (2.17) can only be i acquires the term i W ij j . (Here the results of appendix B are useful.) ijk = 2 i lr2Zijkl + 3i lW [liW jk] + 4i l@[l ijk] : This cannot be made to vanish since one cannot choose a covariant Zijkl to cancel the other two terms. What one can do instead is to arrange for ijk to generate a trivial symmetry of the action by taking ijk / ijklmnp l S0 ; mnp so that S0 = 0 automatically. This requires (pg (GG)1=3F ) 2i F[ijkHl] : The rst term is the variation of the Kahler term with respect to . The F ^ H term ^ W combines with the W terms in (3.17) to give the variation of Chern-Simons action with respect to . This leads to the full expression (3.4a), and agrees (as it must) with the result when Hi vanishes. We have not yet justi ed the forms of F and Gij . These will come from requiring invariance under transformations, but we can already make a few comments about Gij . Because it can only be built from Fijk, G, and Hi (no derivatives may appear if the action should be two-derivative), Gij = (GG) 1=3 c0(x)gij + c1(x)FijkgklHl(GG) 1=3 + c2(x)HiHj (GG) 2=3 (3.20) where the form factors ci are complex functions of jHj2 (2.19). (The factors of GG give the correct Weyl weights.) Spacetime parity further constrains c0 and c2 to be real and c1 Deriving the form factors and the expression for F is a relatively straightforward if laborious task. A number of ingredients aid in this. Collected in appendix B.2 are the general variations of the zeroth-order actions. The variation of the Chern-Simons action is naturally written as (B.5) in terms of the covariantized variations of the potentials (3.4). The Kahler term is more naturally written as (B.6) in terms of variations of the eld strengths themselves. These are derived from the variations of the prepotentials (including the Kaluza-Klein prepotential) and (focusing only on the terms) are given by 1F = 2 LG F + 4 r 1 2 (GG)1=3 i 8 r U + 2 (F ^ H) + r (F ^ H) i 4 r 4 r i 2 iHi G i 2 r _ ir _ Hi G i W W G ; 2i W F ) + iG W _ r _ F 1G = 1H = + i i i 1 2 2 (LG + 2 LG H + 1 (W W _ ) + c.c. An analogous expression can be given for W ij , but it does not appear in the Kahler action so it is not as useful. In casting the result in this form, we have pushed derivatives onto as much as possible. Note that the rst term in each expression (3.21) corresponds to a uniform internal di eomorphism, so this variation just leads in the Kahler action to a total internal derivative. A tractable approach to checking invariance (which we followed to completion) is to work order-by-order in W ij and W immediately, while the linear ones lead to the form factors (3.12) i. The terms quadratic in these eld strengths cancel c0 = F^ 1 ; c1 = 6iF^ 1F 0 = 2i F^ ; c2 = 0 ; (3.22) HJEP05(218) and conditions (3.10). The explicit calculation is not particularly enlightening, so we do not reproduce it here. Instead, in section 6 we will brie y sketch a more covariant approach that should lead to a simpli cation of this check and also permit computation of the gravitino terms to higher order. Because of the complexity of the actions and transformation rules, it is easy to lose sight of an important fact: under very mild assumptions about the basic structure of the and transformations, the complete rst-order transformations of the matter elds and the gravitino have been determined, as well as the function F in the Kahler term. It is clear that this should not be otherwise | we are striving to describe 11D supergravity and that theory is unique | but it is heartening to observe that the consistent Noether coupling of the additional seven gravitini does indeed determine the action and transformations uniquely. In the next section, we will show how to make the connection more transparent. Connecting with 11D superspace 4 The and transformations associated with the extended supersymmetry of the gravitino super elds describe a rather large gauge group consisting of a chiral spinor and an unconstrained complex super eld . As we have reviewed in [16], these transformations permit a Wess-Zumino gauge condition on where only ve components survive: the extended gravitini vector ym i, and a spinor i . 