Supersymmetric E[Subscript 7(7)] Exceptional Field Theory

Supersymmetric E[subscript 7(7)] exceptional field theory

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Citation Godazgar, Hadi, Mahdi Godazgar, Olaf Hohm, Hermann Nicolai, and Henning Samtleben. “Supersymmetric E[subscript 7(7)] Exceptional Field Theory.” J. High Energ. Phys. 2014, no. 9 (September 2014).

As Published http://dx.doi.org/10.1007/jhep09(2014)044

Publisher Springer-Verlag

Version Final published version

Citable link http://hdl.handle.net/1721.1/91259

Detailed Terms http://creativecommons.org/licenses/by/4.0/ JHEP09(2014)044 , 7(7) Springer July 4, 2014 SU(8) and , : × August 20, 2014 eory for E : September 8, 2014 3) , : and c Received 10.1007/JHEP09(2014)044 Accepted Published oup SO(1 doi: , riance. We establish the con- 6-dimensional space. We give bridge, me subject to a covariant con- nder these symmetries and close, eralized geometric formulation of f Technology, Hermann Nicolai Published for SISSA by nstein-Institut (AEI), b , (internal) generalized diﬀeomorphisms. The [email protected] , 7(7) Olaf Hohm, exceptional ﬁeld theory a 7(7) . 3 d [email protected] Mahdi Godazgar, 1406.3235 , a The Authors. c String Duality, Supergravity Models, M-Theory

We give the supersymmetric extension of exceptional field th , [email protected] = 11 supergravity. ´ 46, d’Italie, allée F-69364 Lyon cedex 07,E-mail: France Ecole Normale de Supérieure Lyon, Cambridge, MA 02139, U.S.A. Max-Planck-Institut Gravitationsphsyik, für Albert-Ei 1,Mühlenberg D-14476 Potsdam, Germany Universitéde Lyon, Laboratoire de Physique, UMR 5672, CNRS DAMTP, Centre for Mathematical Sciences,Wilberforce Road, University Cambridge, of CB3 Cam 0WA, U.K.Center for Theoretical Physics, Massachusetts Institute o [email protected] [email protected] b c d a transform as scalar densities under the E ArXiv ePrint: the fermionic field equationssistency and of prove these supersymmetric results inva withD the recently constructed gen Keywords: supersymmetry transformations are manifestly covariant u in particular, into the generalized diffeomorphisms of the 5 which is based on astraint. (4+56)-dimensional The generalized spaceti fermions are tensors under the local Lorentz gr Henning Samtleben Open Access Article funded by SCOAP Abstract: Hadi Godazgar, Supersymmetric E JHEP09(2014)044 1 6 12 17 20 24 30 31 33 36 38 39 43 45 12 23 38 = 8 supergrav- ] clear evidence 2 N metries are specifically s ction t more general properties of metric concepts in string and ard differential geometry was the low-energy effective space- s in maximal ], and in the recently constructed 13 = 11 supergravity [ – 9 D ], a line of development which was continued 4 – 1 – , 3 = 11 supergravity D ]. Somewhat independently of these developments, an 8 – maximal theories, possibly even providing clues towards a 6 exceptional geometry exceptional field theory uncompactified 7(7) 7(7) E × ] a recurring theme has been the question of whether these sym ] and taken up again in [ 1 5 4.5 Connections and fermion supersymmetry transformation 4.1 56-bein4.2 and GVP from eleven Generalized dimensions torsion 4.3 Hook4.4 ambiguity Non-metricity and redefinition of the generalized conne 3.1 Connections 3.2 The supersymmetry3.3 algebra Supersymmetric field equations already given in thein early [ work of refs.important [ insight has been theM-theory, which emergence enable of generalized a geo time duality-covariant formulation theories, of as manifested in double field theory [ tied to dimensional reductionthe on underlying tori, or whether they reflec for the existence of hidden structures beyond those of stand ity [ Ever since the discovery of ‘hidden’ exceptional symmetrie 1 Introduction C The supersymmetry algebra D Non-exceptional gravity E Covariant SU(8) connection A Notations and conventions B Useful identities 4 Exceptional geometry from 3 SU(8) Contents 1 Introduction 2 Bosonic E better understanding of M-theory. Starting from JHEP09(2014)044 ], 3 EFT r the ] and 15 objects, ]. More 7(7) , 23 14 7(7) (as well as of sed on a 4+56- ] to also take into -covariant theory. 8 56). The theory is , SU(8) tangent space 7(7) 7 e original work of [ × 7) splitting [ ,..., = 11 supergravity as the decomposed in a (4 + 7) 3) it is remarkable that the our-dimensional maximal cal SU(8) subgroup, and , yond those of the original embled into E D lar and vector fields in the 56-plet of vectors combining = 1 nstruction of the E (1 ] for a generalized geometric iations of the fermions found -covariance as identified here. ], however, these formulations btleties involved in making a econd, we relate the resulting e included from the beginning = 11 supergravity that retains 3 ity equations for the 56 vector ing point of the analysis) and 19 ansformations into a covariant M eleven dimensions – ansforming under this group, an 7(7) D ], are manifestly SU(8) covariant. ] to a fully E ng the full coordinate dependence in 16 3 3 representations initially depending = 11 supergravity. By contrast, the 7(7) D 7(7) ]. Such an extended spacetime has later = 8 supergravity; their supersymmetry 22 N – 2 – ]. See also refs. [ ]. The purpose of this paper, then, is to bring to- = 11 supergravity have been given for the analogous 15 ]. As one of our main results we will demonstrate the , 21 8 D , – 14 = 11 supergravity is thus guaranteed at each step of the 6 20 (with fundamental indices , D 3 M in a particular truncation of -based exceptional geometry, takes Y ], the recent construction of complete EFTs in refs. [ ] suggest that a formulation that is properly covariant unde 7(7) 8 7(7) 5 ], which has been extended and completed in [ , and 3 7 ] are precisely the ones required by E µ 3 x ]. In contrast to the original approach of ref. [ EFT, which is a natural extension of double field theory, is ba 24 , 7(7) 19 -covariant dynamics of the bosonic sector) was lacking in th , 18 7(7) The E The approach of [ The results of ref. [ the E of all fields (however,theory unlike are in introduced). EFT,their no supersymmetry The transformations, extra already fermions coordinates given transform in be [ under the lo symmetry manifest. Tosplitting, this as end in the Kaluza-Klein compactifications, fields but and keepi coordinates are construction of refs. [ finally, the present work extend the formulation of ref. [ dimensional generalized spacetime, with fields in E recently, similar reformulations of truncations, casting the theoryform and [ their residual gaugeare tr not immediately applicable to the untruncated ‘empirically’ in ref. [ exceptional groups should include extendedidea coordinates that tr also appearsbeen in implemented the for proposal E of ref. [ and has only emergedcombinations of with SU(8) the connections recent in the advances. supersymmetry var Nevertheless namely a 56-bein encoding thethe internal field components 28 and electric a and 28 magnetic vectors of Moreover, those parts ofdimensionally the reduced bosonic maximal sector which supergravity lead can to then sca be ass detailed comparison. account aspects of the E starting point and reformulates it in order to make a local SO approach in the sensegether of these strands refs. of [ development: first we complete the co compatibility of these two formulations, and explain the su on all coordinates only the internal coordinates and field components of the (4 + the on-shell equivalence with construction, a proper understanding of the role of E transformations can be showngauged to supergravity. While take in the this(with precise approach the the form fermions supersymmetry of ar variations the constituting f the start by giving the fullytheory supersymmetric to extension the by formulation fermions; of s [ ‘exceptional field theory’ (EFT) [ given by an action along with non-abelian twisted self-dual JHEP09(2014)044 (1.1) covariant ules for the ]. In partic- for the local 7(7) ]. The status 36 M , 27 ω 35 , ] that implies that , 15 and [ 18 19 µ -generalized diffeomor- , ompare with the usual ω . 18 µνM nce is already implied by 7(7) l give further credibility to B n the following scheme ], one has to pick a partic- ], the section constraint has is the formulation including ms of the 56-bein and other riations. for the local SU(8), respec- 7 or gauge transformations. In olely in terms of the fields of d ndent of the four-dimensional metry. We give compact and 26 . We use the opportunity to ture [ , me to 4+7 dimensions. After ption is consistent by virtue of atures generalizing the geome- on. Furthermore, we determine shell invariance. To this end we lized diffeomorphisms. This is he generalized diffeomorphisms = 11 fields that can be thought , 3 M MCD EFT and give the supersymme- ry underlying the E ponents ]. Importantly, we find that the V Q 25 D 30 = 0 -generalized diffeomorphisms. Cru- – 7(7) ABCD µ µ 912 M 28 and | 7(7) Q K Q µ SU(8)-covariant form, showing that they ≡ P MN Q × Γ AB M V µ 7(7) – 3 – D -covariant section condition [ β EFT also encodes, as 7 components among the 56 µ e 7(7) ]. For the internal, 56-dimensional sub-sector, such a αβ M 7(7) 34 = 0 π µ – ρ M ] = 11 supergravity and type IIB supergravity. After solv- ≡ ω ω µν 31 [ α µ Γ , D e 9 M D ] give the full dynamics and supersymmetry transformation r 19 , 18 3), with all geometric objects being also covariant under E , -covariant expressions for the internal connections in ter In this paper we introduce the fermions of the E = 11 supergravity, and thus with the results of refs. [ 7(7) close, in particular, intoin analogy the with external the and supersymmetrization internal of genera DFT [ try variations of all fields in a manifestly E phisms. The various connection components are summarized i formulation by introducing connectionstry and of invariant double curv field theory [ SO(1 covariant objects. One ofexternal the main and results internal oftively, connection this and paper similarly components external then and internal connection com cially, the theory is subject to an E ular, refs. [ truncated theory, where the fieldsexternal and parameters coordinates, are in indepe give terms a of complete such andE geometrical self-contained objects presentation of this geo the fermionic field equations andhave verify to supersymmetric further on- develop the generalized exceptional geomet geometry is to a large extent already contained in the litera supersymmetry transformations of allEFT, fields in can particular be the written 56-bein,of s as without parametrising recourse these to structures the in a GL(7) decompositi of this field mayconsistency appear of the somewhat EFT mysterious, gaugethis but symmetries. field its by In appeara showing this that paper it we has wil consistent supersymmetry va can be interpreted asaddition, conventional diffeomorphisms and and in tens analogytwo to inequivalent type solutions: II double field theory [ the fields depend onlyD on a subsetular of solution coordinates. of thissolving In the constraint, section order which constraint, to reduces the c the various components spaceti of t ing the section constraint, the E fields. The fields transform appropriately under E gauge vectors, dual gravity degreesa of covariantly freedom. constrained This compensating descri two-form gauge fiel JHEP09(2014)044 ) ], ese 42 1.1 – etries 40 ] and later coordinates. 33 – hat unlikely to 7(7) 31 , 28 , 9 ) that appears in general c EFT constructed here contrast to conventional e term ‘symmetry’ in the ge transformations of the D y [ ction components in ( resting when one considers pplications and extensions. n and all (supersymmetry) of this idea. merge from a single ‘master e seven ‘dual’ internal diffeomor- y. Indeed, the full non-linear ed along the lines of [ he higher-dimensional theory ) connections once we include ions of the section constraint, imposing covariant constraints, articular non-trivial solution to der all gauge symmetries except uge symmetries manifest. (including dual fields) yielding type constraints satisfied by the ot given in terms of the physical f spontaneously broken GL(4) symme- f Kaluza-Klein compactifications y depending only on the physical and our conventions will be given ], in which connections carry ‘non- 8 , miraculously’ drop out in all relevant formulae, 7 , 3 However, the new structures exhibited here 1 ]. As in double field theory, however, this is ] this fact is explained by the gauge ‘Stückelberg-like’ . – 4 – 15 36 = 0. µνM , -covariant form. Evidently, the true advantage of that depend also on the ‘internal’ E B 35 N µ ∂ 7(7) , Nevertheless it is remarkable and significant that the x ⊗ 18 = 11 or IIB supergravity, exist. Such solutions would give 2 M D ∂ -extended directions. ] for the truncated theory. We also clarify the relation of th identified here is analogous to the GL( MN ) 19 α 7(7) t 7(7) = 11 supergravity or IIB supergravity have any new local symm D ]. In the formulation of ref. [ 8 theories (but see below). Although such solutions are somew new ], but the present scheme should not be viewed as a realization imply that 39 – A distinctive feature of generalized geometries is that, in A second question concerns the utility of the supersymmetri One obvious question concerns the precise significance of th 37 It is an old idea to interpret the graviton as a Goldstone boson o The only new local symmetry would be the one associated with th not 1 2 in a more general perspective.The first Here application we concerns see the twoother non-linear main consistency than possible o torus a compactifications. These can be investigat The structure of the various diagonal and off-diagonal conne invariance associated with the two-form field exploiting the present formalismin and a the form fact adapted thatKaluza-Klein to it (gauged) for ansätze casts lower those t dimensional higher-dimensional supergravit fields besides those corresponding to genuinely do internal diffeomorphisms can beform combined fields with and the theirthe tensor duals reformulation gau would in only an become E fully apparent if solut components along the E consistent with the final form offields, the as (two-derivative) theor the undeterminedvariations, as connections shown drop in out [ geometrical of structures the to actio metricities’ the that formulation can of be absorbed, [ as we will show, into SU(8 connection’, whose introduction would finally render all ga geometry, the connections are not completely determinednecessarily by featuring undetermined connections thatfields, are as n firstextended discussed to in exceptional the groups geometry [ of double field theor hints at a larger geometrical framework in which they would e try [ exist for the case atinfinite hand, dimensional extensions the of situation the may become E-series. more inte phisms, but the correspondingas transformation parameters shown ‘ in ref. [ connections. The precise definitions ofin the the various main tensors text. The formulationfor is the manifestly external covariant un diffeomorphisms of Here we also indicate the corresponding covariant torsion- beyond the ones already known. relativity, and is ‘spontaneously broken’ whenthe one section picks constraint a ( p present context. The E JHEP09(2014)044 . ]. ent 41 7(7) O(8) ional ), rel- which ω in E ; eometric , for which = 11 since 5 28 subblock not x, Y S ( D × × 5 -dimensional an- → V ) re very similar to four- = 11 theory in ref. [ x, Y tant, there would be no ( ) (1.2) D V = 11 and type IIB theories are isting’ the 56-bein relative ond the extension to other x ]. ] would lift to an analogous me dimension ( en to also depend on the 56 d be the first example of a ix acts on a 28 ersymmetry transformations een obtained in this way for D eory would no longer be on- ions in this paper has lead to he d, a study of the ambiguities ]), ], obtained by performing an us be obtained by performing 50 V 49 ravity on AdS ic vectors, which is 45 49 ! ories, the U(1) rotation above is to be ω ω ], and hence would correspond to 2 ] or the present version with the IIB sin cos 47 ω ], show that these gaugings cannot orig- ω 8 sin ] may lead to a higher-dimensional under- cos ]. Specifically, the deformed theories can be = 11 diffeomorphism and Lorentz invariance; − EFT [ 2 48

D – 5 – 6(6) ], as well as a more explicit argument based on ≡ ], as well as for general Scherk-Schwarz compactifi- ) -deformation of ref. [ 44 ) symplectic deformation [ ω 44 ω , ; R – 6 , x ( 42 = 11 supergravity, the main outstanding problem here is , 6 D → V ) x = 11 supergravity of ref. [ ( Apart from the non-linear for ansätze higher rank tensors, V D 3 , in precise analogy with the deformation of the four-dimens ] in 1978, and it would be a remarkable vindication of the pres . Because of the inequivalence of the corresponding gauged S ]. V 2 M 45 = 11 supergravity that is encoded in a suitably generalized g Y = 11 supergravity of ref. [ ], where uplift for ansätze sphere reductions of the ], that is, by making the replacement D D 6 46 compactification [ The present reformulation naturally suggests that a higher 7 4 S 56 matrix ]. × × 4 49 Secondly, the fact that the supersymmetric EFT has a structu See also ref. [ In fact, in the context of four-dimensional maximal gauged the 3 4 either the supersymmetric extension of E Partial evidence presented in refs. [ of the 56 framework transcending conventional supergravity. deformation of genuinely new maximal supergravitythe in discovery the of maximal ref. space-ti scheme [ if such away theory to reconcile could this be deformed theoryin shown with other to words, exist. the four-dimensional Equally impor a non-trivial deformation of that theory. In fact, this woul cestor of the deformed SO(8)an gauged analogous supergravities ‘twist’ might of th the 56-bein of EFT (see also ref. [ inate from the obtained from the standardto SO(8) the gauged vectors supergravity [ by ‘tw dimensional maximal gauged supergravitystanding [ of the newelectromagnetic SO(8) U(1) rotation gauged of supergravities the 56 of electric ref. and magnet [ the higher-dimensional embedding tensor in ref. [ solution of the sectioninherent constraint in might defining be generalized employed.(and connections hence Indee and the how theory) the remainan sup invariant understanding under of such the redefinit hook-type ambiguities observed in t cations with fluxes [ can now also benon-trivial deduced compactifications in of a straightforwardto fashion, extend and these bey results to the compactification of IIB superg understood as part of a more general SL(2 conjectured using similar ideas. the AdS scalar or vector fields in the compactification have already b supergravities in four dimensions,shell it equivalent is to clear the that such a th ative to all vectorsextra and tensors, coordinates where the 56-bein is now tak theory [ in all formulae, where each element of the U(1) rotation matr JHEP09(2014)044 , ] 8 ur N 56, , X 7 (2.1) (2.3) (2.4) (2.5) , (2.2a) (2.2b) 3 MN ,..., -covariant -covariant , = 1 = Ω 7(7) 7(7) in the funda- = 0 M M , X B , 2 7(7) = 11 supergravity N − M Y 1) D -covariant exceptional A ∂ , ′ M , 7(7) ∂ al variables, respectively, ent of the E ∗ , + (6 1

