JHEP07(2021)032 Springer July 8, 2021 May 14, 2021 : June 14, 2021 : : . 2 G Published × Received Accepted supergravity linearized ) 8 C / = 1 auxiliary fermions to the phys- , SL(2; N 56 components. These components and Published for SISSA by https://doi.org/10.1007/JHEP07(2021)032 376 + 376 [email protected] , auxiliary bosons and 201 [email protected] , . 3 2101.11671 The Authors. c Space-Time Symmetries, Superspaces, Superstrings and Heterotic Strings

We derive the component structure of 11D, , [email protected] [email protected] George P. and CynthiaTexas Woods Mitchell A&M Institute University, forCollege Fundamental Physics Station, and TX Astronomy, 77843, U.S.A. E-mail: Open Access Article funded by SCOAP ArXiv ePrint: ical supergravity multiplet fortheir a transformations total are organized of into representations of Keywords: Abstract: around eleven-dimensional Minkowski space.tries This closing theory onto 4 represents of 4 theIt local may 11 be spacetime supersymme- interpreted translations as without adding the use of equations of motion. Katrin Becker, Daniel Butter, William D. Linch III and Anindya Sengupta four off-shell supersymmetries Components of eleven-dimensional supergravity with JHEP07(2021)032 25 17 – 1 – However, it is generally the case that this require- 10 10 21 13 29 . By definition, these supersymmetry transformations form a 22 22 9 1 vice versa 4 15 20 2.1.1 Connection2.1.2 superfields Auxiliary superfields miscellanea 2 G B.1 Linear algebra 2.1 Eleven-dimensional superfields 1.1 Results On- and off-shell supersymmetry. ment fails and —the strictly supersymmetry speaking algebra. — If thethat the fields extend action do the is invariant Poincaré not underto symmetries furnish the be to a fermionic a a transformations linear representation? consistent representation An superalgebra, of important how observation can in the this fields context fail is that the failure of the Poincaré subalgebra. Ideally, thislinear supermultiplet representation of of the fields full furnishesfor supersymmetry a the algebra. commutator (possibly of For reducible) this twoonto supersymmetries to — be the as true, translations realized it on on istransformations). a necessary that component field field — (up to close to other bosonic symmetries and possibly gauge Supersymmetric field theories areto invariant fermions under and a setLie of superalgebra transformations extending taking thealgebra. bosons algebra By construction, of the Poincaré fields transformations in to the theory a are super-Poincaré linear representations of the bosonic D Gravitino descendants 1 Introduction B C Prepotential transformations and partial invariants 4 Toward eleven-dimensional off-shell supergeometry 5 Epilogue A 4D, N=1 superspace identities 2 Prepotentials 3 Dynamics Contents 1 Introduction JHEP07(2021)032 ] (see 1 auxiliary . It is consid- only, and the set close on-shell in theories with eight or more . ; only that the supersymmetries 0 . When we compare these off-shell the off-shell problem ≡ PB } Under supersymmetry transformations, S, S ∼ { This should not be interpreted to mean that the – 2 – δS . . In a canonical treatment of symmetries, in which there is off-shell supersymmetry Just as the translation part of the Poincaré algebra is repre- any non-real representation as functions of both the even and odd coordinates. The compo- ]): for trivial symmetries 2 on-shell supermultiplet In this context, the supersymmetry is said to superfields 1 itself. They are trivial, in the sense that S . In two-derivative actions, they have algebraic equations of motion (no derivatives A considerable amount of effort has been devoted to solving the off-shell problem. The problem of extending on-shell formulations of supersymmetric theories to off-shell In the ideal case, this complication does not arise: successive supersymmetry transfor- Such terms are called 1 nent fields of asuperfields supermultiplet in then the appear odd as variables. coefficients in thea Taylor expansion Poisson bracket of on the action the space of fields, these symmetries are canonical transformations generated by the superfield representations. Manifest supersymmetry. sented on fields by coordinateto derivatives, translations so too in can fermioniccan the consider directions. supersymmetries be Introducing geometrized coordinates for these directions, we suitable component fields andalso transformations section was provided 2.4.3 by