5 m i, a complex anti-self-dual antisymmetric tensor tab i, a complex At the end of the day, two remnants of the super eld transformations survive: the extended supersymmetry transformation i corresponding to the bottom component of ij and a bosonic transformation with parameter r under which ym i transforms as a gauge eld. Here let us denote this parameter zi and normalize it as the weight-less quantity i 4 zi := (GG)1=3 2 i : r j This bosonic gauge transformation played a critical role in arriving at the correct linearized action. There are 16 extraneous scalar elds encoded in the N = 1 super elds not present in 11D supergravity. Two of these (the bottom components of G) are Weyl and U( 1 )R compensators, which are natural from the point of view of N = 1 superspace. The other 14 turned out to be pure gauge degrees of freedom that could be removed by the bosonic gauge transformation involving zi. While this leads to a consistent description of the underlying physics, it is slightly puzzling. After all, we could have imagined descending from eleven dimensions directly and de ning our four dimensional elds. How would these 14 degrees of freedom appear in that dictionary? In this section, we will answer this question by descending directly from 11D superspace to 4D N = 1 conformal superspace. Eleven-dimensional superspace was introduced in [2, 3] to describe (on-shell) 11D supergravity. (Our conventions here di er slightly to admit a closer connection to normalizations and conventions used in 4D N = 1 superspace.) We denote 11D vector and spinor 5Actually, contains only the gamma-traceless part of the extended gravitini. The spin-1/2 components are encoded in some of the fermions in the matter elds. (4.1) indices with hats. An 11D vector decomposes as V a^ = (V a; V a) and a 32-component Majorana spinor decomposes as ^ = ( I ; _ I ) where and _ are two-component chiral and antichiral spinor indices for SO(3; 1), and I denotes an SO(7) spinor index, which can be raised or lowered with IJ . The 11D charge conjugation matrix is real and antisymmetric, C ^ ^ = 0 IJ ^ = 0 _ _ IJ ^C ^ ! Spinor indices are raised and lowered as Spin(10; 1) gamma matrices ( a^)^ ^ to be pseudo-Hermitian, obeying and ^ = ^ C ^^. We take the ya^ = a^ . These where the SO(7) gamma matrices ( a)IJ are imaginary and antisymmetric. The superspace is described by a supervielbein EM^ M^ A^ B^ valued in SO(10; 1) so that M^ ^ ^ = 14 M ^ a^^b( a^^b)^ ^ and M^ ^^b = M^ a^ ^ = 0. The constraints on the torsion and curvature tensors imply that the geometry is completely on-shell with no auxiliary elds present. For example, the non-vanishing tangent-space components of the torsion tensor T A^ = DEA^ are given by A^ and structure group connection T^ ^c^ = 2( c^)^ ^ ; Ta^^b ^ = 1 84 ( c^d^)^ ^ Ta^ ^^ = D ^Fa^^bc^d^ ; 1 36 Fa^^bc^d^( ^bc^d^) ^^ 1 288 ( a^^bc^d^e^) F ^bc^d^e^ ; in terms of a super eld Fa^^bc^d^ which turns out to be the supercovariant 4-form eld strength. As a consequence of the Bianchi identities, F is covariantly closed. The components of the super-Riemann tensor RA^B^ C^ D^ are also completely determined by the Bianchi identities, but we won't need them here. One can make the 3-form apparent in superspace as well by introducing a super 3-form CM^ N^ P^ with super 4-form eld strength super eld F = dC. Aside from its top component, which must be the same super eld Fa^^bc^d^ appearing above, the only other non-vanishing component of F is F^ ^c^d^ = 2 ( c^d^)^ ^. The full super 4-form is then given by F = 2 1 Ec^ ^ Ed ^ E ^ ^ 1 4! ^ ^ E ( c^d^)^ ^ + Ea^ ^ E^b ^ Ec^ ^ Ed^ Fd^c^^ba^ : (4.5) Closure is straightforward to check using (4.4). Let us now descend to 4D N = 1. The key question is how to identify the N = 1 gravitino 1-form E possibility is E I = in terms of the 11D 1-form E ^ = (E I ; E _ I ). The most general I E + where I is some complex SO(7) spinor, which may depend on all coordinates, and the ellipsis denotes the other seven additional gravitino connections, which we will ignore. Actually, it is convenient to factor out a modulus from I so that it is normalized to I I = 1. Thus, we identify E I ! I 1=2 E + and E _ I ! I 1=2 E _ + : (4.6) (4.2) (4.4) In order for the N = 1 torsion tensor to be canonically normalized, we must introduce a factor of into Ea, that is Ea^ ! Ea! Ea is a conformal compensator because