MN according to − two inequivalent solutions, in AB the E Ω d working out the supersymme- 2) M articular the 56-bein parametriz- , , µν M V , however the latter dependence is is the symplectic invariant matrix B } . Its analogue in the internal sector , , we review the bosonic E = + (6 7(7) we construct its supersymmetric com- να α 2 0 e MN 3 M AB is the vierbein, from which the external = 0 α , µν 1) V µ , α B , 3 M AB e B µ ′ − , V e N AB = M with respect to the maximal subgroups GL(7) + 7 M + (20 µ – 6 – A ∂ µν 1 g A , . Here, Ω {V M − +1 ] for a summary of index notations and conventions. , 7(7) is the representation matrix of E ∂ ] (to which we refer for details) and translate it into ]; in section = 2) , A AB N AB ′ 15 3, and the 56 internal variables [ A, B 15 N + 21 MN , , M M ) M M ) V α 14 V ′ +1 14 V α + (6 t t , ,..., ( = 1 α = 11 supergravity after an explicit solution of the section , µ +2 + 21 e AB D 1) , , SU(8). To begin with, all fields in this theory depend on the fo = 0 M , we discuss how this theory relates to the reformulation [ +3 / 4 V 7 (6 µ = 0 exceptional field theory , 7(7) µ A x −→ −→ N . The tensor ( 7(7) ∂ 56 56 M SL(2), respectively. The resulting theories are the full ∂ MN × Ω M MN ) X α t ( = We refer the reader to appendix The outline of the paper is as follows. In section N X which we use for lowering and raising of fundamental indices mental representation. Thesewhich constraints the admit fields (at dependaccording least) on to a the subset decompositions of seven or six of the intern exceptional field theory is given by is the complex 56-bein for any fields or gauge parameters field theory, constructed in refs. [ which we describe in the following. The field and GL(6) and the type IIB theory, respectively. The bosonic field cont of the fundamental representation of E transforming in the fundamental representationstrongly of restricted E by the section condition In this section we give a brief review of the bosonic sector of (four-dimensional) metric is obtained as satisfying external variables constraint is chosen. the variables appropriate for the couplinging to the fermions, coset in space p E 2 Bosonic E exceptional field theory, of refs.pletion [ upon introducing the proper fermiontry connections algebra. an In section of the full (untruncated) JHEP09(2014)044 ], in 3 L ] ν (2.6) (2.9) (2.8) (2.7) A (2.12) (2.10) (2.11) (2.13) N ∂ A, B, . . . K µ [ A . . is related to the one . The reason is that, KL ]). However, in accord MN group-valued matrix Ω Ω 40 µνρσ ε EF GH , MN . 7(7) , δp Ω N GH , , A, B, C, . . . , V − = 0 MN representation and collective δ MCD AB Ω al variation, MN 8 V AB CD . KL AB δ M E , ) ] V ρσ α MEF D t ], we here revert to the notation of ref. [ = 0 = i = i Ω V ( F N MN ABCD V 15 ABCDEF GH cation involves Killing spinors as ‘conversion Ω N AB δ ǫ ised and lowered by complex conjugation. AB ] CD δp MN V stinguish between the SU(8) indices ) E µνρσ ’ SU(8) indices in ref. [ N 1 N ] NCD 24 + α , + V V V t e ε in ref. [ ] has been constructed. NCD C = ] MB 4 1 V AB AB B V AB 15 δ 24 ( µν N N − M ABCDEF GH M AB M – 7 – M C B [ V ε ], such that our tensor density V V 1 2 V AB . in the internal sector is the positive definite sym- A in the four-dimensional compactified theory. These are only [ [ ABCD 48 1 in the fundamental A 24 δ − V [ ) are subject to the first order duality equations − i, j, k, . . . M MN , µν C MN 2 3 µν AB 4542] MN − V 8 g . Ω M AB M 2.3 Ω F ] Ω δq 1 2 = = = ν i, j, . . . 2 1 N AB − µνρσ [1312 ≡ V A The fact that the 56-bein is an E − V in ( ,..., , δp iε ≡ = N 5 α MN MN MN . ∂ A = AB M Ω Ω Ω MN = 1 B N ¯ µν µ AB 28 M µ B [ δq A M V M − 2101] CD CD µν AB + ··· . N A − V , NCD N N F ∂ 2 δ V V V ], where SU(8) indices are denoted by the letters 28 = µνρσ [0705 δ δ δ 8 − ε , B MN A, B, 7 ) M AB M A AB ] α V ν ] by t MAB δq MAB M A 15 V M V V µ [ 12 ( µν ∂ 2 − labelling the ≡ ≡ F 6 N M A particular consequence of the group property is The 56 gauge fields µν While the SU(8) indices were taken to be We use the space-time conventions of ref. [ µν AB 5 6 F F the same for the torusmatrices’ compactification. (hence the Any other distinction compactifi between ‘curved’ and ‘flat when considering non-trivial compactifications, one must di with eleven dimensions, and the SU(8) indices employed in ref. [ metric real matrix index is most efficiently encoded in the structure of its infinitesim The analogue of the external metric in terms of which the bosonic sector in ref. [ with previous conventions, fundamental SU(8) indices are ra also employed in refs. [ given by Here, the 56 non-abelian field strengths are defined as This is equivalent to with SU(8) indices JHEP09(2014)044 (2.18) (2.15) (2.17) (2.16) (2.14) (2.19) , , d tensor he correct L , KL Λ Ω K , = 0 µν MN ). The Λ-weights M N F Ω B U M 1 P 24 2.14 ∂ M , where we have also Λ 1 ∂ + − ) P L ∂ , Λ KL ) MN ralized diffeomorphisms in ). The full bosonic theory 2.1 ) . ) L M U table ] he Jacobiator of generalized α α f non-abelian gauge transfor- ( ∂ ν t t derivatives ( λ ( 2.13 K A ρσK δ om that are on-shell dual to the + group structure. The generalized , µν us to ( F MN K ) N µ [ α 7(7) U −F α µ N t NK ] , , . A L ν Ξ L ] µ Ξ ν +( Λ M A KL A K = 0 = 0 K N ) L MN L δ δ ∂ α MN µ Ω A A [ t K − L L ) take the form of the twisted self-duality M Ω ( µ 1 2 N N δ A [ µ K ). The projector on the adjoint representation ∂ ∂ ∂ − – 8 – ] A M N 1 + U 12 2.11 ∂ ( M M L µνρσ = M M 51 α λ B B K ), transforming in the adjoint and the fundamental + ∂ P , µν µ µ ∂ L N i eε Ξ F − D 18 δ 2.3 12 2 1 MN MN N K L K K ) M ] ∂ − Ω Λ L δ ν α = ) t , the latter constrained according to ( A M ( α 1 24 δ t KL MN M from ( of weight U ) ) µ M M α ) is compatible with the E µν K α = ∂ t M t ), equations ( ∂ F ,Ξ L K µν N U K + 48 ( α µ N + ( , , 2.16 [ B , Λ ) µ 2.10 , respectively. The latter form is a covariantly constraine , α M A α ] ] + 12 ( α t , ≡ ν ν ,Ξ AB ]. In turn, after solving the section constraint it ensures t ( = 0 = 0 7(7) α Ξ Ξ µν M K M M M µ µ µ 15 N N KL [ [ B Λ e V U M B B µ Λ Λ D D . ) Λ ) and ( 6 M α L L D M + Ω L t B B ( 2.9 = = = = 2 = 2 ≡ α 7 α M MN µ MN AB ) µ N e µν µνM Ω α L Λ M A B t B δ ( V Ξ Ξ Using ( The bosonic exceptional field theory is invariant under gene Its presence is necessary for consistency of the hierarchy o M , Ξ , , See footnote Λ Λ 7 Λ K Λ δ δ δ δ P whose commutator precisely closes into the field strength ( diffeomorphisms also give rise to the definition of covariant the internal coordinates, acting via [ ensures that the action ( field, i.e. it is constrained by algebraic equations analogo of the various bosonic fields and parameters are collected in is invariant under the vector and tensor gauge symmetries equations and duality covariant description of11-dimensional those gravitational degrees degrees of of freed freedom. representation of E on a fundamental vector with the 2-forms with parameters Λ mations and can bediffeomorphisms inferred [ directly from the properties of t JHEP09(2014)044 on and f the (2.24) (2.25) (2.20) (2.23) (2.22) (2.21) α A µν ǫ varying the B , 1 4 gian” in the ). The shift A µ ψ N after . rs. 2.14 µν β index in partial ] F ν 1 4 e N ABC − 8 αβ χ µνM µ [ F ω , ert term carries the Ricci , . ngian explicitly introduced) that , + MN µ M β M f their field strengths. They α ρ N ] N ed later. Note that e M ν formations that leave the field ,Ξ 1 2 µν K M e ) are to be imposed e , see equations ( just as the µν M B ∂ µν Ω K 1 8 N µ µν M 2.15 Ω [ N , , i.e. αρ µν N − B e MK ), the remaining equations of motion N A M B ∂ ) Ω M 7(7) K α 2 = 0 ∂ 1 2 MN α t µν 2.11 µ α 1 2 − F ) − of E M + ( ν N − M α ,Ξ . 1 α ] + N D α ) α µν ] µν ω M 912 ν – 9 – [ µν Ω µν ) which is the one appearing in the vector field B e µν Ω µν MN N αβ B K , g ) only via the combination/projection KL ∂ Ω M ∂ ( µν 2.20 M ) ∂ M K µ MN R α µ ∂ − 2.13 M t [ MN D ( ) M = = ), and the two-form ≡ α ( ,Λ µν 1 2 t α − A obtained from the covariantized vanishing torsion conditi M 2.1 e g α e V µ αβ µν ] µν M 12 ( ν αβ ) of the vector fields, this is understood as a “pseudo-Lagran A B µν 1 − B e ] such that the duality equations ( µ 48 Ω − b µ Ω R ω [ δ 52 δ 2.15 + top ∂ L b AB R ) should be understood as the tensor gauge transformations o 0 ≡ e M + . Λ-weights for the bosonic and fermionic fields and paramete V α = 2.21 ] ν e α is a parameter living in the µ , see equations ( 1 2 µ [ EFT is a parameter constrained in the index e N α D L ∂ Table 1 M M obtained from contracting the modified Riemann tensor N µν 0 = λ b appear in the field strength ( R µν field Other than the first-order duality equations ( Due to the self-duality ( 8 µν M with the spin connection derivatives As a result, we observestrengths the invariant following additional gauge trans B Lagrangian. Let us present thescalar different terms. The modified Einstein Hilb of the bosonic theory are most compactly described by a Lagra three-form gauge potentials ofalso the act theory on (which the weprecisely two-forms have drop due not out to in thestrengths. couplings Stückelberg the o projection ( transformations ( and Ω where Ω included the Λ-weights of the fermionic fields to be introduc sense of a democratic action [ JHEP09(2014)044 l , ), µν g 2.10 (2.33) (2.29) (2.28) (2.31) (2.32) (2.30) (2.26) (2.27) N ∂ µν g NCDEF M p ∂ . . M MN , along the compact NK CD ABEF στ N M V M µν M F V , 1 4 MCD p g , L ρ M V M ∂ N ∂ − ndary contribution of a D EBCD ∂ 1 ∂ AB g − N KL M N µν e note that in terms of the p NCD µν V ∂ g ]. ABCD µν V A 1 F M ABCD 3 N AB µ [ M F µ − , ansformations. Thus, we will in V M P ∂ AB i AB P ∂ ME g g M q µν ) with the internal metric ( ≡ = MN as follows M V 2 1 F µνρστ MN NCA ∂ C ] e 1 ABCD V M − 2.13 ), the potential takes the form B µ Y ε M µ − 1 4 refers to the covariant derivative defined 1 4 + 4 g 2 1 P M M 56 D µ ABCD − − ABCD d V 1 6 D + M µ g = MN eK A − [ ABCD P ( Z N NBC N N KL ∂ M M x = µ C p 1 , p V N ABCD 5 1 4 ∂ – 10 – µν p − i 1 d M Q M 3 F 2 − 5 KL − ∂ N e + g g Σ ∂ NCA = Z M M ] Lie algebra µνM V µ ABCD MN ∂ AB B ) and where KL i 1 F M 24 D A M to denote the resulting SU(8)-covariant derivatives. The CD M − ∂ M µ p 2.6 7(7) M V N g µ = N KL Q µ MN V M ) is given by Q D ∂ MN g ∂ ), it is easy to observe that the first two lines of the potentia MN M top NBC M ) + AB µ [ ≡ M L V e M µ 2.23 M MN ∂ D M i 1 8 2.32 1 Y 3 4 6 1 1 D 2 2.23 V AB − 1 M 56 48 − g − + ≡ d ≡ M 1 2 1 48 µ V B Z µ − ) = 4 D − ). This moreover defines the composite SU(8) connection D x µν MA 4 q d ) = , g 2.18 5 µν Σ AB ∂ , g Z M V ( MN V M ( V and non-compact parts of the E Written in the form of ( in equation ( simply contracts the non-abelian field strengths ( vector kinetic term in ( indicating that the 56-bein transformsthe under following local use SU(8) tr reproduce the corresponding terms in equation (7.5) of ref. while the topologicalfive-dimensional term bulk is integral most compactly given as the bou expressed via the standard decomposition of the Cartan form in terms of the56-bein internal and and modulo external a metric. total derivative For later use, w Finally, the last term in ( The scalar kinetic term can be equivalently expressed as where we have introduced the coset currents according to the decomposition ( JHEP09(2014)044 ) . α r- ees MN µν K 2.23 ] B (2.34) (2.39) (2.36) (2.35) (2.37) (2.38) ν . The M , L A g ∂ ξ δ from com- . K KL . ) α νρσ M µ [ α H ). Finally, the t A KL νρσ diffeomorphisms. M .... H uations of motion M ∂ , 2.15 µνρσ ( + M − ε 1 ∂ K µνρσ − ] N ε ν e ] µνρ N KL 1 ρ A − 1 H ξ 12 A e M e Einstein-Hilbert and the δ ν µ jection with ( M ∂ = = M D MN µ N ∂ nal coordinates acting as Ω s to second order field equations µ J [ 1 K 24 1 2 µ CD [ A + N AB ... , − V A M − M M V ∂ + . ) fixes all relative coefficients in ( M α µ c − b ∂ L . J − ] ] c ρ β µνρ λ ρ 2.34 N N AB e ] A ξ − e H ρ V ν N AB ), and where the gravitational and matter M = M ∂ N A V ∂ , ∂ ∂ ν K µ α ∂ µ [ L τλ MCD – 11 – [ 2.15 ] D ρ g ABCD ν M . Variation with respect to the two-forms V MN e A µνN 3 1 µ ) ∂ , A µν ρ α F ξ N AB ν ν g t + δ KL µ ξ D [ N ) − P K ) gives rise to the first-order duality equations describing N M V α A στ MN 12 ( µ t µ ∂ − D [ g , b J − 3 A M i 3 ( µν α 2.37 αβ MCD g e ν ν = − ) with the Bianchi identities µνρσ − V e ω KL ν ν ) are separately gauge invariant under generalized diffeomo µ ABCD ) M α M ξ denote the non-abelian field strengths of the two-forms ] M ] ] MN AB ) show that the two-form fields do not bring in additional degr α ie ε D P µ 2.15 ∂ t , 2 νρ νρ νρ N ( e D ), and ( 2.23 M F B B V − ν ) gives the field equations for the scalar fields parametrizin − 2.39 µ µ µ + [ [ [ β + N MN µνρ M e α ∂ α 2.35 D D D µ H M µ 1 2.23 3 M α e − µ µνρ M µνρ ), ( e νµ ν = 3 = 3 µ ABCD 2 ) may be compared to the second order field equations obtained D i e F H H D and P 2 α ν ν µ ρ ρ ) in the internal coordinates. In addition, the full set of eq 2.15 ξ ξ ξ ξ ξ α yields projections of the first-order vector field equations ≡ ≡ − = 0, this reduces to the action of standard four-dimensional 2.35 µνρ M = = = = = µνρ M 2.19 µνρ M M M H N α H α µ µ ) M ∂ H µ b µν M µ J α e MN J µν t B µν M ξ Variation of ( All five terms in ( A ( B δ B ξ ξ M ξ δ ξ δ δ δ Remarkably, the invariance of the theory under ( When and Strictly speaking, the second equation only holds under pro Equation ( of freedom to the theory. the dynamics of the two-forms bining the derivative of ( currents are definedscalar by kinetic the term respective contributions from th and thus uniquely determines all equations of motion. variation of the action with respect to the vector fields lead phisms ( is invariant under generalized diffeomorphisms in the exter first-order equations ( after combining with the derivative of ( and the Einstein field equations for where Combining ( JHEP09(2014)044 ). (3.1) (3.3) (3.2) (3.4) . 2.16 SU(8), A bosonic ǫ e covari- × − ) of SU(8), = , 7(7) ) carries the ). Under ‘in- A ǫ position of the 5 3) spinors, and ABC 3.1 γ 2.1 , AN χ , pace-time tensors X , ), the fermions are and K BN 5 2.3 BN γ lish the link with the X KM X B B ondition ( e been introduced in the µνρσ )Γ ) acts on Lorentz indices, vative in the internal vari- ǫ 1 MA M X µ A ory are SO(1 m the representation content − iantized) vierbein postulate ( Q e ∇ Q λ epresentation of E 2 1 2 1 = 2 3 zed internal diffeomorphisms, lo- + + . − µνρσ γ AN which in addition to ( AN (the matter fermions = 0 AK X X ), and the spin- and SU(8)-connections µ α X ρ 56 αβ ∇ AN αβ e K γ γ ρ X 2.18 ) under generalized diffeomorphisms ( αβ αβ µν MN αβ µ X M Γ γ Γ ( – 12 – ω ω ) they transform as scalar densities with weights λ − αβ − 1 4 from ( 1 4 α M µ + ν 2.16 + ω ], i.e. in particular e BN ) and in the D µ 1 4 48 A µ X -covariant geometrical formalism for defining derivatives AN D AN ψ B + X X µ 7(7) M MA AN D ∂ Q X = ), respectively. By construction, these connections ensur 1 2 = M indices, and has the form ∂ + AN 2.28 AN exceptional geometry 3) Lorentz transformations. This will allow us to couple the = , X 7(7) X µ (the gravitini . M D AN 1 8 7(7) D . As usual, for covariant derivatives on four-dimensional s ) and ( X E = 8 supergravity, or equivalently from an appropriate decom -covariant derivative The main new feature is that, like the bosonic fields ( M AN × 9 N 2.25 ∇ 7(7) X . The most general such derivative (denoted by µ M D Y is a generalized tensor of weight -covariant exceptional field theory to fermions and to estab For the internal sector, we similarly define a covariant deri For the external derivatives, the relevant connections hav We use spinor conventions from ref. [ 9 = 11 gravitino, it follows that the fermionic fields of the the X 7(7) respectively. transform in the D ‘ground up’ approach to beof described maximal in the next section. Fro E with the E SU(8) indices and E the covariant derivative is defined as 3.1 Connections In this section we setthat up are the simultaneously E covariant withcal respect SU(8), to and generali SO(1 if Likewise, we use defined by ( Christoffel connection defined by the standard (though covar here taken to depend onternal’ 4+56 generalized diffeomorphisms coordinates ( moduloas the given section in c table ables previous section. On a spinorial object in the fundamental r 3 SU(8) we may also introduce the covariant derivative ance of JHEP09(2014)044 B αβ ial also M (3.9) (3.6) (3.8) (3.7) (3.5) MA b ω (3.10) (3.11) (3.12) P Q MN , AB and its derivatives er can be obtained K ) by demanding that V ]. Thus, Γ V . 15 K , ) is given by [ 2.27 Q , , respectively. 2 1 Λ 3.3 biguities. In addition one MN e SU(8) connection , AK b P ∇ dified spin connection − . The required transforma- Γ ν β ∂ P X e . K . Under generalized diffeomor- − elated by a generalized vielbein Λ M Q α µ α ] ∂ P and te covariance properties, to wit: ), let us first require that the in- C Λ e ] ν ∂ MN Q e P b B , D , N K ∂ 3.3 3). As a consequence, P as parts of a single big vielbein. We ] M N ∂ , β µν K M V 3 2 µβ D µ M ∂ F e N A µ e [ [ µ + P Q ). In the following, we will discuss the α in a fashion reminiscent of Kaluza-Klein M A αβ SO(1 e P ∂ / M MN MC 3.3 M N N α [ in terms of the 56-bein π Q + 12 K µ M µν MK e P P + = , and 1 4 N F – 13 – Γ = 0 = = Λ α ] − AB MN µ is defined by analogy with ( β AB L MK e µ = 12 αβ M N Γ αβ e = M αβ V V M Λ M M , D ω L M M M ω α D AN ω ∂ µ = α [ e ≡ X MK µ = N e M Γ ) for the internal sector. In analogy with standard different αβ ∂ Λ δ M MK 3.2 AB b nc Λ in the last term of ( ω Γ N Λ and has the inhomogeneous transformation ∆ 3 2 V δ living on the coset GL(4) 1 2 M ) − αβ ( ≡ ∇ M 0 π ), the non-covariant variation of the first term in ( = 2.16 αβ M π . While in ordinary differential geometry, a unique such answ In order to discuss the remaining connections in ( The internal spin connection V M by imposing vanishing torsion,would like here the there resulting remain expressions further to am satisfy all requisi ∂ will denote the corresponding covariant derivatives by theory, whereby we view fields and the generalized affine connection Γ Later, it will turn out to be convenient to also introduce a mo including the non-abelian field strengths and This implies in particular, which is the analogue of ( geometry one would now like to solve this relation for both th ternal SU(8) connection and the Christoffelpostulate connection (or are ‘GVP’, r for short) definition of the internal spin- and SU(8) connection. with carries a weight of explaining the factor for the derivative without the Christoffel connection Γ tion rules for the connectionsphisms ( are determined by covariance where we recall that the covariant terms carry a weight of JHEP09(2014)044 tion (3.18) (3.17) (3.14) (3.15) (3.19) (3.16) nsforms as a K ], one can verify Λ , d as a generalized , ] 53 K 0 (3.13) Q ∂ ] K 15 R ≡ ] QP 36 ∂ KR Γ , 18 M N connections by requiring , Γ inglet under local SU(8). N -covariant variation of the 35 M W N nc Λ P ) ). As we will explain below, , V iantly under generalized dif- , and in terms of an explicit α K ∆ t en in ref. [ L . PQ 18 in a covariant way, as we will raint [ D ction satisfying these combined ( M P Γ 3.17 ng the fermions. s gauge covariant means that it R − P R V al derivatives according to P should transform as a generalized ). Indeed, it is a straightforward ) PQ M valuation of this condition yields M M α ∂ 6480 P Ω N + 4 t W . P Q + 2.17 . ∇ V 6 ( 4 MN MN E L , and − − PN Ω = 0 0 ≡ Γ 912 V − = = Q ≡ + K = 912 P L e