ofsupersymmetries Siegel [ (e.g. and hypermuliplets), Roček any [ infinite potential solution number to of thisdimensions fields. problem requires of an The off-shell proof Fock representations is of essentially the a supertranslation counting algebra argument to comparing those the of (In the case oftry most to interest study to self-interactions useffective in action.) in the this form of paper, higher-derivative we corrections intend to to theAn use M-theory immediate off-shell indication supersymme- of the inadequacy of any step-wise approach attempting to add be integrated out of thechanges path the integral supersymmetry (or transformations set which to now their close classical only values), on-shell. supersymmetry but removing by them adding auxiliaryered fields important is for known various as theory reasons of ranging supersymmetry from algebras the to purely phenomenological mathematical representation supersymmetric model building. Auxiliary fields and thethe off-shell new problem. component fieldsfields transform into the obstruction.on them), Such so fields they are do called not contribute dynamical degrees of freedom. In principle, they can mations close properly onof the fields fields furnishes irrespective a ofemphasized linear any by representation. field referring equations, to When it andtheories this as the to fortuitous the collection situation on-shell ones is describedto realized, previously, closure we it with notice is that extra they fields:give avoid rise the roughly, obstruction to the obstructions terms are in replaced the with transformations new of fields. components that are imposed. is referred to as an component fields in thedo theory not are close required to on the be off-shell on-shell realization. supersymmetries to close on the fields is by terms that vanish if the equations of motion JHEP07(2021)032 , , Φ → mnp 1) , C Φ = 0 D Spin(10 , 3-form a m e ]). In this formulation 1 , because they imply that ]. In previous work, we applied a 14 s can generate bosonic derivatives D ]). In these reductions, new shortening ] or on bosonic ghost-like variables (e.g. 4 – 13 2 , ]). This approach can be generalized and 9 12 – 7 – 3 – to impose covariant constraints such as shortening conditions D . In other words, we cannot take such a manifestly An attempt to classify the possible structure group re- ]. in superspace it further implies the superfield equations of 10 than the maximal one. Examples of this include light-cone ] or pure spinor [ satisfying no constraints other than the reality and shortening con- 6 , ]. Besides being the most familiar and phenomenologically relevant, 5 20 , these superfields contribute a finite set of auxiliary fields guaranteeing superspace. – α 8 15 m , so imprudently-chosen constraints generate kinetic operators generalizing [ ] and its generalizations (e.g. [ / . However, these are generally (highly-)reducible representations, so we must ψ 2 i∂ 11 2 G = 1 , etc. These are often called × ) N } ∼ − C manifest , Alternatively, we can attempt to work around the problem using a superspace with Much work in the superspace literature has been devoted to finding clever shortening The problem of finding irreducible representations of the super-Poincaré algebra thus By construction, superfields are field representations of the super-Poincaré algebra (just Φ = 0 smaller structure group 2 D,D the component fields of eleven-dimensional supergravity areprepotential embedded superfields into a set of so-called ditions (i.e. chiral superfields) allowedand off-shell. gravitino Besides the 11Dthat frame 4 of the 32 supersymmetries close off-shell on the set of physical + auxiliary fields. 11D, ductions appropriate to 16 superchargessimilar was approach made to in eleven-dimensional supergravity [ SL(2 based on the reduction this superspace has the significant advantage(and of has requiring the only finitely-many largest auxiliary super-Poincaré fields symmetry with this property [ conditions can be definedgroup, that but are also not do not manifestlythen imply covariant be any under dynamical combined the equations.albeit original into Such not symmetry a off-shell manifestly. representations supermultiplet may representing the larger bosonic group linearly, Lorentz harmonics [ systemized using supercosets [ a superspace [ using such a representation is moot. conditions by enlarging thecircumvent superspace the counting with arguments appropriate by