it introduces a new Weyl symmetry has weight 1, Ea has weight 1, and E has weight 1=2. Given the constraint on the norm of I , any variation can be written as I = i! I + yi ( i)IJ J and I = i! I yi ( i)IJ J ; (4.8) in terms of a complex GL(7) vector yi and a real parameter !. We have written the SO(7) gamma matrices as i = Eia a, so that they possess a GL(7) index. The 14 components of yi should evidently correspond to the complex gauge symmetry associated with zi. The real parameter ! describes a local U( 1 )R transformation, because it can be absorbed by a phase rotation E = i! E . We can esh out these statements by constructing explicit expressions for the superelds G and Hi. These are naturally encoded as the lowest components of their corresponding 4-form and 3-form eld strengths in N = 1 superspace [15] F4 = Ec ^ Ed ^ E ^ E ( cd) F3i = Ec ^ E _ ^ E i( c) _ Hi + G + c.c. + ; These should arise from the decomposition of the 11D 4-form eld strength (4.5) under the identi cations (4.4), (4.6), and (4.7). The lowest dimension term of (4.5) exactly reproduces those above provided we identify G = 2 3 ; G = 2 3 ; Hi = 2 ( i ) 2 : Because we are being somewhat schematic with the reduction, we should probably not trust these results beyond lowest component. In particular, they may develop -dependent modi cations. But this suggests that the bottom component of Hi is associated with the complexity of : if I = I (up to a phase), Hij would vanish. This is consistent with the interpretation of [14] that this eld can be gauged away in the component theory. We still need to identify how the internal metric coming from 11D supergravity, which we denote gij , is related to gij (F ). It is clear these cannot be exactly the same because gij (F ) transforms under zi while gij must be independent of . Indeed, under the zi part of the transformations we nd gij = pg = 1 3 3 gij H^kzk + H^(igj)kz k 2 pg H^izi + c.c. 6i F 0 H^(iFj)klgkmH^mzl + c.c. ; Here and below we use the Weyl-invariant combination H^i := (GG) 1=3Hi. Whatever the internal metric gij is, it must be invariant under zi transformations. To nd it, the (4.9) (4.10) (4.11) (4.12) In addition to det g, there is another scalar z-invariant, G1=3G1=3 ^ 1, which carries Weylweight 2. The only such invariant scalar in the 11D theory is the conformal compensator, F so we identify 2 = G1=3G1=3 ^ 1 F =) F^ = ( 2 2 )1=3 : This identi es F^ in terms of 2 can nd that . Now from the explicit equations (4.10) and (4.14), one x := jH^ j2 = 4F^ 4 4F^ 1 : additional relations are useful (recall eq. (2.19)): x = jHj2 = 4 (F^ 1 + x)H^izi + c.c. ; 1 3 F^ = 31 F^ H^izi + c.c. (4.13) Using these results together with (4.10), one can check that there is (up to normalization) only one symmetric rank-2 tensor that is z-invariant and Weyl-invariant. This should be identi ed with the internal 7D metric, gij = F^gij + det g = det g F^4 : (4.14) HJEP05(218) This is nothing but the quartic polynomial (3.10), here derived as an algebraic equation when x and F^ are both expressed in terms of 2 . To con rm these identi cations, we should verify that one can consistently write down a relation between the parameters zi and yi in (4.8). G and Hi vary under (4.8) as G = 2 3 + c.c. = (4.17) We want to compare this to the transformation under 1 . For G, this is straightforward from (3.21), but for Hi we must correct for the gravitino super eld. In the Wess-Zumino ij ! 0, r ij = G Gij r 2 j j + : : : by (3.6). With gauge (cf. appendix C of [16]) r this taken into account, we nd This can be identi ed with (4.17) if G = G (GG) 1=3 ziHi and Hi = 2(GG)2=3Gij zj + c.c. yi = 2 F ^ 4 zi 4 1 gij H^j H^kzk + 2 F i ^ 2gii0 gkk0 Fi0jk0 H^kzj : For additional con rmation, let us try to identify the bottom component of Fijk. As with Hij, this contains scalar elds not directly present in 11D: 28 of its 35 degrees of freedom arise from the internal metric via the G2 relation (2.7), but 7 degrees of freedom remained. It turns out that Fijk has a remarkably simple interpretation expressed by i (4.15) (4.16) (4.18) (4.19) (4.20) This can be checked by verifying that the yi-variation of (4.20) matches the zi-variation in (3.21a) using (4.19). Further evidence is provided by checking that the G2 relation (2.7) holds