. Hence, the generalized connection lives in the W ) translates into [ K K – 14 – K 56 M N KP M ∇ 7(7) = Γ V P KM 3.13 MN denotes the generalized Lie derivative with all partial L Γ 12 ∇ K NK 133 = Γ ) and the fact that all other terms in (A.3) vanish by L − M M e ⊗ N T 3.7 M M P ∂ 56 = 1 NK 12 where − M e 2 1 ) − ) and using the cubic identity (A.3) of ref. [ ), the last two indices in the generalized Christoffel connec 4 3 = Γ N 3.6 M 3.12 V,W PM ( Γ NK T ), with the opposite sign. Hence, the generalized torsion tra T of weight nc Λ 3.14 ∆ V,W take values in the adjoint of E should transform as a proper connection under local SU(8) an the adjoint projector defined in equation ( K representations P B N ) From equation ( The first step in reducing the ambiguities is to constrain the M MA 7(7) generalized tensor. The factcan that be the set generalized consistently torsion to i zero. this trace must drop out in all relevant expressions involvi that the vanishing torsion constraint ( the section constraint. Thefinal term last in term ( is of the form of the non computation to show thatfeomorphisms. this From combination ( transforms covar calculation for the SU(8) connection in appendix the generalized torsion to vanish; this amounts to the const fixes affine connection under generalizedHowever, parallel diffeomorphisms to DFT and it is as notcovariance possible a requirements to express s as a conne a function of only also confirm in terms of a simplified example in appendix vector under generalized diffeomorphisms, while Γ In addition, requiring density compatibility of the intern Q with where we have used equation ( derivatives replaced by covariant derivatives. Explicit e for vectors where the second equality is obtained from contraction of ( Using the explicit form of the corresponding projectors giv E (Γ JHEP09(2014)044 B B V M ∂ MA 1 (3.22) (3.26) (3.21) (3.24) (3.27) (3.20) (3.23) (3.25) q CD,A − u ) for the V , ) 3.12 , , L , P GH P GH AB K has been given in V V CD N LN P ) are not sufficient K V N Γ ) EF V AB D M le with vanishing gen- ( K Q . M 3.21 M B A ( CDEF V δ ∂ , e Cartan form V NK B N M EF BCDE 8 1 p B ), ( Ω K D N MA p − V BCDE N AB PCD AB MK diffeomorphisms. The remain- W 3.20 N V ) is equivalent to the following , V CD,A P MEF i p K 0 ], but rather constrain it to the u + M V P AC ) and thus remains undetermined. Lie algebra. We note that M AC ≡ D B 3.17 V B A 36 ≡ P AC V − V transforms covariantly under local ABCDEF GH δ CD CD , V ǫ C MDE 3.21 8 1 N 7(7) M DE MA K V 18 1 KCD V KBC 24 U N D − ABCD ABCDEF GH N AB ), ( V V AC ǫ V : + − V CA,B V + ABCD − M ) N + 1 4 ] + M AC A + B M 3.20 V Q KBC N p Q ( − CD , u CD V BC V , p + MA M P – 15 – P 0 P ] CD,B + V R D NCDEF V V BC u ≡ p CD K ), it is now straightforward to solve ( + N ACDE M AB NCA ), the condition ( ) in the internal sector. Unlike in the external sector BC [ P A AC p D V ] V B P P AB EF V K K N ACDE V MCD + M 3.22 V K M p BC V V 3.12 2.25 CD ∂ V K MA V satisfies D i q K M CD,B [ D of SU(8). It is now straightforward to check that = DE V V Q Q B u = = NBC B M NBC AC NCD V AB in terms of the GL(7) components of [ B B A V QK QK V V Q K δ NDE CD,A , i K B MA 1280 V 3 4 3 2 2 Γ 2 Γ V u given by ( MA V V U B A M AC NBC δ Q − − − MA i V B ≡ , ∂ V i Q V 27 = 6 = 6 7 27 20 B i i ) are given by MA 3 8 4 27 P − + BC Q CD MA q ≡ ≡ K K 3.22 MN V V B B Γ P P D D MA MA R ]. With W AC 18 K K AB Next, we work out the most general SU(8) connection compatib V V SU(8), but neither transforms asing a pieces vector in under generalized ( transforms properly as a connection while along the compact and non-compact parts of the E Here and thus belongs to the drops out of the vanishing torsion conditions ( ref. [ which constitute the analogue of ( An explicit form of affine connection and standard geometry, theto vanishing fully torsion determine conditions thefollowing ( form internal SU(8) connection [ eralized torsion. Using equationconditions ( on the internal SU(8) connection and by are obtained in the standard way from the decomposition of th where the SU(8) tensor JHEP09(2014)044 , ) B B in ce B B MA 3.28 MA MA (3.28) (3.29) U MA U Q E U (we have covariant . Ξ B ] Ξ E B C Ξ B Ξ are Ξ orm, and thus MA s ABCD AB representation of [ W ABCD M affine connection, ). Indeed, , p ABCD M M ] p V ABCD αβ M C p ED M M K Ξ 1280 p CD M ω kable how the supersym- ED citly verified using equa- V M AB KM [ can then no longer be expressed CD tion, and as a consequence mes from M V 3 2 Γ isely those appearing in the M articular contractions ( B V tion M 2 1 xplicit expression for ed diffeomorphisms all terms ations. More specifically, now torsion-free − V en after imposition of the zero guities. Namely, in all relevant V 2 3 MA + 10 te this with a number of explicit D K 2 1 U − (Ξ) Ξ C ] 11 λ + , Ξ D C KM = 11 supergravity in the following section 2 3 Ξ C C Γ B ] D Ξ − Ξ C 1 6 MD a generalized vector, but the resulting q C MB 1 6 appear only in combinations in which the sponding + q B MD ] AB M q [ MB C M AB , + q D ) corresponding to the MA Ξ V M ] AB B M AB [ Q C we will derive the unique expression for V K – 16 – Ξ V AB Ξ M [ 1 2 3.22 E M AB 1 2 V , and with fully covariant derivatives, we have KM V − AB DE [ 2 1 λ + Γ ] M AB 2 1 alone. M C B V − p DE V of SU(8) were shown to be insensitive to the ambiguity Ξ + ] Ξ K M M C ), we have, for example M cancel out. Modulo density contributions resulting from B M ∂ (Ξ) 56 p DE ∂ Ξ ∂ that makes Ξ λ 3.4 M KM M M 3 2 M DE V and AB Γ V ∂ and [ ∂ of the connection is projected out and for which the covarian − M 2 2 1 1 M V 8 M M AB ]. In appendix ∂ AB V B 1 2 [ V V + + 36 1 2 of the connection ( M AB M , = = MA V V + + ] B and U B 18 C = = Ξ V Ξ ] MA B M C M U Ξ ), in equation ( , under generalized diffeomorphisms, the above combination D Ξ D M M M p ). In other words, despite the non-covariance of the Cartan f ∇ 3.22 AB ∇ ). This, then, is the most general expression for a [ M AB M V 2.9 3.25 AB and [ V ]. M AB M In view of these subtleties it is therefore all the more remar 18 M V By contrast, the connections to be derived directly from Such projections onto the q V 10 11 the affine connection would no longer be an SU(8) singlet. connection will no longer transform as a proper SU(8) connec in terms of only with second derivatives ofthe Λ non-vanishing weights of the fermions (see below), the p under generalized diffeomorphisms because under generaliz does not survive intions any ( of these combinations, as can be expli SU(8) represents the irremovable ambiguitytorsion that constraint remains [ ev ignored the possible appearance of the internal spin connec of covariant derivatives withsupersymmetry the transformation 56-bein rules turn and outalso fermionic to allowing field for be equ a prec non-trivial weight of where the part where the piece involving the trace of the affine connection co using ( in a covariant way in terms of ref. [ do satisfy the required covariance properties, but the corre metric theory manages to sidestep theseexpressions difficulties the and internal ambi covariant derivatives undetermined part equation ( expressions that will be useful in the following. Using the e under generalized diffeomorphisms is manifest. We illustra JHEP09(2014)044 . c . (3.30) (3.31) (3.32) + c ], upon 3 . B ] c . ν c ψ µ [ − γ A ], there is no term ¯ ǫ ) is the algebra of 3 below. MCD AC ponents (external and V Q ). The supersymmetry 3.31 3.3 V . AB ution to the vector field N 2.25 ., e, in the bosonic field equa- y obtained in ref. [ c rmore show in the following V , PBC tion ( . n the building blocks for the eed fully covariant under both wn in ref. [ V on ( e of the algebra on the 56-bein 2 MCD + c V √ ., F GH c = 0 . µ ABCD ) in the very same way as the closure χ µB + 2 ABCD must cancel. This fixes the weight of P E µ µ 912 M ] ψ ¯ | ǫ e Q K K Q A ) + c 3.13 ≡ P ǫ N MN 2 ¯ BCD Γ AB ∂ KM χ √ M 1 V again dropping out from this contraction of ABCD − µ µν ν D γ + 2 – 17 – ) take the same structural form as in the four- i e B P A [ ABCDEF GH ¯ ǫ µν 2.3 ε + 2 β MA g ABC µ ( e 1 U 24 χ ) as we shall discuss in section αβ KL N . µ QCD M + ) uniquely, and in agreement with the weight assignments γ = 0 )) in the supersymmetry variations of the fermions. Con- ∇ π V , ] µ N M ρ M ] ] 2.23 C ≡ ω ν ω ¯ ǫ A µν M [ α 3.29 µ 3.19 ∂ A µ Γ ), ( P AB ǫ e BCD ψ NCD δ V M χ α V KL AB D γ A M [ 2.15 (cf. ( N A ¯ ǫ µ M [ ǫ AB V e µ PQ A ) M ) can be expressed as + ¯ M D α V ∂ MN t 1 i 1 µA 24 MN MCD 2 − 2.36 2 ( ) ψ 2 Ω e V − − 3) and SU(8)) of the full spin connection as follows α α √ 2 , √ t γ ( 2 3 i = = √ A ǫ − − − M µ = = = ¯ = 2 J α α M µ AB µ e µν We summarize the structure and definitions of the various com Similar ‘miracles’ occur in the bosonic sector. For instanc ǫ M A B δ ǫ ǫ V δ ǫ δ δ covariant derivatives. with the undetermined connection tions, we findequations after from some ( computation that the scalar contrib transformations of the bosonicdimensional fields theory ( 3.2 The supersymmetryA algebra nice illustrationsupersymmetry of transformations. the properties Inhinges of particular, on the the the vanishing closur full of theon spin generalized the torsion connec ( vierbein requires the vanishing of the external torsi section that these expressionsimposition do of the agree section with constraint. the ones alread proportional to sequently, the density terms proportionalthe corresponding to spinors Γ in ( given in the table. Inlocal summary, the SU(8) above expressions and are generalized ind diffeomorphisms. We will furthe The various components ofbosonic its field generalized equations ( curvature contai As we will see in the following section, and as originally sho internal, SO(1 JHEP09(2014)044 ), α . µ c . e 2.19 , (3.35) (3.36) (3.37) (3.34) (3.38) (3.33) M + c ) consis- b M ∇ AB ] Λ Λ B 1 3.33 τ A ǫ 2) ψ A BCD ρ 2 χ N AB γ ¯ gauge ǫ ↔ 1 δ A . − µν ¯ ǫ ] (1 γ + V C which is invisible in . A ǫ M AB − [ M + + c.c. , ¯ ǫ B V M D i A N b ∇ 8 AB µ β B Λ στ µν M is confirmed by a standard M he supersymmetry algebra e 1 Λ g − ǫ B PCD and this variation is indeed α AB metry algebra. The algebra µν [ V eory, α αβ ry compact form given by µ . ransformation with diffeomor- αβ AB Ω ) + c.c. µ d gauge transformations ( e ˜ M M Ω M , , γ e , A SU(8) N µνρσ B ∂ V 1 A 1 δ 1 D α ) + ǫ i ǫ 2 , − B µ 2 ¯ B ǫ µ α A M i eε ǫ AB 1 γ γ 1 µ ) + √ L µ ]. The remaining (constrained) gauge M ), introduced in the previous section, ( ǫ ǫ P e µν ] 3 A γ µ ≡ V ν b ǫ + 2 ∇ A V ( 2 Ω 12 48 M ( γ M ǫ 2 A 2 ¯ ǫ b B 3.31 ǫ − M Λ ∇ − B 1 A 2 δ ] 3 . In particular, the variation ( ǫ α √ ǫ susy b ν B ∇ M D 4 + 2 ¯ K γ δ A 2 ), ( 1 ǫ ∂ ψ ¯ ǫ µ gauge + ¯ ǫ A [ µ A δ M AB + µ − 1 2 2 [ µν M 1 γ B M A ¯ ǫ V ǫ – 18 – γ 1 A M AB B 3.11 ] + b µ ǫ ∇ M P AB ν A ) + V µ γ αβ ∂ α ¯ ǫ i M ψ γ A µ αβ Ω 4 µ M AB M b ABCD 2 A − e [ ∇ µ M ) on γ − V . Note, that all SU(8) connections cancel in the vari- ǫ 2 µ γ V i − Ξ L A ¯ ǫ are not present in the four-dimensional theory and will M C ] i 4 B P C , 2 ∂ ν 4 A ¯ ǫ 1 − D ¯ ǫ 2 2.14 N ǫ ǫ α − = 2 ¯ A M µ A − µ A ǫ √ BC ] M µ Lorentz δ 2 2 ν D ξ Q A 1 A 1 δ ¯ ǫ (Ξ µν KBC ǫ ǫ ψ − M V µ α V ∂ + M AB M ) + Λ AB γ = 2 = = 0 . ,Ω D µ K M α V P D A α gauge α µ ν A µ P AC [ µ D D 2 V δ M [ γ e ¯ ǫ µ M V ψ µ β ∂ ν γ A A ǫ ξ + ABC ξ 2 δ µν A ( µ i e χ ǫ ¯ ǫ PQ NP ǫ 4 K AB µ D ) 2 ¯ δ KL Ω ,Ω i V )] = α D − i t 8 2 ( ǫ 8 3 = = 2 = 16 ( µ M − + Ω 8 3 − α , δ ), such that the external index is carried by = µ ) = = e 1 ] ǫ α N 2 ( ǫ 3.33 µ δ Λ µν M [ Ξ , δ ), with parameters B Closure of the supersymmetry algebra on the vierbein In terms of the full spin connection ( 1 ǫ ǫ δ δ [ 2.21 calculation: be specified below. The first term refersphism to parameter a covariantized general coordinate t The last three terms refer to generalized diffeomorphisms an compatible with the constraint ( parameters Ξ ( the fermionic supersymmetry transformation rules take a ve ation ( tently vanishes when again, as specified by the four-dimensional theory [ the four-dimensional theory canand be yields deduced from closure of t It is then straightforwardtakes the to same verify structural closure form as of in the the supersym four-dimensional th The supersymmetry variation of the constrained two-form as we show explicitly in appendix JHEP09(2014)044 . ) ) of .. GH c 2.34 3.40 ) can 70 . Λ (3.40) (3.42) (3.41) (3.39) ) with K N . ., Λ + c 3.38 ∇ c K L ) 2.21 . ) has been D 1 K EF K ǫ aviour ( V . Remarkably + c N of Λ Λ µν onto the 3.20 C V A N 2 1 γ N 1 ∂ V ǫ C ∇ ( 2 δ µ K ¯ ǫ 1 γ L N − , C 2 N ¯ ǫ V M α Q weight ] reproduces not only the P ν PBD orm fields can be verified P y algebra on the 56-bein. , e AB . P V P µ ABCDEF GH [ K 7(7) to appendix ǫ M Q M thus confirms the above su- under generalized diffeomor- ation V ] ] µν M D CD E orphism, the remaining terms i Lorentz transformation, upon ning gauge parameters on the 4 D ν α ., B M M ξ c µ + c.c. to a local SU(8) transformation. M CD CD − . s V ] D ] e 1 1 B − − Λ P AB 1 + 2 δ CD CD + c V V ǫ BC V 1 ) α Λ K − M µ 1 D AB AB ) in the four-dimensional geometry. AC N [ [ e V ǫ V b ∇ ν N P P M AB ∇ µν 3 1 . Furthermore, the first term in ( V V D V 16 αβ γ 2.25 − ν 1 i AB γ AB [ ) in the internal space. Specifically, ξ [ C V − 3 A 2 ( ) with the vanishing torsion condition in ( 32 N ¯ ǫ 2 + M ) but also the shift transformation ( + 12 + + 12 ¯ ǫ V A 1 V ν – 19 – 3.20 − ǫ i ξ AD K 3.40 µ µ 2.19 γ M ∇ ), and Lorentz transformation given by B 1] +6 D M AB ǫ V , cf. table A ABCD ABCD ABCD ] α V 2 ν 2 1 µ µ µ ν ¯ ǫ K i γ 3.37 e P P P 8 M QCB µ µ µ ABCD − ξ ξ ξ D V µ + Λ M ) = = P + = = = ∇ α AB µ ), we obtain ν µ ) and ( [ A ξ P e ) in the second equality. The second line of ( αβ γ 1 ν V CD ˜ ǫ Ω ξ = A [2 3.34 ( ): M ¯ 3.36 ǫ M ) reproduce the transformation of PQ 3.20 µ V ) D ] CD AB D α µ 2 t 3.35 ǫ and finally their rather unconventional transformation beh 3.38 M γ N ( i V , δ V A 2 N ] 1 ¯ ǫ ǫ 2 3 32 ǫ 32 δ i [ − − , δ µν M 1 ǫ = 8 = = δ [ terms in ( M AB α N 1 µ M M M − Ξ V An analogous calculation shows closure of the supersymmetr Closure of the supersymmetry algebra on the vector and two-f µν M AB µν M 1 Ω Ω − SU(8), since the remaining part willUsing entirely transformations be ( absorbed in action of generalized diffeomorphisms ( parameter Ω under external diffeomorphisms.persymmetry Consistency transformation of rules the and algebra determinesright the hand remai side of ( absorbed by the weight term associated with the non-trivial (and necessarily for consistency), closure on the two-form replaced by the corresponding condition ( We concentrate on the projection of the algebra-valued vari V be rewritten in the standard way into a sum of (covariantized)making diffeomorphism use and additional of the vanishing torsion condition ( While the first term iscan the be action rewritten of in the complete analogy covariantized diffeom to ( phisms as scalar densities of weight with parameters from ( where we have used ( by similar but more lengthy computations, which we relegate The Λ JHEP09(2014)044 ). ). B ), a ǫ as it 3.31 2.22 ation. (3.44) (3.45) (3.46) (3.43) (3.47) 2.15 ) trans- ]. Hence, ABCD , , ν ) 3.43 19 C P X M ABC ∇ χ αβ µ γ µν ) and [ γ . γ ( , i.e. satisfies ( αβ , B N ǫ 3.28 B reduced theory and the Mρ b ∇ . After some calculation, ǫ sections to spell out the 912 b MCD of motion obtained from the second term refers to R M y they transform into the V inger equation ( AB b i ∇ es and can be derived from . µ µν CD NBC ) M AB F V xternal spin connections ( αβ + 4 V 2 n into the components of the cur- tive M ν BCDA 1 4 √ ) that contain an even number of vector B b ω i P ǫ E ρ + ). The Rarita-Schwinger equation is σ ABCD ] ). It is straightforward to check that lives in the − γ 2 ( ν C ω 3.44 [ α P BCD ) are such that the undetermined part ). ] B − X 2.23 σ ρ 3.43 χ M ψ A µ D ρ − D ǫ µν + 2 ), ( γ , DE 3.43 3.24 γ ν ν γ satisfy the required algebraic constraints anal- M M αβ γ ρ V M 2 ∇ 2.15 N – 20 – µν precisely drops out, cf. ( [ ω 6 ) b ν ∇ √ µ ABCD ν γ M ) and ( M P γ ∂ − µν M Q M ABDE AB 3.1 EFT ( ≡ ρ ), we find AB σ A ∇ Einstein M P E ψ M B αβ V ), and upon using the first order duality equation ( ρ M 7(7) ǫ and Ω V 3.10 D = ( ∇ Mρ ν b 1 2 ): one can verify that all SU(8) connection terms above (which 3.34 γ µνρσ A R µ M ε µνρσ µ MCD − 1 ) ε V 1 ψ − ). It is instructive to give a few details of this computation ) and ( 2.14 i µνρσ = , which should combine into the second-order vector field equ − E ε ( i e A 1 ǫ C i e ǫ δ + 4 − 2.15 2.11 2 X e ] = 4 ρ − γ D ≡ − , ) is fully defined via ( A M µ ∇ ) even # [ 3.43