introducing bosonicgeometrizing field spaces. the dependence R-symmetry on These bosonic transformations coordinates [ naturally { those of d’Alembert anddescribed Dirac. above. In But this notemotion event, that the directly on multiplet is thesupersymmetric on-shell representation representation again off-shell in in any the sense; the sense construction of an action principle ated by fermionic covariant derivatives D does not depend on some of the odd coordinatesbecomes thereby one shortening its of Taylorbecause expansion. understanding the the superalgebra possible implies shortening that conditions. successive This is non-trivial as the component fields areto so be for the Poincaré subalgebra),reduce and the them supersymmetry in is a said can way impose that is combinations compatible ofconstraints compatible with reality with manifest conditions, superspace. supersymmetry. gauge In To symmetry, the this and latter, end, we multiplet-reducing we use the supertranslations gener- JHEP07(2021)032 - θ (1.1) )). 1.6 , a gauge these con- a m e , ]. Our result does is sufficient to put ), and ( derive 28 bcde – αβ ) At the end of this 1.5 G c 26  (Γ 2 ), (  ]. ∝ 1.4 c 23 bcde – αβ Γ T a 21 Γ −  -independent components are pre- a θ Γ bcde 3Γ  4! – 4 – 1 · the set of torsion constraints for the supergeom- 6 on these superfields. Besides mixing the bosonic − D β ) derive  . Under linearized supersymmetry, they transform into α bc m ψ ] (Γ c ψ bc a a ab [ ψ ω b Γ 1 2 ]. This observation allows one to postulate an “off-shell” formulation of 11D ¯  Γ ¯  − 24 . As a realization of supersymmetry, these transformations can only ]. However, there is some disagreement about this point [ a m = = 3 = 25 δ b β a a abc is the 4-form field strength, and we have flattened all the indices with the e ψ  C  δ  superspace derivatives δ , and a gravitino δ dC ) ). In this paper, we will explain how to fix this mismatch and give explicitly the C := mnp , 1 C G As a result of this analysis, we In this paper, we will construct superfields whose SL(2 It is known that imposing just the dimension zero torsion condition 2 modifications to these supersymmetry transformations (cf. eqs. ( 11D superspace on-shell [ supergravity (or at least parametrize higherthe derivative Bianchi corrections) by identities relaxing [ thisnot constraint and speak re-solving directly to this issue, because we attempt to keep only 4 of the 32 supersymmetries off-shell. where background frame close on-shell: the degrees of(cf. freedom table needed for the off-shell counting to match are missing 1.1 Results The component fields of3-form eleven-dimensional supergravity consist ofeach a other frame as with the solution to astraints set of by constraints acting we repeatedly do notSince with yet the the know, prepotentials superspace and can with derivatives then to which on terminate we the without start ever unconstrained imposing are fields. apotentials, unconstrained, dynamical field this constraint. strengths, process Along and is the Bianchi way, guaranteed all identities the are physical derived. etry with this reducedpoint structure group. for As solving itlengthy the is and curved traditionally technical process done, superspace (which thisat Bianchi we set the do identities is solution not [ the to carry the starting outthe constraints in situation which this are we paper), the described one off-shell above finally prepotentials. and arrives Amusingly, will then, follow in in this paper, we have essentially started fields (or, more precisely,independent the term). superfields that The4-form have supergeometrical field the strength, tensors auxiliary torsion, of and componentsby curvature the taking as 2-form) the eleven-dimensional may their curls then theory of beforms the (i.e. defined a(n gauge in off-shell) connections. the representation By naïve of construction, way the this reduced spectrum structure of supergroup. superfields cisely the physical 11D gauge connectionsifest (i.e. supersymmetry the frame, transformations 3-form, of andthe gravitino). the The component man- fieldsgauge are fields, effected this by action acting generatescomponent the with “tensor auxiliary calculus”. fields We in use a this form familiar to from explicitly the identify standard the spectrum of auxiliary JHEP07(2021)032 M (1.3) (1.2) by . 2 components structure to G 2 et cetera × G 128 . , | ) 3 . 3 , 7 C 2 1) , 128 , 1) 1 −