upon inserting (4.20) and the expression for gij in terms of and gij from (4.14). The holomorphic structure of Fijk suggests that one should identify the bottom component of the chiral super eld ijk as ijk = Cijk ijk ; 1 2 where Cijk is the 3-form descending from 11D supergravity. Matching transformation rules con rms this, which means that the 3-form Cijk de ned by the real part of ijkj actually di ers from Cijk, just as gij de ned from Fijk di ers from gij . In both cases, adopting the gauge where is real (equivalently, where Hij vanishes) they become equal. Let us end on one particularly interesting result that we have not completely understood. The Hermitian metric Gij appearing in the gravitino transformation possesses an inverse ( G 1)ij = F^ 2gij 41 F^2H^ iH^ j 2 i F ijkH^l : Remarkably, it is this inverse, rather than Gij itself, which has an elegant interpretation in 11D. We nd simply (4.21) (4.22) (4.23) ( G 1)ij = F^ 1 gij + ij : 5 The conformal supergravity supercurrent Until this point, we have been treating 4D N = 1 conformal supergravity as strictly yindependent. We have also implicitly assumed that the super-vielbein was invariant under the extended supersymmetry transformations to the order we are working. That is, we have assumed 1EM A = 0 (equivalently, 1H _ = 0). This is to be expected, since the component vielbein always transforms with second-order gravitino weight, that is, e into the component gravitino with SUSY parameter . This means we have not yet actually determined that the rst-order gravitino coupling is consistent with supergravity at the non-linear level. To remedy this, we will introduce the prepotential super eld H _ to describe ydependent uctuations around the y-independent background vielbein EM A. The schema for introducing prepotentials to deform a background (non- at) geometry can be found in refs. [27, 28]. (See [29] for the particular case of N = 1 conformal superspace.) The prepotential H _ is now subject to the gauge transformations H _ = r _ L r L _ ; (5.1a) where the superspace derivative r is de ned in the y-independent background EM A. One must assign L transformations to the other potentials. The right choices can be deter0V = 2i r2(L r F ) ; L W 2i r2(L H) ; L _ W _ + iL { W F iL _ {W _ F ; 0X = r (L G) + r _ (L _ G) + iL { W H iL _ {W _ H ; 0V = L W L _ W _ : In addition, as in the linearized case [16, 30], we must assign an L transformation to the gravitino, 1 mined following for example [29], and correspond to the (covariantized) transformations i 2 1 2 We have labeled these by gravitino weight. In principle, each of the equations (5.1) may possess higher-order gravitino modi cations on the right-hand sides. We should also mention here that L itself possesses a certain gauge-for-gauge symmetry where it can be shifted by a chiral spinor super eld; this shift is countered in (5.1g), for example, by a shift in . The rst order coupling of H _ to the non-linear action is SHa = d x where J _ can be interpreted as the supercurrent. We derive it directly by requiring gauge invariance to lowest order in the L transformations. This is a long calculation that can be split into two parts. The rst arises from the minimal coupling of H _ to the Chern-Simons action and is naturally written as a 6-form, (5.1b) (5.1c) (5.1d) (5.1e) (5.1f) (5.1g) (5.2) (5.3) (5.4) J CS_ = i 4 r F ^ r _ F ^ H W ^ W _ ^ F W ^ {W _ F ^ F W _ ^ {W F ^ F 3 {W F ^ {W _ F ^ F ; 1 2 i 2 2 + 8 i 4 3! 3! ijklmnp r Fijkr _ FlmnHp : or equivalently as a density J CS_ = 1 pg W ij W _ klF~ijkl + i pg W ij W _ kFkij + i pg W _ ij W kFkij +2pg W i W _ j gij 1 + The contribution from the Kahler term is more complicated and given by J K_ = 2 [r ; r _ ] (GG)1=3pg(F^ 3 2 F ) 3 2 (GG) 1=3pg F 0Hi[r ; r _ ]Hi (pg F (GG)1=3) [r ; r _ ]Fijk + (GG)1=3pg F^ r _ log(G=G) 3r _ (GG) 1=3pg F 0Hir Hi + c.c. F 0(GG) 1=3 3HiW _ j W ij 3HiW j W _ ij 6iHiW j W _ kFijk : (5.5) The two quantities J _ = J CS_ + J K_ , given above, and J i in (3.8) describe the two supercurrents of 11D supergravity written in N = 1 language. J _ is the N = 1 conformal supergravity supercurrent, and de nes the rst-order coupling to the y-dependent uctuation super eld H _ . Similarly, J i is the extended gravitino supercurrent, describing the rst-order coupling to i. When the covariant N = 1 super elds obey their equations of motion, these currents are subject to the conservation conditions r2J i = 0 ; Gr (Gij J j ) = iW _ ij J _ j ; (5.6a) (5.6b) (5.6c) These conservation equations are a direct consequence of the gauge transformations (3.6), (5.1a), and (5.1g). 