ψ A E µ ) ψ M AB Let us first collect all terms in the variation ( Under supersymmetry ( E V 0 = ( ( ǫ δ -matrices acting on using in particular ( These are the terms that carry precisely one internal deriva ogous to those given in ( vature associated to the various blocks of the internal and e γ forms as illustrates the embedding of the bosonic equations of motio the contractions of covariantfrom derivatives the in internal ( SU(8)equation connection ( somewhat lengthy computation confirms that the Rarita-Schw In this section wefermionic employ field the equations formalism andbosonic set sketch field up equations how in of under theof the supersymmetr previous the E form Moreover, the parameters Ξ would obstruct these constraints) mutually cancel. 3.3 Supersymmetric field equations into the Einsteinvarying and the the action second ( order vector field equations The commutator of covariant derivatives can be evaluated as where the first termthe describes ‘mixed’ the curvature of mixed the SU(8) spin curvature, connections and As required for consistency, the parameter Ω where the first twosecond terms can line be captures read theverifying off the dependence from supersymmetry on the transformation dimensionally the of internal ( variabl JHEP09(2014)044 ), e B 2.23 ǫ ). We (3.52) (3.54) (3.53) (3.51) (3.48) (3.50) (3.49) N ) , A e ∂ AB N ABCD B.5 ǫ B p ǫ M M N ∂ V 2 ∇ , µ − M ρτ e γ B µ g ABCD , y of these terms ( ǫ b , J N M i B M p MN ǫ ∇ AB , ∇ µ N . -matrices in the vari- νσ + 2 ) M γ g ∇ MN )) B 4 3 BC M C ǫ M B ǫ N M αβN ǫ + ∇ vector σ ∇ V 1 6 E e F γ µ CD N B στ ( γ ǫ N g ere we just focus on the addi- − ∇ − sms. Comparing the explicit 2 ( ] ∂ N MN mponents νρ N νµ M MBC d from varying the action ( ), we see that they are related by B A M ∇ γ δ V ρ F ∇ ∇ ∂ M ≡ − + total derivative , 1 γ ( 2.32 − B M ν ( MN e µν ǫ , B NCB A EBCD M g ∇ ) D ) + 8 δ [ V V N M ABCD ∇ N ν C p µ ν ǫ MN ν β ∇ + γ AC CD γ e A N P NM and combining into µ N M M µν ABCD 2 ∇ α vanish due to the section condition ( g V CB V 2 3 B ν A e µ M M R ME ) − δ 4 1 γ q M A P NBC 1 4 – 21 – N ∇ ( N − ǫ ] B V 1 2 ∇ , µν − ǫ V N + MN M ρσ g MCD ∂ − ( B A AB Ω F ∇ AC V M δ MN ). Putting everything together, we find for the M i µ M AB M ∂ M b ν + R ( J [ + V + M AB ABCD ∇ NCDEF V e 1 2 1 D 2.36 N N p ) using the fact that the following combination of covariant 16 νµ ABCD 1 4 4 1 V CB − 16 N F ] µνρσ e p MCB p ε MCD N = = 3.51 ≡ 1 M V ν M ] V V ABEF ∂ − R − MBC D ∂ i 1 from ( µν e + 4 AC terms in this variation gives rise to M e V 2 − AC 8 ν ρσ p [ N e ] − − Mν A + 4 M M V [ ǫ − M b µ R = = b V CD b N CD R J = 16 8 γ N N NCD ∇ + 2 V e V V V M ∇∇ 2

) ] ∇ AB AB AB A [ even # NCB µ 19

M M ) M V ) which should combine into the Einstein field equations. Man 3.45 A ψ V V V µ E ) 6 4 4 ). AC ( ψ ǫ 3.44 − − δ M E ( V It remains to collect the remaining terms with odd number of ǫ 3.30 δ 6 R ≡ − gives rise to the definition of the curvature which is invariant underexpression for generalized the curvature internal to the diffeomorphi scalar potential derivatives [ evaluate the full expression ( cf. ( arrange precisely as intional the terms dimensionally carrying reduced internal theory. derivatives H variation ( showing that all double derivatives Collecting all with the current Evaluating this curvature in particular gives rise to the co reproducing the second-order vector field equation obtaine ation ( JHEP09(2014)044 ). B ǫ , ρτ 3.44 g B ). A (3.58) (3.55) (3.59) (3.57) (3.56) ǫ ǫ N ρ ν ∂ precisely γ γ 2.23 mbiguities N νσ µν ndependent µν g , we present ∇ ) reduces to T ar field equa- ν M γ ∂ , ≡ T , M 3.50 στ B A g ǫ ∇ ǫ µν . ν ν as well as the single γ ) with respect to the γ δg A µνρ µνρ ǫ N A γ uation of the spin-1/2 γ ν ]. The operator on the ǫ µν ∇ γ ρσ 2.32 N A T 54 g ǫ M ∂ A N NCB ǫ NCB = ρσ ∇ M A µ e ial ( g V V ∂ ǫ tein field equations, cf. ( e ∂ γ µν N N δe emaining terms in µν γ AC M 1 AC e g e ∂ ∂ e ∂ − MN M N M 1 e N µν M M V − V g ∂ ∇ M NCB e ∂ ∂ e 8 2 1 N N V N ), and noting that the variation M µν − ∂ +2 + − − g ∂ ∇ e e A 2 e ∂ AC ǫ B ση M µν + − + 3.54 ǫ µ g MN M M g e ∇ e γ ∂ N N V ση 1 M ) can be written as N + M ∂ g ∇ − ∂ e ∇ e MN µ MN e ρτ N – 22 – γ M g N 1 4 ∂ 3.50 + 16 ∂ M ∂ M M M MN M 1 ρτ − e ∂ ∂ N M g − ∇ ∂ e A M 2 3 δ M 1 ǫ τη e ∂ ). In summary, the supersymmetry variation of the e 2 ∂ ν − g − B ) DFT case discussed in ref. [ 1 4 A M γ 1 4 e δ − τη − ∂ = + 2 1 3.57 g d, d ) and the expression above, the variation ( . Together with ( ρσ MN − ρσ MN R − δe g − e g µν 1 4 N M g e δ N M R 3.52 ρσ ρσ ∂ ∂ g N − g + ∂ ) correctly reproduces the full Einstein equations from ( N A M ρσ N ) = ǫ ) is such that the double derivatives ∂ ∂ g ∂ µν ν CB g γ M M ) ρσ 3.43 e V ∂ ∂ νρ g M N 3.52 disappear by virtue of the section constraint, and also all a terms are also absent in these terms. These terms, which are i g − ∂ V M MN ( ǫ ). Rather than going through the details of this computation 2 1 A µ ∂ δ νρ µσ , which under supersymmetry transforms into vector and scal ǫ M g γ g M AC ∂ − M ]. N µ µσ M 2.23 ∂ ( A ∂ g γ MN MN ABC ǫ 19 V χ µν µ M M g γ MN MN 16 8 2 1 1 M R ∂ M M Finally, a similar discussion can be repeated for the field eq The remaining terms in expression ( 4 + + 1 2 1 2 8 2 1 1 − + + of the ambiguities, can be further evaluated to give come from a variation of is a total derivative, we find that the variation of the potent with respect to the metric gravitino equation ( external metric is given by Indeed, ignoring the first term in the expression above, the r left hand side of ( showing that and precisely coincides with ( in a form analogous to the O( and gives part of the scalar matter contributions to the Eins derivatives drop out [ Together, using equation ( tions from ( fermions JHEP09(2014)044 ] of 7(7) 8 This = 11 = 11 , (3.60) correc- 7 D 12 ]; the full D , 7 3 M µ ABCD , 5 ∇ P ) , ow how they s theory after 4 D ν ], and has been ψ 4 µ ion; the detailed ABC and SU(8) struc- , γ χ 3 ν µ γ γ 7(7) etail in the two fore- ( ], where this strategy 1 N ABC = 11 theory in which the b y lifted from the dimen- letion of the bosonic La- ∇ ¯ ia [ χ duality group [ particular duality group in of refs. [ C e µ tension of the bosonic E D t the E ¯ y groups directly manifest in ψ 2 is subtle, not only because it 8(8) SU(8) covariant form. ymmetry variations of lly reduced theory). One main √ ities exhibited in the foregoing e of the reformulation with the ., ally establishing the equivalence ach of reformulating the higher- 3 1 c × n. . N AB of the g non-linear Kaluza-Klein ansätze − ding to IIB theory (with only six internal V f the E 7(7) + c i e objects and where the supersymmetry 2 ABC ], which have succeeded in providing an √ χ F GH 8 2 7(7) µ – χ 6 D − M ]. µ theory similar to the one presented in this section. b γ reduction remains to be established; this would in fact ∇ 56 B σ = 11 supergravity 5 – 23 – ψ S 6(6) ρ ABC γ D × CDE ¯ χ 5 ¯ χ e M 1 6 b ∇ − ν M AB γ -covariant expressions and those originating from V A µ σ A ¯ ψ ψ ], for dimensional reasons they are insensitive to 7(7) ρ ) of the section constraint relates to the reformulation [ 55 = 11 supergravity. D AB ν M γ D 2.2a A V µ ¯ ABCDEF GH ψ ). In the rest of this paper, we shall discuss in detail how thi µνρσ e ǫ In this section, we briefly review these developments, and sh i ε ), given by i µνρσ 18 2.23 2 ε 13 − = 11 supergravity is guaranteed at each step of the construct -covariant EFT is constructed in ref. [ − − 2.23 ), ( = D 8(8) ] and other references alluded to earlier, and described in d 2.15 15 ferm While the section constraint does admit a solution correspon There exist partial results along similar lines for the case o L 13 12 up to terms quarticsionally in reduced the theory [ fermions. The latter can be directl require a detailed analysis of supersymmetric E Independently of the constructionref. of a [ field theory based on a dimensions), the full consistency of the AdS tures become manifest (following the work of Cremmer and Jul tions. We have thus obtainedEFT the ( complete supersymmetric ex sections play a key role in establishing the precisebosonic relatio E tie up with the resultsof of the the two two foregoing approaches. sections, eventu involves As various we redefinitions, will but see, also the because full the identification ambigu was applied first in theadvantage restricted of context this of procedure the isoriginal dimensiona that the on-shellcomparison equivalenc between the E on-shell equivalent generalized geometric reformulation reformulation is achieved by startingsupergravity, from and the then known rewriting supers the theory in such a way tha taken up again recently in a series of papers [ the explicit solution ( the full (untruncated) 4 Exceptional geometry from supergravity is also anfor essential all prerequisite fields. for derivin bosonic degrees of freedom aretransformations assembled of into E the bosons assume a manifestly E grangian ( the final result in the compact form of the full fermionic comp going sections, there isdimensional the theory reciprocal in (‘ground such up’) ahigher way appro that dimensions. makes This the role approach of goes dualit back to the early work JHEP09(2014)044 , (4.7) (4.6) (4.5) (4.3) (4.1) (4.2) (4.4) AB # 7 p 6 AB p q , Γ , when con- Γ # q ! 7 ), and thus is 5 p AB q AB p 6 4 m p ]. Γ = 11 theory to 2.9 p with the bosonic 8 7 3 ords, the 56-bein A , ! p 3 5 D ! A p −V AB 5 2 4 7 p p p p = 4 1 3 p p ··· 3 , 3 mp p f the A p AB AB 2 5 Γ A , A hen to directly express this 8 identities ( p p , 2 2 asis of symmetry considera- 1 2 4 along the seven-dimensional m p p 7 4 p 1 1 qp √ m AB Γ with the one introduced in the ⊕ A 3 V mp ms of certain components of the mp ≡ V p + 2 , 2 A A 4 mnpqrs 5 p 21 √ p 2 1 2 AB AB nt to those used in [ p ··· 12 ⊕ ,A √ − 1 √ m A in eleven dimensions mn 5 2 + V p mp 21 In particular, it can be explicitly verified V 5 14 mnp √ , ··· p ] yields the explicit formulae . A under its SL(8) and GL(7) subgroups 1 ⊕ , a 7 ··· ,A + 126