, , − 1 ,..., D 2 p − ( , 55 D 320 120 D · = 0 , these components embed ( = SL(2 = 1 c off-shell D 2 2 D = 1 ijk 175 + 320 ). Using the 1 2 b F G α i, j a, m 2 2 × 1) 1 , they add up to , and and with indices, and as a label for 7 additional 1 − 3) 2 arising from the variation of chiral fields, these ijk , 2) G 2 − j , . . . indices will be doing quadruple duty as ¯ Spin(3 ], as summarized in table D i αi − C
D , j 2 ( 16 e i, j D p − → · [ , 44 84 ψ 128 D , 1 j – 5 – m ij 1)( α on-shell m c− j for a stable 3-form 128 + 128 C 2 − D e , , b ij ψ D , 2 a h Spin(7) ( i 1 2 αi mn i for the 7D part. Under the decomposition of the tangent × m e , Y C a 1) ψ , prepotentials , , m e α m mnp fermionic components. ψ C Spin(3 = 3 320 → → → -form p frame → p gravitino a gauge field α 1) m transform canonically. Their (double) curls are field strength, torsion, total (D=11) m , mnp ψ e The components that arise in the Taylor expansion of the fields are listed C 3 for the 4D part, and bosonic and Spin(10 X . Counting of degrees of freedom in D dimensions relevant to Poincaré supergravity. (For on-shell, but off-shell (modulo175 gauge transformation), we find a mismatch between Gauge fields and curvature 2-form components. Using table The prepotential superfields are unconstrained superfields or constrained only to be To get a representation on which 4 of the 32 supersymmetries close off-shell, we de- Such constraints are innocuous: besides factors of • 3 fields are unconstrained assloppily refer integration to variables them in as a unconstrained.) path integral. (For this reason, we will sometimes real or chiral. under the superfield. They have been separated into three groups: To avoid introducing too much notation,7D the coordinate indices, 7D tangentgravitino space fields. indices, (We collecttrade the the 7D index graviton definitionsinto polarizations in a table set of superfields called For notational convenience, we willfor label 11D, spacetime, polarizations, indices, space, Table 1 conformal supergravity, some (gamma-)traces should be subtracted.) compose the 11D component fields under JHEP07(2021)032 ) in C.2a ( G , but the details representation, and Y X 2.1 M 2 label G and denoted Poincaré supergravity itself, tangent, = 1 fermionic degrees of freedom all of N SO(7) 56 4D spinor 11D spinor description 4D tangent 11D tangent 4D coordinate 11D coordinate 7-component label coordinate, – 6 – conformal compensator 7 3 3 10 10 32 , , , bosonic and , , , 2 2 2 Once the eleven-dimensional superfields correspond- , , , GL(7) 1 1 1 ··· ··· ··· ··· , , , range 201 , , , 0 0 1 0 0 1 have been defined, we are in a position to compute their 3 . β . . . . α, ,... ,... ,... n b β are gauge invariant, but do not propagate. Due to this, it is easily , , , suffer Stückelberg shifts under “pregauge” transformations. They do index i, j, . . . a, b, . . . a α m, n, . . . m α, β, . . . , ). This reduction is crucial to the representation of the off-shell supersym- 1.2 -dimensional representations ( . Legend of indices used in this work. To avoid having even more indices, those in the 7 verified that they contribute which are off-shell. Compensators not appear in the spectrumany of gauge(-invariant) degrees superfields, of so freedom they do at not all. contribute Auxiliary fields Relatedly, the auxiliary fields that appear in the Taylor expansion of the prepotentials The compensating fields are artifacts of the reduction of the structure supergroup • • are not important here. Supersymmety transfomations. ing to the components insupersymmetry table transformations. Inponent superspace, fields supersymmetry result transformations from of infinitesimal translations com- in the odd directions — in other words are — strictly speaking —ever, not the invariant under parts all of ofare the the again pregeometrical pregeometrical artifacts transformations symmetries. of under the How- tation. structure which group they Therefore, reduction are it having notsome no is covariant parts eleven-dimensional possible interpre- of the to other covariantize prepotentials. these This auxiliary is components done by explicitly in mixing section in not know.) This situation iswhich precisely is analogous described to in 4D, termscoupled of to conformal a supergravity scalar (whichthis superfield is paper. called not the a physical symmetry) implied by ( metries, but it haspregeometrical no meaning, geometrical but eleven-dimensional the meaning. off-shell pregeometry (Presumably for