6 Toward higher-order terms and a more covariant formulation The ability to couple i and H _ to the non-linear action at rst order provides a strong check of consistency. Their couplings correspond to the N = 1 supercurrent and extended supersymmetry supercurrents of 11D supergravity. The associated lowest-order extended SUSY transformations completely determine the function F in the Kahler action and lead to a number of consistency conditions. In this section, we will describe how these couplings could be taken to all orders. One key feature that was useful in determining the gravitino supercurrent J i was the simplicity of the part of transformations of both the gravitino (3.6) and the other prepotentials (3.4). These are di erential form transformations involving neither the metric gij nor the function F appearing in the Kahler action, and so the Kahler and Chern-Simons actions must be canceled separately. If this feature holds to all orders, it would mean that in order for the Kahler action to be canceled by gravitino terms, it must be possible to construct new eld strengths Fijk, Hi, and G by introducing -modi cations of the old eld strengths so that invariance is manifest. To lowest order, this is precisely how the Kahler part of the supercurrent (3.8) arises. Indeed, the supercurrent (3.8) can be rewritten as 3i 2 144 J i = + W i i 2 1 ijklmnpFjkl (G r Fmnp 3 HmW np) : ! 3pg GG 1=3 F (6.1) Thus, one simply makes the following shifts in SK , G ! G = G + G{W ; H ! H = H F ! F = F + ^ W i 2 2 r (G ) + 2 r _ (G _ ) ; _ ^ W _ ; which are -invariant to lowest order, and expands to rst order in . This observation cannot be extended simply by exponentiation. Remarkably, however, it is possible to construct higher-order modi cations that ensure invariance. For example, (6.2a) (6.2b) (6.2c) (6.3a) (6.3b) (6.3c) (6.4a) (6.4b) (6.4c) G H F G = G{W H = F = + + i 2 4 2 r (G ^ W + O( 3) ; G 16 r 2 4 i 4 i 1 G {W _ (r _ ) + c.c. + O( 3) ; ^ H 8 r ^ r _ H ^ W 2V = 2 1 2 8 r ( ^ H) : _ G ; correspond to modi ed -invariant eld strengths, provided we introduce higher-order transformations of the form6 Proceeding in this way, one determines the higher-order transformations order by order. To apply this logic to the Chern-Simons action, recall that in the component 11D theory one can write the integral R C ^ F ^ F over 11D spacetime M as the integral R F ^ F ^ F over some auxiliary 12-manifold whose boundary is M [31]. This can be extended to N = 1 superspace by taking Y to be the boundary of an 8-manifold Z and integrating a super [4; 8]-form on X Z. The requisite 12-form was computed in eq. (5.49d) of ref. [15]. Being the superspace version of F 3, it involves only eld strength super elds so that we can proceed to apply the procedure above. Carrying out this program generates a large class corrections necessary for invariance. However, at the linearized level [16], there are quadratic terms involving r _ that are already -invariant, and we should expect corrections to these terms in the nonlinear theory. In principle, they can be determined by requiring L and/or invariance. 6There is some ambiguity in these transformations corresponding to the ability to make O( 2 ) eld rede nitions of . In this paper we have given the construction of eleven-dimensional supergravity in 4D N = 1 curved superspace to rst non-trivial order in the elds with 4D spin to all orders in the remaining elds. More precisely, we formulated a gravitino super eld expansion of eleven-dimensional supergravity and solved it to leading and next-to-leading order for the action and gauge transformations. The consistency of the construction relies on a powerful set of local superconformal symmetries arising from the foliation of the spacetime by N = 1 superspaces. This formulation is well-suited to backgrounds in which the spin 32 components are Y -independent but otherwise arbitrary (assuming vanishing vacuum values for the