1 qp 7 m a e MAB + 60 A 7! m 7(7) mp V AB mn AB p AB e

) → A × – 24 – V Γ 2 AB m , 2

5 of E Γ p √ √ 28 7 2 9! mnp ··· AB p 6! M AB 1 56 ⊕ A m ··· p + 56-bein + 3 − 1 2 are the internal components of the three-form field, and Γ V p " √ ≡ V 28 2 (Γ 7(7) / ≡ " 1 mnp → + 6 2 − / A 1 M AB ∆ 56 − AB 5 V p M AB , ∆ ··· V mn 7 1 p Γ m AB ··· Γ 1 2 2 mnp p / / η η 1 1 − − 5! 7! 1 1 · · ∆ ∆ 1 8 4 4 8 1 viz. is the siebenbein, = = = = a the internal components of the dual six-form field. In other w m AB AB AB e m m AB mn mn V V The notations and conventions used here are slightly differe V = 11 fields and their duals. The calculation [ V 14 mnpqrs whose existence in eleventions dimensions in was the postulated previous on sectionD the is b here given concretely in ter A four dimensions; this 56-beinprevious will sections. be Decomposing eventually the identified sidering the embedding of GL(7)56-bein into SL(8). in The terms main ofdirections, task components is t of eleven-dimensional fields where we have the following decomposition of the 56-bein degrees of freedom that reduce to scalars under a reduction o The first step is to identify an E 4.1 56-bein and GVP from eleven dimensions where we will often employ the simplifying notation where ∆ is the determinant of the siebenbein that the 56-bein defined by the components above satisfies the JHEP09(2014)044 - . ,m lds 7 7(7) ] ! 7(7) , 8 (4.8) (4.9) m – ...n 7 1 ! 6 n , as ex- of E 5 ation as , 6 p µn 3 n 4 f the last p AB A 56 A 3 p 5 M n that incorpo- V A 4 δ ., n covector. From 2 3 c M , m . p n µ 7 1 n A 7(7) in the obvious way A qp 6 + c ) n 2 A , q n A EFG 1 µ ) χ 5 µB B pn D pmn ψ ¯ ǫ e variation A ...n = 11 supergravity, due to − expression above. However, A p A 1 existence of suitable actions (as well as an undetermined are here obtained by ‘E e the precise form [ p 2 ǫ µ f the six-form potential in the reduction to four dimensions, mal supergravity into a single p pn µ cation of the EFT formulated D ). To show that that it is more 2 ¯ 1 B ,m ation of the new field A B R vector fields 7 √ e, the components in a GL(7) de- the last seven components of the µp p − µ − A ) directly, or invokes eqs. (14), (17) 2 ...n B n 1 + 2 7(7) 1 2 2.6 = 11 supersymmetry transformations, µn − 4 ABCDEF GH µmn µn √ 5 ε A A D A ABC ( 1 − 24 χ ...n 2 1 2 5 µ p + 12 √ γ √ µn , ··· ] -plet of E 1 C A . ¯ ǫ A transforms as a generalized E qp = 3 – 25 – 56 µ ) + , which is introduced to complete the A BCD 1) ( V ψ m q χ α 7 ,m µ AB − 7 γ A µ mn [ pn B c N A ¯ ǫ A ...n ǫ A V 1 − p = 11 supergravity can only be achieved at the level of the µ + ¯ 5 µn + (3˜ p MN B D A ··· ], this matrix corresponds to an element of the coset space can be explicitly written in terms of eleven-dimensional fie µA 1 ,m MCD 7 − 7 ψ 2 Ω V , µp α dual graviphoton m 3 2 √ 7(7) ...n A 7 γ i 1 √ A

µn ǫ − µn ), such that after a local SU(8) rotation the direct identiﬁc 5 of E A ]. While A ( = = ¯ = 2 ]. Strictly speaking, the supersymmetry transformations o ...p 7 6 7