it 11D has is precisely some the thing we do Table 2 various label for seven gravitini), have all been identified. JHEP07(2021)032 R 128 376 | | sdim 128 376 (1.4) (1.5) off-shell 140 α ijk | ) χ , 70 | ijk ijk ij 70 f Φ h 63 chiral , + of the gravitino ijk 63 ( C supergravity multi- i α . λ 8 c . / 84 | ) α ij + h = 1 42 χ ij ij 21 | , ). The real superdimension k V d real N ¯ 42 λ + a ij 4 2.5 a ( C 63 ( σ . c . ijk αijk 00 αi 0 0 | ϕ χ η αi | ab i , appears in only one place in the i 7 . + h 1 6 Σ 21 C c chiral i . i α H . . c ¯ − λ ψ and c . . a 0 k | ] ) + h 0 α j αij 1 0 i + h η X | abc iσ ¯ + h ψ , X χ 0 d ) ψ real a C ) + G + b 1 ab i,j ( σ ( ¯ ψ . i a i [ 0 a c – 7 – σ ψ | . . ( k ) 2 ψ c i to include a Wess-Zumino gauge transformation . 7 0 iδ i i | i a 2 i i α − iσ 4 . d V h  η + h )) ensures that + real 14 i − − δ ] ] + h 56 − b c 21 | 2.6 l ( = 2 = = ¯ ] k ψ ψ k a a ) for the explicit expression): [ ij ab ai 201 ab are encoded in [ ψ α i ψ h ) h h   ρ    , σ iσ βj δ ij δ δ 56 2.8a [ i 6 4 l ijk αβ i ψ βj α i − − ϕ ϕ + t 14 a | , Ψ 0 ). 84 a i ψ = = = = spinor ( y | and , 1.5 i α 98 β ijk abc ab i a ij λ i , or eq. ( C C C ψ C     3 δ δ δ δ ) and ( β 12 | a ) a 2 a 1.4 4 ψ | , d U 2 real ab + h 6 ( spinor index contractions are implied.) The supersymmetry transformation of ) . The unconstrained “prepotential” superfields of the 11D, C , Finally, we present the gravitino supersymmetry transformations. These allow us to Some of the gravitino components can be shifted by the gauge-invariant dimension-1/2 auxiliary field 4 , and this type of correction could also have appeared in the graviton transformation. Because this i α SL(2 λ field vanishes on-shell, thisof leads to the an gravitino ambiguity wetransformations in make ( the below definition (see of eq. the ( off-shell gravitini. The definition — that is, asuperfield field-dependent multiplet. gauge transformation that involves unphysical termsidentify in the the dimension-1 auxiliarylegibility, fields we needed suppress to the close ubiquitouspensating spin the Wess-Zumino connection supersymmetry gauge terms algebra. transformations. in For addition (The to complete including expressions com- are presented In these transformations, we define the 3-form acquires amultiplet correction (cf. by table the dimension-1/2 auxiliary field ( — from acting withthat the the fermionic graviton transformation covariant derivatives. is unchanged The from result the of on-shell rule: this calculation is Table 3 plet and their physical,gravitino auxiliary, of and the compensating type contributed component by content. each The superfieldthe components is auxiliary computed of field modulo the space gauge adds transformation. up to The superdimension of on-shell off-shell physical auxiliary prepotential compensating JHEP07(2021)032 . D and ; cf. j (1.6) i 2.1.2 B ). This Z ). abcd β B  and com- G 1 2 1.6 -form com- jbc 3 -dimensional t 3] − β ,   14 d ) [1 , we consolidate ¯ σ ak ¯  4 appearing here are y j i abcd − Z iε ak fields in table + Γ i c ( ¯ σ ijk , and geometry (appendix ¯  parts of the jk a i f i 2 ab , ϕ Γ 1 -component multiplet of eleven- η G i ij 6 , 4  d + R i 376 6 , | a i ) + 27 − y a i , j 376 Γ d X a klm β β d ) ) G a a ( ¯ σ j i ¯ σ   R h (¯ (¯ – 8 –  β i , and the ) i β 12 a ) − ¯ σ a  − ab ij ¯ bj σ (¯ representations of the 4-form field strength i  ¯ i G 2 Γ (¯ d ) and the essentials of 2 i  β β G − j a  A )). −  i ¯ b y 2 j d ab of intermediate calculations combining the prepotentials of − β − 2.9d ) ab i σ j a C 8 5 ab y G ab i 4 t σ 1 + − β π ) j a + ab β ).) For the latter, the invariants ab ), and ( ) Γ i η ab  ab σ  β ( η B.18 -form component  σ ( 2.9b i i ( 1 4 5 i 2] 12 12 2 i , .) . (The explicit definitions of these projectors are given in appendix ), ( − [2 D This concludes our description of the superspace (appendix ' − ' − ' − ' − ) and ( a ijk 2.9a β a β i , we describe the dynamical equations that result from the off-shell action derived into building blocks with simplified transformations that are used to define all the βj βj G i ψ ψ a = 1 2 3   ψ ]. These equations relate auxiliary fields to physical components, thereby demon- δ δ ψ B.4b   What follows are four appendices with conventions beginning with basic results for The results are organized as follows: in the next section, we give the steps starting N δ δ 16 section other fields throughout thissuperspace work. derivatives Finally, in we all presentThis