gravitini) such as warped compacti cations with uxes. 32 and The new local symmetry can be understood by comparing it to eleven-dimensional superspace. Reduction of the 32 supersymmetries to 4 is parameterized by a complex spinor. Dirac bilinears in this spinor de ne the expected G2 structure on Y , but additionally we nd deformed chiral and linear super elds corresponding to conformal, U( 1 ), and special conformal compensators. In the super eld description, these compensators are mixed up in multiplets containing physical elds. Because of this, interactions of the latter can be determined exactly by requiring invariance under the gauge transformations of the former. On the other hand, the gravitino super elds share compensating Stuckelberg-like super eld transformations with some of the physical elds for which the action is determined exactly. Covariantizing under this part then introduces gravitino corrections to the eld strengths appearing in there, which we constructed explicitly at next-to-next-to-leading order. What is more, the four-dimensional part of the superconformal symmetry appears in the gravitino transformations through its Y dependence, again in a Stuckelberg-like shift that can only be canceled by similar logic. This time it involves the 4D N = 1 conformal supergravity prepotential Ha. As a rst step in this direction, we have explicitly constructed the 4D N = 1 supercurrent responsible for this mechanism at lowest order. However, working order-by-order in a this prepotential is vastly more complicated, since it couples to everything. Instead, one might attempt a more covariant approach in which this eld does not appear explicitly. This would be in close analogy to how we have treated the Kaluza-Klein gauge covariantly, hiding it in the superspace connection and manipulating only its eld strength W explicitly. In such an approach, the new covariant object Xia replacing @iHa and analogous to the aforementioned eld strengths would appear as the curvature components in the superspace commutator [rA; ri] where rA is the conformal superspace connection [22]. This supergeometric approach is currently under investigation. eld V Acknowledgments We thank Sunny Guha and Daniel Robbins for discussions and collaboration during the early stages of this project. This work is partially supported by NSF under grants PHY1521099 and PHY-1620742 and the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University. We also thank the Simons Center for Geometry and HJEP05(218) Physics and the organizers of the September 2017 Workshop on Special Holonomy, where results from this work were reported. A 4D superspace and supergeometry Our conventions for 4D N = 1 superspace follow [22], where N = 1 conformal superspace was introduced for describing conformal supergravity. The 4D supermanifold is described by local coordinates zM = (xm; ; _ ) and is equipped with a set of 1-form connections that gauge the N = 1 superconformal algebra. These are the super-vielbein EM A, a spin connection connections FM tive rA is given by M ab, a U( 1 )R connection AM , a dilatation connection BM , S-supersymmetry and FM _ , and special conformal connections FM a. The covariant derivarA = EAM M abMba BM D AM A FM AKA (A.1) where Mab is the Lorentz generator, D is the dilatation generator, A is the U( 1 )R generator, and KA = (Ka; S ; S _ ) collectively denotes the three special (super)-conformal connections. The algebra of these generators with each other and rA can be found in [22] and matches the global N = 1 superconformal algebra with rA identi ed as the supertranslation generator PA = (Pa; Q ; Q _ ). The presence of a non-vanishing super-Weyl tensor W deforms the algebra by introducing curvatures in the (graded) commutators [rA; rBg. While the lowest anti-commutators are unchanged fr ; r _ g = 2i( a) _ ra ; fr ; r g = 0 ; fr _ ; r _ g = 0 ; a dimension-3=2 curvature operator is introduced a la super-Yang-Mills, [r ; r _ ] = 2 W _ ; [r _ ; r _ ] = 2 _ _ W ; where W i W ( ab) Mba + i 2 r W S + i 2 r _ W ( b) _ Kb : The vector-vector curvature [ra; rb] can be found in [22]. The super eld W mal primary (annihilated by KA) and chiral (annihilated by r _ ) and contains the curvature is a confortensors of conformal supergravity. As