1 -valued matrix one either verifies ( α 56 7 M , µ AB µ ...n ] where it is shown that 3 , 1 objects are found by analysing the ...n 7(7) mnp δe 8 M A n 1 η A, B, . . . m δ V n µ 2 δ η cA component, there appears a new field 7(7) B √ 1 2 + ˜ ), related to the c µ m = = 6 = 36 A SU(8) in a specific gauge (where the local SU(8) is taken to act m / mn These E In the same manner, one identifies a µ µ m µ A A 7(7) A plained in refs. [ as discussed in ref. [ where a compensating SU(8) rotation has been discarded in th the point of view of refs. [ rate the degrees of freedomcombining corresponding the to 28 vectors electric under and the a representation 28 that now magnetic live vectors in of eleven maxi dimensions.composition of As the befor the absence of acovariantization’. non-linear The supersymmetry formulation transformations of of vector dual field instead gravity, determine but the supersymmetry transform seven components of the vectors cannot be derived from and (18) of ref. [ E The components of the six-formin potential the appear again in the which in the SU(8) invariant reformulation were found to tak specifically an E indeed an element of the most general duality group Sp(56, constant ˜ given above is lost. Noteexpressions, also the as appearance a of consequence componentsin o of the whose previous presence section and the the equations identifi of motion (which,for of either course, formulation). does not preclude the on the indices JHEP09(2014)044 ) , ) by 3.25 ] ] (4.10) (4.11) (4.13) (4.12) (4.14) C ǫ 3.24 bcdef CD ] µν Γ C γ ǫ , AB a [ B AB mn AB AB [ ǫ [ Γ Γ µ ) and ( m γ e DE β B ′ ǫ ν m 3.23 e , αβmn B P oson fields as derived m F A mabcdef ABCD s ( 2 ∂ µ F DE / ′ m 1 Q m 5! ore proceed differently by µ β AB abcdef P the GVPs (see below). − 2 e e · ntial that is related to the Γ + ations of the fermions were 2 ∆ t. Furthermore, the direct 2 √ e. This is because the latter hus require an analysis of the ing sections is thus manifest, CD i A 2 56 3 √ √ 2 ǫ m even-dimensional fields cannot B dom. The link of the particular 3 2 e ǫ 32 √ − M αβ µ 1 2 ] f freedom from a four-dimensional ertheless, our ignorance regarding ∂ − m AB − γ γ mabcdef + ] Γ − M D νρ F 2 αβ bc CD C ǫ µ | / ] γ µ ), and C Γ ǫ 1 6! with those of the previous two sections ǫ β C ω 2 nAB − · ρ = 11 supergravity yields the expressions µ αβ 2 1 4.4 m AB Γ e √ γ − A a [ ∆ γ µ n i | 14 ′ m D D m C Γ µ + ν B ∂ ∂ Q = + B A AB [ ) ǫ ′ m B ν β ≡ ). However, this would lead to extremely cum- mabc AB e m [ − Q αβ AB abc – 26 – F for a simplified calculation), whose relation with ∂ m m G 4.7 m Γ mAB ∂ 2 e m 2 e µ D µ 2 32 m √ m AB √ , )–( B m AB µ e 2 = B = 11 fields. Of course, ignoring the ambiguity ( √ e E mabc m 1 4 + B 3 ǫ 2 1 4.4 ∂ − F ] + 6 D given above in ( AB 4 1 − − − µ 2 D ] m ∂ b CD µ ǫ ) + 48 B ( √ − e C ∂ µ Γ ǫ AB ν B ABCD ǫ ] µ γ ǫ m + ′ β M m m γ µ ∂ AB e a [ ∂ γ P V µ m ( αβ Γ ab AB α µ γ [ AB Γ m DE [ ABCD e B viz. ∂ 2 µ m m m ab AB / e e − ], P 1 p 2 2 m ab 3 2 − αβ µ 4 3 q ∆ ∂ G √ √ m AB √ i 8 1 2 2 e ) in terms of the − 2 − + 2 + 3 − + = = ≡ − 3.22 = = 2 B A µ m A ′ δψ ABC αβAB Q mABCD ′ G δχ While the agreement in the supersymmetry variations of the b P where is easily seen by noting that is just part of the 56-bein comparison with the fermion transformations of upon taking the canonical solution of the section constrain comprises the contribution from theexpressions spin involving one the degrees Kaluza-Klein of vectors free bersome expressions (but see appendix the ones given belowstarting would be ‘from far the from otheralready obvious. end’. derived We in The will [ theref supersymmetry transform for the moment, wesubstituting the could explicit simply formulae ( try to work out the expression tensor hierarchy point ofbe view, determined. its direct This relationthree-form is potential to in via the an stark el explicitthis contrast duality field to relation. is the Nev compensated six-form by pote the fact that it does not appear in for the vectors, is clearly related to dual gravity degrees o above and the exceptional fieldthe theory agreement approach in of the thedepend fermionic forego on variations the is connections, much and moreconnection a detailed subtl ( comparison would t JHEP09(2014)044 V m ∂ ] this (4.17) (4.18) (4.15) (4.16) 8 m ABCD this con- P e we have AB , and M representation , V B well as providing 35 is the SU(8) con- abcdef AB . CD Γ m A ] B M Q V ]. Here we concentrate n the four-dimensional bcdef CD 3 ′ m A Γ ) Q b otation as in the previous | tly different choice below, mabcdef ed in the previous section. AB n ly covariant under internal in the approach of the pre- a [ ) and F e ] is the so-called generalized ): mponents of Γ are equations satisfied by the 8 m ABCD m 6! s generalized connections and | 2 , P heories and the embedding ten- · ein which are analogous to the ∂ = 11 supergravity allows for a 4.14 7 equisite covariance properties, as n √ ] in the form and its internal derivatives = , a s first structure equation takes the form 14 8 ( 3 D C e , ] V by zero. As explained in ref. [ mabcdef ]), where the explicit expressions for + 7 8 F ≡ , MB 5! abc AB 7 2 m ab transform as proper vectors under inter- · V Γ p m ab √ A [ 56 C m not mabc and − Q , p ] F ] – 27 – do b | + 2 ] they can be checked explicitly on a component n bc CD m ab 48 8 e √ Γ ) (see refs. [ ω m ab ). However, there appears to be no way to reproduce , m | + p 7 defined in equations ( AB ∂ , and this is one of the main difficulties in establishing N AB a [ , n 2.27 V V Γ 3 4.18 a ab AB [ N and e Γ M mabc ≡ = 11 fields were already given in ref. [ m F m ABCD m ab m ab ′ Γ D ) and ( q 2 ω P m ab 32 − transforms covariantly in the complex self-dual 2 1 √ q 4.17 = = and ), differ in their components relating to the siebenbein sinc ), which satisfies all covariance requirements. B M AB B V . ′ b m A m ABCD 4.18 m 4.18 e m A ∂ P by the spin connection Q ′ ∧ m ABCD ] that defines them from a higher-dimensional perspective as Q b P a in terms of the 45 ω m ab µ q , ) and ( + P a 42 e 15 , 4.17 The external GVP, which gives the dependence of the 56-bein o The other important feature of the reformulation [ 8 and Note that in this paper our conventions are such that Cartan’ = d µ 15 a sists of certain differentialusual equations vielbein satisfied postulate by in56-bein differential the geometry. and 56-b The in GVPs the approach of [ agreement between the above expressions and the ones obtain without ‘breaking up’ the matrix nal diffeomorphisms. Forsee this ( reason we will switch to a sligh coordinates is given by equation ( diffeomorphisms, because of SU(8). However, as written, these connections are not ful nection, while defined above, ( these covariant expressions in terms of the 56-bein replaced change is required if the connectionsis are indeed to the satisfy case all for the ( r Q on the internal part of the GVP which was given in [ generalized geometric structures thatused to can construct be a interpreted generalized a curvature tensor. by component basis, whilevious they section. appear as genuine Moreover, postulates the direct comparison with vielbein postulate (GVP). When evaluated on the different co T direct understanding of four-dimensional maximalsor gauged [ t where Notice that are the components of thesection. GL(7) Cartan These form, objects with analogous transform n properly under local SU(8 where JHEP09(2014)044 α sh m Γ 7(7) (4.20) (4.21) (4.19) (4.22) (4.23) ) of the ) involves ). Finally, 3.27 . , ( 6 3 p 3.12 2 3.12 p tail below, this ...m 1 1 p | part of the SU(8) and will thus drop pm m F Ξ ), see below) differ in ero term proportional 3 4.3 1 7! p 2 1280 orm as ( p − 1 4.24 ] p constraint, ( 6 . 4 n m , grams, and thus encapsulate mbol, indeed transform with be expected for a generalized 5 ··· -form and the six-form fields. transforming as densities, are n the 1 = 0 m n mn n s. 4 ] Γ 6 η m . 2 4 3 A 1 N ...m ] √ We will show below how to absorb − 1 6 M 3 m m ) (and also ( ) | = = 5 m p we can solve for the affine connection 16 α [ takes values in the Lie algebra of E 4 8 2 . m t 8 n 4 Ξ ( m ) 4.16 N = 11 supergravity; we use boldface letters 1 ··· m α 1 m m M A n m [ Γ D ) 3 p ( m Γ ABCD F m m different from the ones identified in – 28 – Γ m 2 = 1 Γ 4! ) and ( P m ( ), 1 N , − m [ M , 3 p 3.12 4.18 and m m , F = 0 2 6 Γ 2 ) involves only the seven internal dimensions with index ] p B m pmnq 48 , can appear in the supersymmetry transformations of the 1 √ ··· F 1 m mnq p [ m A | 4.16 q mq | m + 1 4! ) and ( p ] for details). Another noteworthy feature is that they vani [ A Γ m Q Γ 6 8 p p n Ξ − Ξ m = 11 supergravity allows to solve for the components of δ D 6 4.17 ··· 1 4 generalized non-metricity p 1 D = 11 fields; the non-vanishing components are ··· mnq m + 2 1 )) and the hook ambiguity described in section 2 A A D √ np in terms of the fields of p p p mn η D D 2 − Γ N 3.25 √ ≡ ≡ 6 mM ≡ − = is the usual Christoffel symbol, and where m Γ p n p mnq ··· | 8 n 1 p ) ) mn m Ξ . The second distinctive feature is the appearance of a non-z m m | on the right-hand side of the GVP. As we will explain in more de p Γ Γ m ( ( Ξ m The internal GVP as given in ( Given the coefficients We would like to thank Malcolm Perry for pointing this out to u = P 16 where Γ term corresponds to a to directly in terms of here to indicate that these coefficients are previous section. With ( M this non-metricity, and therebythe connection bring coefficients the GVP into the same f out in all relevant expressions. two respects. Firstall of 56 components, all, whereas and ( prior to imposing the section otherwise only sensitive to the local SU(8). fermions only via their traces, because the fermions, while The comparison with coefficients Fortunately, the apparent discrepancy turns out to reside i connection (see ( One notices that thesesecond objects, derivatives of like the the tensoraffine usual gauge connection parameters, Christoffel (see as sy ref. would [ under full antisymmetrization: Therefore, they correspond tothe hook-type non-gauge Young tableaux invariant part dia of the derivatives of the three JHEP09(2014)044 B M ear ∂ and and, m A (4.25) (4.24) ishing 3 Q 189 . Γ ⊕ have all the , we will first 21 3 P s a generalized ). In this form that the section onents only along mN transforms covari- CD Γ ]). That is, 4.16 N 8 V hibit their precise rela- and contrasted with the one . Hence, taking ABCD transform as generalized the local SU(8) has been ore “natural” from a gen- m 7(7) P ike the partial derivative representations, respectively; see ref. [ truncation of generalized Lie M ABCD s). will become the ally be required if we want to ABCD ts had non-trivial components ctions in section (8) and generalized diffeomor- P r to achieve full agreement with ABCD w of the derivation given in the of E thermore, the generalized affine int is why all other components m , 48 ized affine connection m = with the proviso P 8 we can formally write the internal P , 56 of SU(8). We will also see below C m ), ] B and = NB and m A 3.12 V M 1280 Q B A 210 [ if C M m A Q Q to part of a – 29 – . However, given that vanishing torsion is taken + m 3 m ∂ 0 otherwise ( ) is crucial for the absence of torsion in the sense of P AB = representation, as has been assumed in section V 4.23 P M ∂ 56 MN Γ − components of the connection coefficients with those that app ]? Indeed, we will see below that the introduction of non-van m ] as a symmetry), Stückelberg-type while N AB 3 V 18 ), with all other components vanishing, gives back ( M ∂ is invariant under SU(8) transformations, and transforms a ), is that, at this point, the connections have non-zero comp 4.16 Γ of SO(7), all of which appear in the 3.12 27 We now proceed to reformulate these structures in order to ex As given above, the connection coefficients A distinctive feature of the internal GVP as given here, to be ⊕ and identifying the in equation ( tionship to those constructed in section transforms as an SU(8)introduced connection in (as ref. is [ obvious from the way 21 after imposition of the sectionGVP constraint. as Nevertheless, by trivially promoting the GL(7) index the formalism of the preceding section. to be an important ingredientconsider for the defining generalized generalized conne torsion associated to the general for example, ref. [ connection components along therecast other the directions supersymmetry variations will of actu the fermions in orde the internal GVP can be compared to equation ( the seven internal dimensions, but vanish otherwise — just l generalized geometry. of the connection coefficientseralized should geometric vanish. point Would of itin view not the if be the other m connection directions coefficien of the constraint also applies to theforegoing connections. section, a However, in natural vie question that arises at this po desired transformation properties withphisms, respect as to can local be SU verified explicitly from their definitions ( In terms of SL(7) these Ξ’s correspond to the connection (with a second derivative of the gauge parameter antly under SU(8) transformations. Both derivatives to vectors with onlyconnection seven vector indices). Fur vectors under generalized diffeomorphisms (for the natural when further decomposed into SO(7) representations, these that the irreducibility property ( given in ( JHEP09(2014)044 P ). mN given 3.14 (4.36) (4.29) (4.28) (4.27) (4.33) (4.32) (4.26) (4.35) (4.30) (4.31) Γ P mN Γ . 8 r 8 The generalized q . 8 r P Γ omponents of 8 MN m independent) definition 8 Γ . q . ) are equivalent in usual 8 st st n y substituting the relevant 8 8 8 indeed vanish. For example, p m m . 3.13 P ] 8 8 , ometry. Here we will evaluate r p n u a priori ) δ Γ q Γ s r S 8 δ , st p m + 16 r ] 8 P . [ r t δ r , which leads to the formula ( ] , 8 ∂ r st u δ 8 3 r = 0 (4.34) s 8 P [ n Γ 8 + ] . pq n stu pqr 3 2 q r m pq δ δ pq r δ 8 m pq r P pqr MN [ 8 1 p | q s P − 1 12 = 0 δ Γ T m Γ p 24 [ ] 24 = 8 − 8 (2 Ξ p r = − 8 8 − 8 = 4 m n = 1 q n [ ∼ 96 S – 30 – 8 is defined as follows pq 8 8 ] 8 st u ] Γ 8 st ) of section n 8 N m 8 n P = 8 pq r r pq r p T 8 8 ∇ 8 8 8 = 2 pq r P s m , n 3.13 m m pq r MN m 8 P [ 8 ]. An alternative (and Γ q pq M T Γ p 48 8 is defined using the connection T 8 Γ 8 P r ∇ = n = [ 8 − 8 M p 8 8 pq P m p ∇ 8 8 T n ) explicitly in terms of the connection coefficients ). Hence, pq r n 8 8 8 m m m T 3.14 4.20 T Γ = 8 p and where from ( ). Finally, consider the following components 8 n S Γ 8 m 4.23 T ], the generalized torsion ] and above. A simple component-wise calculation using the c 8 8 in ref. [ However, the right handcomponents side of of the above equation vanishes b the above equation reduces to Using the fact that While the above definitiondifferential of geometry, this torsion is andthe not that generalized the torsion defined case ( in in ( generalized ge identified above now shows thatconsider the generalized torsion does by equation ( However, torsion as defined above vanishes [ the above equation reduces to In ref. [ 4.2 Generalized torsion for some scalar Using the fact that of the torsion is given in equation ( Next consider, for example, Using the fact that JHEP09(2014)044 ) ], ). 4.23 3.28 (4.41) (4.37) (4.39) (4.40) (4.38) abcdef . D ε . , ] ] and [ ] m ED 5 ED Γ abc m bcdef CD ...t 2 e Γ ) is zero ], [ ), the supersymmetry ab mnt a AB | [ [ ABCE 1 3.14 Γ t ABCE m 4.10 ′ antisymmetrization of the o first equations of ( m tions are insensitive to the , ′ tions based on generalized δP P + 5Ξ he right hand side, is that they hiral SU(8). With the connec- 4 hing of the generalized torsion, abcdef | 5 = 11 supergravity. d torsion can be similarly shown n the following combinations of (7) s involving the trace of the affine abcdef AB nections. We first recognize that m − + 4 ...t ] Γ ) admit a non-trivial kernel which Y D ] 1 , C D C D B + δ mt δ | , ε ] 4.39 n µ ] AB γ AB Ξ [ abcdef [ . | bc CD symmetry conditions are imposed on the = 0 (7) Γ m − ] ] EF in the second equality and equation ( EF q n 5 5 X , = 0 m δ m a AB ′ pq [ [ ′ p ...t ...t [ + 8 P 1 1 Γ r ABCD P n 8 δP nt nt – 31 – 8 m | | a priori ′ m abc AB abc ABCD m EF MN m | m ′ m 8 [ P Γ m EF T (4) r m Γ m Ξ Ξ e Γ Y 5 δP CD abc 3 Γ 5 | ...t + ), is crucial for this argument. m (4) m 1 − + 2 ] ...t + 3 m CD e 1 X Γ D pqt D pq ], or equivalently from equations ( − ] ] b CD 4.23 8 pqt η 3 + − C C Γ n 2 η B 8 m 2 m B ′ √ ′ m ab AB AB √ a [ m C Q ′ Γ Γ Γ δQ m C ′ Q ab AB = 21 = 3 = | ab [ | AB (3) δQ m (3) [ m m pq m e 8 Y X m AC n m AC e 3 8 simply indicates that no = = | m B T + + 0 = Γ 0 = 3 Γ m A ...... ′ ’s other than the obvious ones (to wit, anti-symmetry in [ ′ ABCD m m ′ P ∝ ∝ Y δQ δP A µ In summary, the generalized torsion, as defined by equation ( and δψ ABC ’s and ′ m δχ X where the slash Secondly, the expressions on theis right found hand by side looking of for ( solutions of variations of the eightQ gravitini and the 56 dilatini contai in the final equality. All otherto components be of the zero. generalize ItΞ should quantities be is zero, emphasized equation that ( the fact that the full where we have used the expression for An important property of the expressionsare appearing actually here on insensitive t to certainthese modifications are of exactly the con the same combinations that appear in the tw Let us proceed with the following ansätze 4.3 Hook ambiguity As we have already mentioned,generalized affine the connection, supersymmetry modulo transforma densityconnection, contribution because the fermions transformtions only as under originally given the in c ref. [ Let us emphasize again the remarkableas feature that originally the vanis definedgeometry, on here the follows from basis the of direct comparison very with different considera we obtain JHEP09(2014)044 ) two and ]. 4.17 npqrst is the | (4.45) (4.48) (4.42) (4.44) (4.43) (4.46) (4.47) 41 cancel, (4) m have p mn bb | ] that the ], the first ,Y (3) a (7) 41 41 (4) X and Ξ X X ) there are the npq | ], equations ( and 4.14 . . 8 eomorphisms. The m ] that ) are precisely of the (4) , rm that was obtained = 0 = 0 41 X ) provided that tween the expressions for D p mn 4.22 ] es from the ‘leftover’ term f. [ AB a Γ C abcdef | δ Γ n freely, as they drop out in 4.40 p b he siebenbein and Γ ven-form field strengths) are (7) m bb e | . dy shown in ref. [ (3) a AB a X a [ , which is required here because n Y ) and ( Γ k 3 2 ), and in ref. [ appearing in ( e = 0 bb 1 2 | ). Note that Ξ − − km ] (3) a . 4.21 4.14 = = X ab AB b | 4.39 − . (3) Γ m abcdef | X ) becomes m (7) m ab [ abcdef D | ] 2 3 = 0 ab ’s. Notice that both ) except the last ones involving | ω (7) (3) ab C m a c − ] Y 4.40 Γ b Y + , which was not considered in ref. [ | explanation for the ambiguities found in [ – 32 – 4 3 a ,X = 4.46 (3) [ ab ], equation ( | AB n a ,Y c [ 3 X (3) + m e ab | Γ = 0 (3) m X ’s and m ab ) is indeed satisfied for a torsion-free affine connection. abc | ] ∂ | − Y (3) a b X (4) m ) and ( n D abc X ] | c e ] 4.47 X 2 b C m (4) [ 2 3 Γ 4.44 2 1 geometrical X ) reduces to + = a AB = [ [ ab abc abc AB 4.40 2Γ | | Γ (4) (3) m m Y bc bc X | | (3) a (3) a X X are in the kernel of the supersymmetry variations ( ). These connections are fully covariant under internal diff ) which is just a density term proportional to Γ (7) 4.18 ,Y As for the remaining SO(7) part To interpret the remaining term let us check the difference be 4.46 (7) usual Christoffel symbol. Hence ( irreducible parts: besides the fully antisymmetric pieces the supersymmetry parameter is a density. This is the same te With this identification the second line in ( X The only extra term inin ( the supersymmetry variations then com and ( difference is thus expression in equations ( hook-type, hence providing a hook diagram contributions. Furthermore, it was shown in re GVP remains valid if provided we demand that where we have used the usual vielbein postulate satisfied by t with no further restrictions on the respectively). For the form field contributions it was alrea Whence we read off the condition determined, but the hookthe diagram contributions supersymmetry can variations be of chose the fermions in ( That is, the fully antisymmetric parts (the four-form and se We now see that all terms in ( that appear in the generalized affine connection in ( the connection coefficients given in ref. [ JHEP09(2014)044 . ) ) e k n ]. V ∝ ∝ . 3 and km k k 3.1 4.24 4.18 ABCD km km mn ] can be = 0 for m 8 P , ξ e Γ M ) and ( here can be P cancels with mn k ξ K = , and only then 4.17 andard differen- ]: as given, these km ′ avity. First of all, m 3 ), where Γ M KM e’ as derived from P Q auge. Furthermore, 3.19 f connections (both of transforms covariantly and ations ( nections is possible here tion given previously in . ctions have non-vanishing ecific connections to bring ′ m parameters ). nnection as in ref. [ Q alone. We thus see that the = 11 fields. It is not possible of the generalized vielbein namely the one that accords rtional to Γ D going sections are themselves ABCD ts as inref.