the possible demonstrates result the ways consistency of on offormations acting the the of with scheme gravitino the fermionic and eleven-dimensional components generates gravitino in the components appendix supersymmetry reported trans- above in ( the results for the 4-form,and torsion, conclude and curvature the field body strengths of in the terms of paper. supergeometry, 4D, is followed by an appendix from the basic prepotentialssection and arriving at thein connection [ and auxiliarystrating superfields. the In relation between on- and off-shell field content. In section Outline. dimensional supergravity furnishing areal linear supersymmetries. representation In of the theresult. remainder superalgebra of In with the the paper, four process, wefield will we strengths, explain will and the provide relations derivation explicitderivatives. of between expressions this these for under the the gauge action superfields, by their the fermionic superspace combinations of the more basicponents auxiliary of fields the 4-form(cf. field eqs. strength. ( These relations are given explicitly in section In these expressions, only certain dimension-1 auxiliary fields appear explicitly.part For the of former, these the areponent the eqs. ( in appendix JHEP07(2021)032 (2.1) auxiliary X d 2 ¯ θ ) are invariant superconformal 2 a θ d 2 i α C.4 + ¯ θ ρ = 1 2 . 2 α ). These are collected θ ¯ θ ¯ η ), ( N 2 . α + θ ¯ 2.1 θ . 2 α C.1 ij + θ m d that shift under pregauge 2 ¯ ψ + abi ¯ θ . α t i 2 α ) α θ d η ) 2 m α ¯ θ + ab θ 2 θσ 2 . θ α ij ( ¯ θ θσ 2 ¯ χ ( ¯ + . θ 2 α + α i ¯ ¯ θ θ . αi + ˜ 2 η 2 ¯ η θ + npq α . θ α a i ¯ C m θ + ijk y 2 transforming canonically, there are ψ θ α ] + f α -expansion. This should cause no confusion, as ) 2 α ij ) – 9 – a θ mnpq + θ ¯ θ χ ¯ σ mn i i α α ¯ m θ ¯ + θ ε θ η ( C σ 2 2 α m ( α ¯ θ θ 2 ) θ 2 θ θσ α ijk compensating components ¯ + + θ mn χ + gauge fields + α + αi a θσ θ m ij G , we embed the components into prepotential superfields m m i m 2 + e ψ e C ¯ 3 θ + ( ) ) ) ) ¯ ¯ ¯ ¯ i θ θ θ θ + ijk m m m m H ¯ F G α i 2 θσ θσ θσ θσ θ θ + . They contain abelian transformations of the M-theory 3-form, a non- + ( + ( + ( + + [ + ( ijk C C ∼ · · · ∼ · · · ∼ · · · ∼ · · · ∼ · · · ∼ · · · ∼ i a α i ij αi X V ijk V U Σ Ψ Φ under the linearizedones. hierarchy So transformations we make but combinations thatparameters not are using invariant under under as one or few the two superspace of superconformal these derivatives additional as possible. parameters) the 3-form components we already had from the non-abelian tensor abelian gauging thereof undertransformations, 7D diffeomorphisms, and local extensions 4D, thereoftions). (e.g. extended supersymmetry transforma- in appendix that do not propagate, and These partial invariants are used to covariantize (with respect to the superconformal The field strengths of the non-abelian tensor hierarchy ( We write the linearized transformations of the prepotentials ( The procedure to construct the eleven-dimensional superfields is the following: These superfields are complicated but explicit combinations of the prepotentials that Our main result is a kind of inversion of this embedding: we will present explicit 3. 2. 1. and 3-form gauge transformations).supergravity gauge Concretely, fields, the whereas gauge the fields auxiliary fields must must transform be as invariant. 11D no analogous superfields forno the lift compensating to 11D. components, precisely because these have transform only under the physical partthose of corresponding the pregauge to transformations (by 11D which we diffeomorphisms, mean supersymmetry, Lorentz transformations, expressions for 11-dimensional superfields containingas these the gauge fields leading and term auxiliarywe in fields will their Taylor use expansions. thethat (To component same avoid as introducing symbols the yet leading tohenceforth more term we denote will in notation, always its the mean the component superfield field unless explicitly and stated otherwise.) the There superfield are having We recall that, in additionfields to the transformations so they do not appear in the spectrum. We now begin the tasksections. of As explicitly implied identifying in all table the elements described in the previous 2 Prepotentials JHEP07(2021)032 , . G 3 ij kl (i.e. R ) and γ , ) with cd ab cd 2.3 ( . 