discussed in [22] (see also [27] and [28] for the conventional formulations in N = 1 superspace), an invariant full superspace integral is built out of a scalar function L via where E = sdet(EM A) is the superdeterminant (or Berezinian) of the super-vielbein. Super-di eomorphism invariance in superspace guarantees supersymmetry in components. Chiral superspace integrals are built out of chiral super elds Lc via Z Z d4x d4 E L ; (A.2) (A.3) (A.4) (A.5) (A.6) 0 = Z B B.1 for some V A, it follows that Z Z Z d4x d4 E rAV A FABKBV A ( )a ; (A.8) d4x d4 E rAV A ( )a = d4x d4 E FABKBV A( )a : So if V A is not a conformal primary, there is a residual connection term left over. This actually re ects the fact that in these cases, rAV A is not itself a gauge invariant Lagrangian. Useful variational expressions The origin of covariantized transformations The tensor hierarchy that descends from the 3-form of 11D supergravity is gauged by the non-abelian Kaluza-Klein connection. This is most easily described in di erential form notation where the 11D exterior derivative decomposes as d11D ! D + @ + F . D is the covariant derivative in four dimensions, D = d LA, where A is the Kaluza-Klein connection, @ is the internal derivative in seven dimensions, and F is the interior product on an internal form index with the Kaluza-Klein eld strength. The 4-form eld strength G = dC descends to the set of ve eld strengths G[0;4]; ; G[4;0] as Arbitrary variations of these eld strengths involve varying both C[p;3 p] and the KaluzaKlein vector, leading to (A.7) (A.9) (B.1a) (B.1b) (B.1c) (B.1d) (B.1e) (B.2a) (B.2b) (B.2c) (B.2d) (B.2e) where E is the chiral superspace measure, see [22] for its de nition in superspace. In both cases, the functions L and Lc must be conformal primaries and they must possess appropriate Weyl and U( 1 )R weights; that is, L must have Weyl weight 2 and be U( 1 )R neutral, while Lc must have Weyl weight 3 and U( 1 )R weight 2. Full superspace integrals are related to chiral superspace integrals via Z d4x d4 E L = 4 E r2L : Because of the presence of the special (super)conformal connections FM A, the standard rule for a total covariant derivative is slightly modi ed. Using G[1;3] = DC[0;3] + @C[1;2] ; G[2;2] = DC[1;2] + @C[2;1] + F C[0;3] ; G[3;1] = DC[2;1] + @C[3;0] + F C[1;2] ; G[4;0] = DC[3;0] + F C[2;1] : G[1;3] = D C[0;3] + @ C[1;2] AG[0;4] ; G[2;2] = D G[3;1] = D G[4;0] = D C[1;2] + @ C[2;1] + F C[0;3] C[2;1] + @ C[3;0] + F C[1;2] C[3;0] + F C[3;1] where C[p;3 p] := C[p;3 p] + AC[p 1;4 p] (B.3) are the covariantized transformations of the p-forms. The relations (B.2) can be understood as the general variation of the G[p;4 p] eld strengths consistent with the Bianchi identities. A corresponding set of relations exist for the p-form hierarchy written in N = 1 superspace. There the situation is more subtle because the connections A and C[p;3 p] must be built from prepotential super elds. The superspace analogue of the set of eld strength variations (B.2) is H = G = r ( V W 1 2 V r (W 1 V + i V F r _ i V W F ) 2i W F ) + r _ ( V W _ + i V W _ F ) 1 2 V r _ (W _ + 2i W _ F ) ; i V H) + W i L V G : V r F L V W ; (B.4b) (B.4c) (B.4d) E LCS;F + c.c. ; (B.5) 2 SCS = Z d x i 2 F ^ W Y 4 E LCS;D + G + W ^ W 1 2 ^ W Z d x ^ F ^ V r F + W _ ^ F ^ V r _ F : i 2 1 2 1 2 space piece, [2ex] LCS;F = where the covariantized transformations of the prepotentials was given in (3.4), analogous B.2 Arbitrary variations of the Chern-Simons and Kahler actions The variation of the Chern-Simons action with respect to the tensor hierarchy and KaluzaKlein prepotentials can be decomposed in terms of a full superspace and chiral super[2ex] LCS;D = ^ H + F ^ r W + 2r F ^ (W i W F ) + c.c.) ^ F + c.c.) + F ^ F + H ^ V W F ^ r F + F ^ V W _ F ^ r _ F + F ^ rW F ^ F 1 2 1 For the Kahler term, it is easier to give its variation in terms of those of the covariant objects directly, 2 SK = LK = Z 1 144 Z d y LK ; 1=3 pg F 0 gij Hi Hj ; GG 1=3 pg F 2jHj2F 0 log GG (B.6) and then employ (B.4). 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Katrin Becker, Melanie Becker, Daniel Butter, William D. Linch. N = 1 supercurrents of eleven-dimensional supergravity, Journal of High Energy Physics, 2018, 128, DOI: 10.1007/JHEP05(2018)128