[ er, it is straightforward to see ities identified in section ] because D ns rphisms: for the 7-dimensional m 8 , while the other into fact that they do not transform ′ P m ab B P m A solution where the coefficient of the Q and , as they were originally given in ref. [ on the right-hand side of the GVP ( M ′ m ab P Q ABCD transforms as a proper SU(8) connection (for m – 33 – P B of generalized or exceptional geometry provided we interpolating m A and ) reintroduces the density term proportional to Γ ) the contribution proportional to Γ Q B 4.18 ) just from m A 3.29 Q -covariant manner, as we already saw in the foregoing sectio preserve these covariance properties 4.14 ) and ( 7(7) ]. In other words, even the density term which is there with th 3 4.17 ) to ( the usual vierbein determinant), and the above result, wher e ), it is useful to recall that similar ambiguities arise in st 4.14 by choosing the specific ‘frame’ as derived from D=11 supergr 3.12 V (with . Of course, these statements apply only to the specific ‘fram ∆, is resolved: while in ( all required covariance properties e m m m ∂ Let us point out once more that the existence of covariant con Let us also point out how the apparent discrepancy between ( Let us emphasize once again that the connections given in equ is consistent with the formulae (17) and (18) of ref. [ ∂ 6= 1 = 11 supergravity, that we have adopted here, where the conne 1 − − m ∆ that was absent in ref. [ because we have given the connectionsto explicitly achieve in terms if of alland quantities its are derivatives to in be an E expressed only in terms D coefficients only along the seventhat internal the dimensions. manipulations we Howev arethem now in going line to with perform thefully on constructions covariant these and described sp therefore in the two fore eliminated by shifting back to the non-covariant connectio M 4.4 Non-metricity and redefinitionIn of order the to understand generalized how connection thecan appearance of be reconciledequation with ( the absence in the corresponding rela expressions correspond to objectswith in the a D=11 special theory), SU(8) such gauge ( that satisfy break up e correct weight if the GVP isabsorbed formulated into with a the redefinition usual of affineIn co fact we are free to also choose any the covariance under local SU(8) follows by the same argumen density term changes, as part of it is absorbed into the weight assignments given there, the contribution propo the SO(7) subgroup this is anyhow obvious). Secondly, the two pictures canwhich be are consistent) made simply to reflect agree. the unavoidable Otherwise ambigu the two sets o above with the connections ( switch from ( these objects are alsointernal covariant diffeomorphisms under this generalized is diffeomo manifestlyat true, all while the under the remaining generalized diffeomorphisms with when we apply an SU(8) rotation that moves us out of the given g ξ (and will explain again for a simplified example in appendix JHEP09(2014)044 ition (4.56) (4.55) (4.50) (4.51) (4.52) (4.53) (4.54) SU(8) lure of , CD P V can be absorbed NBC V is referred to as the M C , p P , but the connections ABCD ] b 3 . n M MA e mn [ P b Q ient does not affect the T a , and accordingly redefine = 0 m = C ]). Notice that there is quite ): 3.1 ] der to recover a torsion-free P pin connections. N AB P AB . equation above. For example, p , om in how one defines various . c 57 c + b ed as p p NB p mn where there was none before, in e an be absorbed into a redefinition 4.24 a = 0 (4.49) − V e − V e V ) in section d p T are still non-zero only for the first d ] a ˜ e a A ) Γ n [ p p n n P e e BC can be absorbed into a redefinition of C e e | 3.12 | PCD ] N d mn | M d p mn | c V V MN T c Q Γ mn mnp C ˜ p [ m Γ [ m = T A ( + − P a P b M − p − n and e ). However, by removing the non-metricity in Q − e , p b M ABCD – 34 – P AB B B a p mn m ab P AB V p mn mn m Γ ω 4.17 P P Γ T ω MA AB MA V − N Q Q b + i MN −→ V n −→ −→ a ˜ + Γ e n + b p i e B a − P p mn m ab m + m mn Γ ω ∂ ω T P MA MN is referred to as the non-metricity, as it ‘measures’ the fai + Q ˜ N AB Γ ) in the internal GVP, equation ( . Similarly, the non-metricity can be absorbed into a redefin a MN V ) ab ≡ n ≡ ( Γ M M e mn B ∂ m P Γ p ( m = P ∂ P . The GVP is now of the form of ( MA MN = = Γ b B b Γ Q MN p mn ˜ Γ m ab MA −→ −→ P Q is no longer given by the Christoffel symbols, the Christoffel symbols, there is a more general expression P B −→ p mn P p mn MN MA ˜ so that Γ Γ Q In complete analogy with this discussion, connection coeffic MN Γ p mn the metric to be covariantly constant (see for example ref. [ where Γ of the affine connection and the torsion: connection torsion and seven components given by equations ( T Furthermore, the fully anti-symmetric part of theof torsion the c spin connection the affine connection once more, as follows: are still different. In particular, the the affine connection we have reintroduced torsion in a lot of freedom inthe the antisymmetric definition part of of the the various objects affine in connection the Γ affine connection we follow the same procedure as in section analogy to ordinary differential geometry. Therefore, in or structures such as non-metricity, torsion and the affine and s Hence, in differential geometry there is a great deal of freed with Γ tial geometry. While the vielbein postulate is usually quot so that the internal GVP becomes We note that this shift only changes the affine connection, but into a redefinition of JHEP09(2014)044 ) ], d 8 4.57 (4.57) (4.58) (4.59) ations given ). This con- . in terms of the is required to be P is now chosen to 3.19 in equation ( 3 NCB MBC M MN the ones obtained in V V . W Q C , guity. P AC to be expressible in terms KA to the connections defined and Γ = 0 V 56 components, and this is : Q osing MBC ns will be very complicated, P C B n section eterminant. This remains so ] K V all C MN KAB MA NB KM Γ V Q KA i A . 3 [ − V 2 W C M + BC Q B log∆ K AB M , the modification + M V B ∂ , namely C MA − V 2 3 3 A ), which implies equation ( MA W drops out of Γ K P AB = U – 35 – BC + is given by the determinant of the siebenbein [ V Q Γ 3.18 K M P B Γ alone (see the previous section). V AB transforms as a connection under SU(8), and as a C K KM B MN A MA B V Γ in section b ) ensures that the affine connection Γ satisfies the Γ is torsion-free, an SU(8) singlet and transforms properly U K , which is cancelled by the density contributions in the i MA P − Q + W MA W Q + 3.24 Q B MN AB K ), we have now brought the GVP into the standard form b Γ K N AB MA V V of weighted tensors in the supersymmetry transformations. Γ. We should point out that, with the formulae at hand, we coul KM 4.56 R M ˜ Γ i ∼ = M ∂ = = = b ∇ Γ B K MA ). Note that the part of the fermion supersymmetry transform KM Q Γ , namely 3.19 3 given in equation ( the previous section are all contained in the hook-type ambi of the SU(8) connection necessary for the supersymmetry variations of the fermions the affine connection under generalized diffeomorphisms. generalized vector under generalized diffeomorphisms. The remaining differences between the above connections and The SU(8) connection The connections have non-vanishing components for W Modulo the ambiguity, these connections are now equivalent The trace of the affine connection • • • • = 11 fields. However, after the redefinitions these expressio in section With the redefinitions ( with the following properties: The connection used to construct the exceptional geometry i condition ( dition can be satisfiedappropriately. In by particular the the torsion-free trace connection of by cho compatible with the vierbein density, ( D and by themselves not very illuminating. The in principle proceed to work out explicit expressions for obtain precisely the connection where, modulo the remaining ambiguity covariant derivative by the internal connectiondespite are the independent of contribution the from vierbein d JHEP09(2014)044 ) B ǫ ] µ ] E (4.61) (4.60) (4.62) C γ ǫ C ǫ ( satisfies ǫ µν M µν ] γ , AB ∂ γ C = 11 theory ABCD viz. ǫ , αβ AB AB mn [ AB M D [ B nAB F Γ AB p m ǫ [ M AB Γ e ) in terms of the M µ n V β | γ V DE B DE ν i ) become ǫ β γδ e αβmn m M ν 3.34 4 B m p e F on should yield those X V A 2 ∂ strengths, − 3.34 i / µ M n and in order to modify ABCD 1 ∂ DE Note that B 2 µ β m Q − ǫ αβγδ . e p m ǫ √ µ µ β e + 2 ∆ 2 8 γ me from the supersymmetric i , / h those of the AB 2 4 i e √ A 1 CD 2 B νρ − ǫ 2 3 B √ 2 ǫ αβ 64 m γ ∆ √ ǫ l 3 µ i D e G αβ µ √ − ǫ γ 2 1 ] γ γ + ] − + C νρ ) reduce to the following expressions C | αβ + 3 − M AB D . γ and ǫ ] µ ǫ β AB V ] C nAB C ρ ω MD 5 β ǫ αβ AB C 3.34 ǫ ABCD e Γ q AB 2 1 ρ . Solving the section condition to obtain µ γ n | e ). Using the definition of the covariant m γ M αβ µν ...m ν αβ ∂ + ), transformations ( p AB m D 1 γ C M 3.1 [ F q AB B F ∂ [ m A ) ν β 4.10 M ǫ = CD Γ AB e [ m B m − αβ AB 3.22 ν β 5 V [ q – 36 – , respectively. ∂ M − ), we find that they are identical upon identifying F µν i ], ( m p i e AB m V m 2 3 e ...m 2 F , µ . µ 1 i m AB 2 4 − q + ] αβ √ 2 e . √ B m AB B 4.10 2 2 3 C √ F e 1 4 B 4 E ǫ ) and ( m √ γδm 3 ǫ ǫ − 1 2 − 3 ∂ + + 6 µ − F µ 1 4 with ] AB γ C − [ D ∂ + ǫ 3.11 ′ B ] C ( ǫ ) + ρσ µ ǫ ǫ − C ν µ D P γ µ ] DE γ B αβγδ ABCD ǫ ǫ γ M ǫ , ǫ β m γ C ′ µ m m M ∂ 2 µ e ∂ AB / p γ ∂ p µ γ Q αβ 1 m ( α − γ ρσ [ MB µ − AB [ AB e DE q m DE [ ABCD F B ∆ 2 ∂ AB µ M and / m m 4 1 M ABCD 1 e e 5! − P V V − 2 µ αβ 2 2 · 2 i + ], transformation ( ∆ µ M AB i AB P F 2 √ 3 ∂ √ √ m AB √ A i V 2 4 1 2 64 e 2 16 ǫ 2 √ αβ i µ √ √ ) and equations ( G − + − + + 3 + 6 2 12 , we give the fermion supersymmetry transformations ( 2 D = = 2 ≡ − − 3.3 − − − , however, does not satisfy a twisted self-duality conditio 3.2 A µ = = 2 with AB A µ δψ ABC = 11 supergravity, the fermion supersymmetry transformati αβAB αβ ψ AB ǫ G G First, let us consider the relation between δχ ABC D δ αβ χ ǫ F δ 8 1 Comparing the supersymmetry transformations aboveEFT that with co the canonical solutionas written of in the ref. section [ condition wit In this form, the supersymmetry transformations ( of the SU(8) invariant reformulation [ upon use of the canonical solution of the section condition torsion-free connection constructed inthe section derivative ( In section 4.5 Connections and fermion supersymmetry transformations it so that it does, we need to add to it the Hodge dual of the field The a twisted self-duality condition, which means that on-shel JHEP09(2014)044 ), in. and ′ , 1280 2.13 (4.63) Γ Q ,( However, M ] (see also . , the µν m 18 = 11 theory µ F 4.3 B D . Therefore, at the useful comments and for hospitality during 4.3 of the ment of Energy (DoE) il under the European omponents in the . / ERC grant agreement ENS Lyon, in particular , ng’s College, Cambridge. mboldt prize. We would ather complicated. cannot be dualized in the connection coefficients are 60 and a DFG Heisenberg CD m N NCB ization. efinitions to which the super- µ re insensitive to [ section antly in terms of the 56-bein, V V B r hospitality at Great Brampton representation. These terms are P AB P AC V V P P 1280 : as explained in section p mN mN , Γ q by the generalized affine connection Γ i still contains terms, not expressible in terms i p 3 + is necessary in the definition of 2 and the unambiguous part of the connection of and U – 37 – − ′ U and P B µνM q , Q − ′ B mABCD m A Q − Q p q ]. In this paper, we use a connection that allows us to = = 18 B ], where they are also given in terms of a generalized SU(8) m A by the usual Christoffel symbol associated with the siebenbe 19 Q mABCD P are related to P P and Q and Q would correspond to the field strength of the field dual to m | ). Therefore, the connection αβ 3 ]. In practice, an explicit expression of this difference is r X 18 The fermion supersymmetry transformations of a truncation Regarding the relation between are related to ′ ref. [ precisely the difference between the of the 56-bein and its derivatives, that are in the have been studied in ref. [ symmetry transformations are insensitive, as explained in In both cases, the redefinitions correspond to hook-type red since the first term in the expression above is not exact, Moreover, the discussions. H.G. acknowledges fundingCommunity’s from Seventh the Framework Programme Europeanno. (FP7/2007-2013) Research [247252]. Counc Theunder work the of cooperative O.H. researchfellowship. is agreement supported The DE-FG02-05ER413 by work thelike of to U.S. H.N. thank Depart is B. supported de by Wit, a E. Gay-Lussac-Hu Musaev, M. Perry, and D. Waldram for H.G. and M.G. wouldH.S., like for to hospitality, thank as the wellHouse. AEI, as in the H.G. particular Mitchell and H.N., foundationthe H.N. fo final would stages also of like this to work. thank H.G. KITPC and M.G. in are Beijing supported by Ki Acknowledgments connection constructed in ref.express [ the fermion supersymmetryrather transformations than covari its components. This is donesection by using some of the c level of theequivalent. supersymmetry transformations, the two sets of schematically “eating up” the non-exact terms to allow dual usual way. This is why the new field where P representation, to which supersymmetry transformations a JHEP09(2014)044 (B.1) (B.2) (B.3) s that ] . D ] , B . C δ A C δ M GH [ CD AB E V δ N . EF V EF N CB M EF A N [ V b ω N V N E identities in order to deal V V M ) upon contractions with the EF ation V AB M EF 2.1 1 3 t derivatives: V M M V V + + ] 2 1 + D ABCDEF GH ] + ǫ B . MEF δ 1 48 V C MEF -covariant derivative that is also covariant [ 7(7) V − MCB ] 7(7) ME V NEF V V CD – 38 – NEF ) of E A [ M V . N AB E 56 1 -covariant derivative. V 16 V N 7(7) 1 24 1 2 = 7 spacetime and tangent space indices, respectively. = 4 spacetime and tangent space indices, respectively. V − 7(7) AB [ denotes derivative that is also covariant with respect to 1 3 − D = D N B V = denotes the E ) of E A 1 2 ) and the section constraint ( B M QCB A = 133 V Q µ QCD denote denote 2.17 is the fully covariant derivative, V + Q acts as an identity on the right hand side. Similarly, one find CD β + Q Q P AB is the fully covariant derivative. denotes the E α β P V V P MN P AB µ M α N Q ρ µν a, b, . . . V µ A ω are defined with the modified spin connection α, β, . . . P M ω L Q + Γ + P AB P N label the fundamental ( P M + Γ denote SU(8) indices. + − V and M projectors ( M N M b and µ µ ∇ µ Q D ∂ P D ∂ M D P 7(7) P = = N = = = and labels the adjoint ( µ M M µ µ M P M M,N,... with respect to the local SO(1,3) and SU(8) symmetries. A, B, . . . µ, ν, . . . m, n, . . . D α the local SO(1,3) and SU(8) symmetries. b D • • • • •D • • •D • ∇ • ∇ As a consistency check, we may calculate the trace of this rel confirming that 56-bein. Let us first note the projector identity Furthermore, the following notations are used for covarian The index notation used in this paper is as follows: A Notations and conventions In this appendix wewith the collect E a handful of useful relations and B Useful identities Analogously, and JHEP09(2014)044 ] α ), β )– . ν c µν . , e (B.5) (B.4) (C.1) B 3.13 N 3.32 A + c b ∇ . 1 ectively. α ǫ [ µ C 1 ν A ǫ γ e 1 µ . C 2 ǫ ) and ( γ ¯ ǫ A µ µβ N A 1 e γ ǫ ∂ 1 C 2 K 2.25 ǫ µ ). The third term ¯ ǫ o supersymmetry A µ b ⊗ γ ∇ C 1 γ A 1 2 C ǫ A ǫ K ¯ ǫ KBC M 1 C 2 and two-forms α 2 above that closure of 2.34 ǫ ¯ ǫ , ∂ V N γ ¯ ǫ ∇ µ N b M γ A 2 A ∇ N ) as ¯ ǫ µ 3.2 1 ∂ C 2 D BC µ ry transformations ( ǫ ¯ ǫ γ A N GH MK K µ ⊗ MK ation together with the non- C.1 γ conditions ( C 2 V V ¯ ǫ MN M C 2 citly M , K AB M ¯ ǫ ∂ (trace) A C V A C M δ MK A C MEF M AB δ A = 0 δ + 8 1 M AB CD 1 V V 8 1 ) M ǫ 8 1 V + 8 N + N A µ MK C + ∂ AB − δ ). For the commutator on the external γ V A AB K ⊗ + 32 M 1 8 A 2 BC 1 V N ¯ ǫ ǫ CD BC A B C K AB M M ν N 3.35 V + N δ 1 ∂ M V γ M V ǫ ( D b 1 8 ∇ V A 2 N V ¯ ǫ BC ABCDEF GH K − + – 39 – BC + 16 ∇ ǫ MN K AB B A K AB M D δ A µ V N K we have seen in section V 1 V 1 γ PQ 8 24 1 B ∂ ǫ of the diffeomorphism action ( V ) 1 + BC C 2 + K AB ǫ N ν ¯ ǫ = = K AB AB V ξ A M MN b 133 ∇ V BC 2 N N M N V µ + MN ¯ ǫ K ∂ ∂ BC ∂ 1 V M γ + M AB Ω + ) states that V M AB P M A 2 ⊗ ⊗ µν V ( 8 3 ¯ ǫ V g V MN 2.1 M AB BC M M and BC − ). Let us rewrite the last term of ( M AB V + ∂ ∂ CA M AB K M K α ] + 4 α V V V N µ MN µ µν V µ K BC e BC V g 3.10 M Ω CD Ξ D ) D K i Λ i N N N 8 µ V e M AB ∂ V V are in the symmetric and antisymmetric tensor product, resp BC − D V K + 32 + 32 + 8 K ∂ 1 AB AC [ = = 1 V MN M AB ) − M M ( V e M α K t V µ V and ∂ 32 . A ] + 8 − 2 K ǫ 133 = 12 ( = 32 ∇ µν M , δ ) closes into the supersymmetry algebra ( B 1 We start with the vector fields, for which the commutator of tw The section constraint ( ǫ 32 δ [ 3.34 In this appendix, we show that the commutator of supersymmet C The supersymmetry algebra and can be reduced using ( respectively. Here, we complete the algebra on the vectors ( and internal vielbeine the algebra is a direct consequence of the vanishing torsion In the first line, wecovariant contribution recognize the action of a gauge transform where transformations yields Contracting this equation with the 56-bein, we obtain expli JHEP09(2014)044 , 2) M N (C.2) (C.5) (C.3) (C.4) ) and ↔ ) with µν B 1 . With ǫ µνN (1 F M 2.19 Ω µν 2.19 µ M − α . γ N µ A Λ α A 2 ) in the first . M ¯ Ξ ǫ A µ c ) . 1 N Ξ α ǫ 2) MN ∂ t ) 3.19 µ N γ ∂ α ↔ NDB . t ) + c C 2 + ( V MN ¯ ǫ D ) (1 M α Q ) on the vector fields. 1 MN + ( descending from varia- α ∂ ) ǫ . t M µν ] ρ ] − c α ν ξ . mations on the two-forms C.2 F t D 1 µν γ E ǫ ( ν . Ω 1 KBC cit gauge fields NCD and can be simplified using ξ c tion part in the last two lines ǫ M . M 12 ( V + c ge transformation ( M ρ V + 12 ( µνρσ 1 2 b ∇ b ∂ N γ ∇ M − eε A − M D 1 E D ] + c 1 + ǫ µν B 1 Λ P AC BC M ǫ = ǫ in the second line yields additional F µ ] V N µν µ M D 1 i µν ν MBD γ D γ ǫ BCD ) simply reproduce the corresponding V µν K AB F αβ V D A 2 ρ A 2 ρσK ν ¯ ǫ ρσ + F µ σABCD V + 4 ¯ ǫ µ [ P M µν ξ [ γ ) in the second. Together, we obtain ν F γ P M γ b N γ ω 3.32 + D 1 ξ 1 2 BD A µν µν b ǫ Λ D A A N 2 [ 2 γ γ M Q 2 − ¯ ǫ 3.21 ¯ ǫ NK ∂ AB C 2 ¯ ǫ ∂ − A V [ 2 ν QCD ¯ ǫ ), and where we have used ( K MN ¯ ǫ ξ M A µν V ) AC V MN – 40 – AC 1 BC γ N A α Q ǫ QCD N t Q in the action of gauge transformations ( Qρσ 1 ∂ Q , C 2 3.37 V ǫ V M ¯ ǫ N V V D Ω QCD P AB ): ν b N D ] + ( µν V V MN γ µ N ν µ g BC α PBC γ Ξ P AB A 2 P AC A PBC PCD PCD M 2.15 ¯ ǫ ] V + K PQ V from ( V A 2 V 2 V ) i i V i contribution in ( P AB ¯ ǫ V ρµν µν 6 4 M α MN α , δ g 12 t ) then takes the expected form N H 1 µ ( ] Ω µνρσ + 2 − − + 3 νµ M δ − V ρ ν ε − [ + ∂ 1 3 2 1 ξ F MN MN ). C.2 A ν ǫ M M Ω Ω ξ + + + δ µ PQ MN i [ Λ ) PQ PQ 3 8 α α 3.42 = ] ] µ ) Ω ) M A α ), by virtue of the closure of the algebra ( ν ν i t µ α α D [ + 8 − ( t Ξ Ξ t M ( ( µ µ µ A − [ [ = 3 MN 2.34 4 3 32 3 8 ) A D D from ( ] M α − − − 2 t MN µ ǫ ) = = 2 = 2 A α , δ µ M ] t 1 α 2 ǫ ǫ δ [ µν , δ B 1 ǫ ] δ . First, we note that to lowest order in the fermions the terms 2 [ Next, let us check the commutator of supersymmetry transfor α , δ 1 µν δ [ We observe, that we can simultaneously dropsince the they SU(8) mutually connec cancel. The spin connection contributions which explicitly carry the field strength tion of the ( the twisted self-duality equation ( In total, the commutator ( some calculation the various remaining terms organize into with the last term corresponding to the action of a tensor gau B terms of type ( diffeomorphisms ( We can thus in the following ignore all terms that carry expli parameter Ξ equality and the vanishing torsion condition ( reproducing the parameter Ξ JHEP09(2014)044 M µ , thus τ A , (C.9) (C.7) (C.8) (C.6) σ ξ ρ ξ α ρ ξ P στ µν στ Γ B MK , PCD M Mρ Γ V persymmetry ∂ R ) on the vector M uge fields σ K AB D . P AB V c µνρ C.2 . V . µνστ c AB M . P two-forms D i eε CD . + c i eε V ). The remaining com- K + c .. . CD c V c + 2 ] . + 2 . of the corresponding spin K β ) can be shown by a rather KCD 3.35 e determined. Indeed, clo- τ αβ V ules but also uniquely fixed A A commutator ) above, and the shift pa- λ e V + c NCD + c 1 1 LCD M ξ ρABCD l ingredients here. ǫ ǫ V M V 3.35 ω . ρ ρ ) i M BD P ∂ . LCD c M ρ γ γ 3.37 ue to closure ( . CD ∂ L ] c α D 1 [ . V A 2 A 2 + D ǫ i τ K V ¯ ¯ ǫ ǫ σλ σ e P V g µνρσ γ into ( − AB ) ( A, ω σ [ M eε ρ ) + c )) + c N D 1 D ρ N KCA ∇ ] ∂ MK ǫ D 1 D 1 στ V 2 3 ] V b σ A D ∇ Γ ǫ ǫ ν M µν M ] ] − D L σ γ − D ν ν M ∇ ∂ − D B Mτ ( ] V γ γ CD σ σ D ( ( K AB R B αβ K defined in ( B ) αβ ) in order to reproduce on-shell the transfor- µνρ thus becomes a consistency check of the entire N [ λ ρ ) L ρ V K K b ξ ∇ C V µνρ µνρ ω B 1 α ω V α ρ b b ] ρ ∇ ∇ – 41 – ǫ i eε K C.5 M ] ). Finally, we have used the first-order duality ν µ A γ µνρσ γ [ M M 2 ν ∂ V e M µν M i eε i eε ∂ ∂ C 2 C C 2 γ CD eε b − B ¯ ǫ K D ¯ ( ǫ M σλ . P ρ K 3.42 µ L g b [ ρ ρ N ∇ α V D = 8 = 8 = V γ β β µ b α i ξ ∇ . [ M and Ξ e e ρ µ µ µν ρ 4 c µ [ C 2 [ [ . σ σ ∇ ¯ ǫ γ B e D F γ α α − KCD KCD M σ e e σ A C 2 D 1 ( 2 V V + c ¯ ǫ ǫ CD M ¯ ǫ ] i i = ≡ MN µνρ 3 3 ν µνρ M C 2 i K 2 2 given in ( A ¯ ǫ Ω Ξ eε D V 1 στ i i ), we have made use of µ ǫ = = 3 [ i eε M σ 128 2 64 16 16 64 L D Mτ C.6 D AB − − + − − − + 64 R ρ µνN L γ MP = 2 V A 2 Γ ¯ ǫ ,Ω , we can consistently ignore all terms that carry explicit ga M α ) for the last equality in ( α AB D µν M M M P ) on the two-forms ) under external diffeomorphisms. Together, we confirm the su µν ∂ B µν ] 2.39 B σ 2 ǫ 3.35 2.34 µνρ , δ 1 ǫ δ i eε ⇐⇒ V As for Closure of the supersymmetry algebra on the vector fields and [ 8 which separately organize into the correct contributions d construction with no moresure free of two or supersymmetry adjustable transformations parameters on to b fields. After some calculation, we then find for the remaining mutator for the constrained two-forms lengthy calculation of which we will give only a few essentia rameters Ω equations ( mation ( as well as In the calculation of ( with the gauge parameters Λ has not only determinedall the the supersymmetry gauge transformation parameters r appearing on the right hand side of ( Here the curvature inconnections the second term refers to the curvature algebra ( JHEP09(2014)044 (C.14) (C.10) (C.11) (C.12) (C.13) N ρ ] , ν στ . ) F σ B B µγ µ ξ 1 1 B [ e Mρ ǫ ǫ ] 1 ). This yields ǫ D A 2 A 2 R N µν M ¯ ǫ ¯ ǫ A 2 ρ ρ ∂ .. ¯ ǫ B ξ PCD ξ γ c C.6 . ν V CD . e c K AB MN N M . µν D K AB + c M µνστ V . [ 1 D ) N F ρσ V ∂ ǫ 2Ω ( F ieε ρσ C 2 + c λ AB µβ CD − M 2 ǫ under external diffeomor- B ξ F )¯ e P . 1 K AB D µν N AB − ǫ M V αβ αβ V ] να ] nto the form ] V F N CD AB ∂ ν e ν ν M ABCD ρ CD ) combine into the Ω transfor- | | t is useful to explicitly develop στ M ] N γ N µν M p µν 1 2 ν τλ CD M D M V N AB µν B ). After some further calculation ρσ λτ g F F N AB Mσ b C.6 V B 1 K ( − R R F γ µ F ) of the two-form ǫ V [ CD R ρABCD µ ρ ρ V N β ρ [ M C.6 ( B N ξ ξ ). This can be verified by a lengthy ρ D ρσ µ K CD γ P ), they reduce to 1 ρ ρ ρσ D D ξ e | | ρ γ M ǫ A 2 ] σABCD V ρσ 2.19 K µ µ F ξ ¯ ǫ σ [ [ D στ N F P V 3.42 µν γ B 2.39 8 g ∂ µνρσ αβ µντρ αβ γ 1 1 3 B α KCD MN λτ ǫ 1 + 16 − ] ν A 2 eε V ǫ γ KDA ν ¯ ǫ + ] ieε ieε ieε e ) + 8 ρ B µνρσ 3 2 B 2Ω ν γ V 4 4 2 1 σ γ 1 στ B M γ from ( ǫ [ ǫ − − − 1 M ρσ − A 2 − – 42 – M ∂ A 2 K AB ǫ i eε ¯ ǫ A 2 ρσ b γ ¯ ǫ Mτ N J D PCD ¯ ǫ 2 A 2 = = = , V γ terms on the right hand side of ( µν ¯ ǫ λ V λ R i g µ ξ − β ξ [ 8 K M M . CB M M γ γ 1 2 c µν M ] M M D . D K AB µν KCD − D µν F A 2 ∂ c ∂ ρµνσ N µ K AB ρµνσ V σ [ − ) ¯ ǫ V F F ) ω V γ eε σλ σλ − AB B γ ( ρµν M eε M M g 1 g ρ ρ µνρ A 2 P αγ ρ ∂ A ǫ CD e M H ¯ ǫ D ] V 1 NCB A 2 i i M i ξ iξ K ρ ρ ρ K AD D [ ǫ µν N ǫ 4 4 i eε V 4 ξ Λ (¯ V ω ∂ D D F ] 3 8 − − + 4 + CD − + 2 + σ M β M − spin ρσ ν ] + 2 D = D µν e M M M K ] ] ] µνρ µν M KCD F ν ν ν F Λ M b αβ AB [