2.2 aj ab ij aj kl T ab ( , with the ij b R R R R a , , , ab abc i abcd , e C γ abc γk R G 2.1.1 ijkl ab ab ij ab C G T -corrected versions G T , Ψ a ijk ]. From this, we may G 20 α i ρ . ) and their derivatives into , cd G ab i ijk 2.1 ab t R f , i a , ij , a i R d d d components of the field strength X Γ ). none ijk , d 2 a i , m y 2.8 3 2 , , α i 1 λ . (Hatted quantities are C α ijk α ij β β αi χ i a , b χ j , i ψ ψ abc – 10 – ab i ψ , , , i e b , C a a ij a a i e e e C . ijk 4 C ) from the partial invariants so that they transform as 2.6 ). Taking curls defines the dimension-3/2 torsion ( forms embedded into superfields in [ β 2.7 C i a ijkl ] P b i H , ψ i α are the curls of the gauge fields as described in section α ij E , ∗ , b S a i H p, q , c W W R [ Under dimensional reduction, the eleven-dimensional 3-form splits ijk X ) and ( b F 2.4 G , and curvature i a α i α i ij . Prepotentials and their derived quantities. All derived quantities contain admixtures of ijk T X V V U Σ Ψ Φ the gravitino curvature) and dimension-2 curvature The resulting invariants are listed in equation ( leading Taylor component the physical 3-form in the on-shell spectrum. gravitino superfield they should ( hierarchy. This results in the eleven-dimensional 3-form superfield Finally, we use the partial invariants to covariantize the auxiliary fields in table In the same way, we construct the eleven-dimensional frame superfield 5. 4. prepotential partial invariant gauge field auxiliary field strength successive combinations are made togauge have progressively and simpler local transformations underresults superconformal the presented parameters. here. This isM-theory outlined 3-form. in appendix up into a collection of 2.1.1 Connection superfields We construct the superfields carryingas the their eleven-dimensional leading 3-form, component frame, bysuperfields and combining that the gravitino transform prepotentials as ( connections. Much work goes into this construction in which 2.1 Eleven-dimensional superfields In this section, we definecorresponds superfields to that a have field thetheir property curls) in that or eleven their auxiliary dimensions. leading fields component that These do are not propagate. either physical connections (and These steps are summarized in table components to any oneterms of field the is prepotentialsof are similarly deferred the ambiguous.) to naïve appendix hierarchy Explicittorsion field formulæ strengths.) for The partial dimension- invariants in Table 4 other prepotentials, so these identifications are approximate. (The assignment of curvature tensor JHEP07(2021)032 Ψ ¯ D (2.4) . gauge (2.3c) (2.2c) (2.3a) (2.2a) (2.3b) (2.3d) (2.2b) (2.2d) -forms, D ] p, q [ SO(7) . ) corresponding × i , respectively). ] ¯ Ω X 1) i bcd , − ∂ C i ) in appendix a [  (Ω ∂ . δ and . SO(3 i α δ 1 D.7 2 k α a U = 4 ) U X − i ], this fixes the components 2 ) ∂ ¯ i ijk D 23 τ abcd ϕ S klm + . . to all the naïve components of β α G + ¯ + . F 2 δ i } l . Up to an β α k D τ i b δ b ( . ( H V klm β i a 2 3 . ,X 1 2 α λ ϕ ¯ Ψ − α i α j i i ijkl β − ∂ δ − Ψ i ψ := b X ) , D b ] 1 β i 1 2 . a ijk 36 δ ξ ), but we will not need that result. U j ξ − a ] ϕ ¯ U D V . ∂ α − { β i , ] + Ψ ) + − . ¯ δ α ) of the prepotentials and adding terms so that D Ψ C.9b i = ikl , . ) – 11 – ¯ ij β D ijk D b [ β α and V F , 2.1 ¯ ¯ a ] Φ  D Σ α . D e h . α δ [ jkl h δ + ¯ D ).) 1 2 1 2  D [ ϕ α . ) and ( , α ( 1 2 1 4 − − abc δ ijk α ¯ L D ε The eleven-dimensional frame is determined by starting α D i C.13 2 1 4 [ (Φ := := := := may be thought of as partial covariantizations of the C.9a D 1 2 1 2 b b j j − − i i i a a e + e e b e -expansion ( H := := := := θ α L . ijk . α abc α ij αβ i and ¯ α C C D C a i  C i X − := is the “lower-left block” of the Kaluza-Klein decomposition. This block is a ξ Ψ ¯ D parts of the gravitino superfields (by the derivatives Ψ D We see explicitly from these expressions that the 4D component is as expected, and The construction given here is typical of everything that follows. We will now present Note, however that the corrections by We give the explicit expressions for the local Lorentzthe parameter Kaluza-Klein ( component is the naïvevector one. in We also see that the real part of the complex The translation and local Lorentzare parameters also are shared fixed. among For these the components, translations, so this they gives Eleven-dimensional frame. with the naïve term inthe the resulting superfields transform as choice and a normalization weof fix the by frame matching to to be reference [ to the M-theory 4-form