D V ] Ξ Ξ Ξ ( ∂ , M i eε i eε µ µ µ ] KCD N µν [ [ [ α ∂ 2 2 ν [ M ω V KCD F ρ D D D ) with parameter Ω K µν M D i D e − − V [ M 3 2 [ B 32 γ 8 µ µν [ ] ∂ ν ). Next, we collect all 2 γ − F − 2.21 e 2 + 8 + 8 = 2 = 2 A 2 , δ −→ = = = = ≡ ¯ ǫ 1 2.34 δ [ i αβ 32 Let us start by considering the first five terms of ( It remains to show that all the remaining terms in ( MN R phisms ( but direct computation. In thethe course curvature of this computation, i mations of ( and upon using the first-order duality equation ( This exactly reproduces the expected transformation of and After some further calculation, these terms may be brought i and precisely reproduce the gauge transformation ( JHEP09(2014)044 is . m ab (C.15) µν M p B , . m ac . tive. Under local p m q ] c ξ b νκ r a b [ g ∂ ] p m 2 N + Λ ∂ ∇ le (taken from standard ) − b on, the spin connection is | m cb q µλ = 11 connections, but that The main point will be that e p , as expected. So the spin g nstructing a fully covariant c r . . m ab oses also on the field ms, and we also know that it a a D ibutions are M q ) [ ( ca b qc m ab e bein and its ordinary derivatives ab e ∇ Ω ≡ ( ω p = Λ = ∂ λκ m − q g S a , p ac e 1 2 bc a m ab p m ab ≡ p nb b c − p Ω e ). Now a quick calculation shows that e m nc m = − e − ∂ m ab p ∆ , δp n b 4.14 νκ qc a e ab c g e e ] , p Ω , p − in ( m ac N – 43 – q ∂ ] ≡ ), and decompose this into a symmetric and an q c q ξ ∂ c ab b r [ b p bc m νβ e ∂ µλ e p e p m ab m g c 3.23 m ab m a p 2 1 S e S ∂ µα m + Λ M ] [ − e e b | ≡ ∂ in ( p ≡ q a and = e αβ λκ m cb e V r q g a ab c ∂ c m ab [ 1 2 1 a q e MN mab Ω − m ab − ω q R − = V transform non-covariantly as SO(1,3) gauge fields, while + Λ ≡ = m ab q ab m ab µν m ab Λ q = ω m nc ∂ MN ∆ m ab R = ω such object that can be built from the vielbein and its deriva m ab δq and hence only In standard differential geometry and in the absence of torsi m ab q fully covariant expressions can be obtained in terms of the D Non-exceptional gravity In this appendixdifferential we geometry) will how illustrateconnection the in can difficulties terms be understood encountered of and in a resolved co in simple our examp framework. Under an arbitrary diffeomorphism, the non-covariant contr with coefficients of anholonomy antisymmetric part these cannot be written just in— terms unlike of in the generalized ordinary viel differential geometry. We conclude that the supersymmetry algebra consistently cl Now define the Cartan form and these two contributions cancel in the variation of from which one obtains connection is indeed ais covariant the object under diffeomorphis which is the analogue of These are the same as the defined as so covariant under local SO(1,3). SO(1,3) we have JHEP09(2014)044 (D.3) (D.4) (D.2) (D.1) (D.5) = 0, and ∆ is just m ) give some- (6) ∂ , . 1 , absorbed into and replacing A − m AB D.4 AB ∆ Γ = ] pressions depend- 1 2 q m ab Γ are real (this is true CDEF p − . (3) , ] m aa . BCDE N A , P p aa p [ N 2 1 b CD p p BCDE Γ mAB i mnAB mnAB al to − N o lines in ( 7 27 Γ Γ maAB V p , MEF 56-bein (whose components 2 AB 2 a [ , on appearing in the variation / Γ we find / V M AC + AB 1 Γ 1 ab AB ) V apq AB − ∆ a bb and Γ 7(7) CD ab MDE a AB i Γ p 4 ∆ m ab Γ DE N V Γ p 8 1 1 − a AB m N 27 V m ab a bb ] 3 4 8 q p AC V = = (Γ p q + m 1 2 + − [ N 1 V + 27 p V 1 = = AB m ab 8 − − formalism, replacing the siebenbein by the + ω m mnAB bmAB ∆ 1 2 that is required because the supersymmetry V i V 3 Γ bpq AB NCDEF NCA NCD − 7(7) Γ p V N ACDE V − p mp , a ab – 44 – p = m m , p a ab EF ∂ ∂ N ACDE p AB 5 m AB ] BC p 54 M q mn AB Γ 5 AB V 54 M Γ 2 Γ + / p DE V N ABCD NBC 2 [ 1 + / V V m M − 1 i a a i ab AB p NCD )) simplifies to V ab 3 p ] 2 ∆ ∆ − Γ ]. V 1 q AB NDE are imaginary, while i 1 8 4 3 − Γ 4.7 CD V = = p B A [ a bm ∆ m = = NBC AB δ B i e )–( p i V a b i , and hence are cancelled by the third term in the definition of 4 1 27 6 27 p 7 20 27 mn − . AB AB B A 4.4 m A i 1 3 8 − − 3 δ V q 4 B + − m mn m ABCD = = = 0 = V V = p m C B B B B and q B A A and its derivative, there is no way of getting rid of of exceptional geometry. To simplify things we set m pq m A pq A AB MA 8 R R R is a density, showing again how the density contribution was R V m AC m R AB e ε V M V . This shows very explicitly, that no matter how we combine ex B With this information we can now compute Next we repeat this calculation in the E MA ing only on R which is indeed the correct result. The last term proportion are explicitly given in ( This gives 56-bein this will suffice to make clear our main point. Then the E only in this particular SU(8) gauge). By direct computation As a check on the coefficients weof compute the (this is gravitino) the combinati Note that thing proportional to the density contribution proportional to Γ The last component drops out because for this term the first tw parameter the connections given in ref. [ JHEP09(2014)044 . B V ∂ (E.3) (E.4) (E.1) (E.2) MA q does not, and B B . In other words, ND and its derivative . V q MA V U E , and its derivative AC C , nnection satisfying all , D N V nishing form fields; but iantly under generalized CD ND V B q ND have indeed the required ) for K NA q ms of KCA q V nnection of section 3 can be form field, and this is the CD B EC V MA the explicit introduction ‘by problem by picking precisely E.2 R ovariance under SU(8). g aspect here is that, again, on are needed. It is therefore AC M N CD ), and we make the following R W Λ nted (and the resulting expres- diffeomorphisms, which implies N MA V Λ N S V + on ( s in the transformation of S ∂ − V V W nce requires such a contribution. 3.24 ∂ CD B M D M BD ∂ BC M ∂ ’s that would produce the fully anti- R M MA NA M R V S q , and U − V S ∂ N B ) and ( − V N + − V D K K D NBC P B B MA P 3.23 NE V R and its derivatives. However, the modifications q NC ND MA ’s and , – 45 – q V q NAB R V B NBC V MCD NEC B + V i V V i MA NBC NCD B be constructed in terms of only q MA V V i 3 given by ( 2 U = 8 = 12 MA q MCD B B + in terms of V M AC M AD = B cannot V V B MA MA B B A ABCD q δ W MA i i M nc q MA combination of i p MA 20 2 34 189 189 27 U 2 3 ∆ ) where these terms drop out. nc Q no − − − ∆ , and 3.28 ≡ − B B MA by such manipulations, without ‘breaking up’ the 56-bein R MA U , B m ab ω MA MABCD q → p To see that the full connection can be made to transform covar In principle we could extend the above calculation to non-va Let the SU(8) connection be . However, while the first term V m ab and diffeomorphisms, consider the non-covariant contribution with there appears to be this will be far moresions tedious will than not the calculation be just any prese prettier). Perhaps the only interestin required to achieve this come at the price of destroying the c full covariance cannot behand’ achieved of in the this spin way, connection. but requires made to transform as a generalizeda vector unique under generalized expression for Namely, we will show by explicit computation how the SU(8) co symmetrized (exterior) derivativesreason on why the the hook-like 3-form contributionsvery and in remarkable the the that affine the 6- connecti supersymmetricthe combinations theory ( avoids this These are indeed all the objects that one can construct in ter choice for the undetermined part E Covariant SU(8) connection In this appendixdesired we covariance properties provide yet more evidence that an SU(8) co will therefore violate SU(8) covariance if general covaria q covariance properties of an SU(8) connection, the expressi ∂ JHEP09(2014)044 and EF ., (E.6) (E.7) (E.9) (E.8) (E.5) c EF R . M R V EF q .. .. V R c EF c + c . . EF V R EF S R BD V BD +c +c S V R Λ EF R R R V S S V EF V ∂ Λ Λ V S +7 · + S S N EF MP 5 V EF N ∂ ) ∂ ∂ S REF ∂ S α − + N N V V SEF t V ] M ∂ ) can be simplified to: ∂ SEF ( ed tensor densities of REF V ∂ V V EF , ] | E.6 NQ SEF · SEF NEF REF NEF ) S V V N V V REF V EF α ] V N GH | ition. Now using, REF ∂ REF t REF | V C A C A e transformations of S V V V δ V 4 δ M CD V | | CD REF ∂ REF + ( 9 1 ation ( 1 9 | S B A B V A V N δ V δ − SFD BD SFD N + | CD MEF [ V 1 9 | V V V V 1 MP N 36 B EF [ A [ − Ω V DE δ DE NBD NBD N − C A R R NEC NEC V V MCD δ V NQ BE V V C V A C V A A V B 9 BE R Ω δ δ 40 δ + 10 B A B A A V R B AE AE ] δ δ 4 9 8 9 − δ 9 S V S + 14 8 9 9 8 1 8 ] AE P V Q V − CD ABCDEF GH − | δ S AE − ǫ – 46 – + − = + S ] + ) EF S V M | N 1 · 24 V δ V S | CD CD + N CD REF REF | E.7 + R V R ∂ + = | V V S SDE BE SDE V V [ · M ), (A.3) and P V N V V | R ∂ N δ BD SBE [ EF EF SBE V ∂ SBF SBF V R M S E.4 Q S EF V N BC [ δ M V V V V R AE V N R AE AE V 2 R ∂ ] V V V V EF R AE 8 R AE 1 V CD N NDE NDE 24 NEF N V NEF EF − MAC N V V V V V V V = C C A V NBC C A NBC A A B B A B A δ δ δ δ δ δ V V Q AB 8 9 8 9 9 NCD 8 9 1 1 9 [ 9 P NCD 40 14 MCD V V N + M − + 8 − − + V + − 4 8 V M = P = = . B 2 B B / MCD MCD 1 MA V V − R MA MA 1 3 1 3 U R nc − Furthermore, − contain a weight term. So in fact they transform as generaliz nc nc ∆ ∆ ∆ M where we have used equations ( Similarly, using identities (A.2) and ( weight which can be proved using identity (A.3) and the section cond p and the section condition. Note that the covariant part of th where we have used which holds by identity (A.2) and the section condition, equ JHEP09(2014)044 . 2 , / S 1 Λ EF − S (E.10) R ∂ , ommons V N ∂ EF , ., S ]. c , . V CD , (1993) 2826 = 11 + c +5 N SPIRE V R D IN Λ should be expressible ]. SEF CD S D 48 ][ ∂ V R M M Λ N V Invariance Q explained, this conclusion ∂ S f SPIRE ∂ + REF , redited. IN ]. N V ∂ 4 ][ SU(8) EF supergravity and supergravity in Phys. Rev. NCD B A R , V δ SPIRE V QBE IN in a definite manner, the total SU(8) V 1 36 Invariance EF hep-th/0006034 ][ SO(8) [ B ]. S MCD − V P AE V MA BE − V q B A R R SPIRE δ SO(16) Generalised geometry from the ground up Einstein-Cartan Calculus for Exceptional S V An exceptional geometry for 1 8 IN Q arXiv:1302.6219 ) is a generalized tensor density of weight [ SEF – 47 – ][ AE P ]. − ]. V S P (2000) 3689 Supergravity Theory. 1. The Lagrangian ]. E.1 Supergravity Theory in Eleven-Dimensions V ]. BC 17 + N = 8 REF Supergravity With Local SPIRE CD arXiv:1401.5984 SPIRE [ V V IN N SPIRE N [ IN SPIRE [ SBE . ), which permits any use, distribution and reproduction in V (2013) 077 IN AC [ IN = 11 V B [ ]. M itself depends on The Deformations of gauged d CD 05 V NBD m A M B V R AE + W C arXiv:1307.8295 A V [ (2014) 021 SPIRE δ = MA − V nc . (1986) 363 Supergravity With Local IN JHEP 9 1 (1987) 316 U V ∆ 06 , CC-BY 4.0 (1978) 48 (1978) 409 NBC ][ B − M V − defined in equation ( ∂ NCD This article is distributed under the terms of the Creative C NCD Class. Quant. 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