invariant.hierarchy (An invariants explicit is set given of in expressions ( for this in termsthe of analogous tensor results for the frame and gravitino. They are defined explicitly in ( the gauge 3-formcompensating are fields. needed to With this,so cancel we the may the immediately corrected define local superfields the superconformal field transform strength transformations superfield as gauge of the The combinations and local superconformal transformations. This results in the following set of superfields: extract the superfields for these components, and covariantize them with respect to the JHEP07(2021)032 - )) θ (2.7) (2.6c) (2.6a) (2.5a) (2.6b) (2.6d) (2.5b) of the ] for a B.18 by the j 29 i ij e V that enters ) α i 3 ).) Ψ , j Ω C.2d β i . ) j α D  j 2 representations to form ¯ β i . W αk i − χ 2 ( Ψ  l G 2 structure (recall eq. ( := 14 + 2 V ¯  , see eq. ( D ] π j . The second is the analogous k . (The part of 1 4 βj is the double curl of this. The α i b β jkl ∂ ijk − H ] ] . χ ijk Ψ ij − Φ α d b [ l Φ ¯ h D jkl ϕ c β lmn αk i [ χ ,  + ∂ ψ , these components can be reorganized W a . β 1 6 + 3 .) Together, they comprise 56 spin-1/2 [ α  S B j γ l ∂ lmn L b . H α 2 k − is the curl of the spin connection. These b X H ¯ ijk ¯ ψ D γ D := β jk β α cd 49 = 28 + 7 + 14 1 4 jk ] iϕ χ ijkl D δ i into b of the linearized eleven-dimensional gravitino. cd 2 i 3 ψ − ϕ jk vanishes in Wess-Zumino gauge. See [ + ω h i α i – 12 – 1 2 ab ] a βj b 1 β [ α α j + ϕ 36 Σ i i := δ ∂ R  + e a Ψ 2 3 ψ β i − , [ U  1 12 β ∂ ) β The definition of the eleven-dimensional gravitino suf- ijk − i := 2 F D j a ψ − ]. are mixed together through i 2 klm β + 2 . cd ] β i X  ¯ χ 30 b c D β α ij ( λ with α ab 35 i h 8 j i D a 1 and the gravitino superfield D W [ R 12 h − for the “scalars” of the tensor hierarchy. (This is a partial ∂ β i  2 1 and := := V a := := := := 2 ijk ∂ α ij β β := F i α ijk βj βj 21 a χ projectors given in appendix i components = a χ ψ ψ 2 ab ψ ψ β c G a ω ψ . The Riemann tensor δ c ) b .) δ c 49 a implies ( 56 = 7 + 49 e and = 0 These components are defined by their gauge transformations Using various The symmetrization of this frame defines the linearized eleven-dimensional metric := 2 7 c ab ab and how the original the (In the last component, we see explicitly the Using this, we explicitly define the eleven-dimensional gravitino components as thing for the 35requires spin-1/2 the fields further in covariantization by thefields from chiral the multiplet 4D perspective [ into the The first of theseconformal may shifts) be of the understoodKaluza-Klein 21 as gauge gaugini the field in covariantization the (canceling abelian the vector local multiplet prepotential super- Eleven-dimensional gravitino. fers from various ambiguities.do Below, so, it we is present helpful what to we first consider define the the simplest components one. (which appear To also in table h linearized curvature 2-form curvatures agree component-wise when the spinT connection is taken to be torsion-free, since expansion. (The lowest componentcomplete of discussion of thislinearized frame component.) lying along Wepartial the also seven-dimensional field see space strength explicitly iscovariantization a that of particular the the component imaginary part of part the of the chiral field usually taken to be zero, but in superspace that is only true to lowest order in the JHEP07(2021)032 , the to be i (2.8f) (2.8c) (2.8e) (2.8a) (2.8b) (2.8d) α i V . There ρ d superfields where, as a ).) α i )), there is an . = 1 Ψ , this superfield is C.9e 2.8a N lmn 3 F . k . ∂ 4 qrs -component, we find b f F p ijk lmn ∂ defined in ( . G α i 1 4 , the Kaluza-Klein field ijk a J i + F U lmnpqrs S ε k + 2 i i ∂ b b H H ) ijk P α 2 a ϕ -components are covariantized as ijk lmn ¯ ∂ D ) D 1 6 d ]. In addition to the dimension-1/2 auxiliary 1 + − − − − 16 a i 2 S  – 13 – G ] ( . . α kl D X α i 3 . ( ¯ α χ ¯ D . . Their α ¯ i , + D 12 ¯ ij α D V + ijk D . Covariantizing its auxiliary + [ := 2 b i a auxiliary is that of the chiral scalar field, of which we have F 1 6 of dimension 1/2 that can be added to three of the four 2 ijk α i X α kl a λ + Φ D i α S χ = 1 ) ∂ a λ α 2 2 3 := U are invariant; they are the dimension-3/2 torsion components. ¯ N D D β  − equation of motion. γ ijk + D kl i α 2 α i ab f 2 ij ¯ W D Ψ T ψ D β α ( 1 1 4 32 D D 1 8 1 3 − − := := := := is the partial covariantization of the scalars i a ij X -expansion. These “auxiliary superfields” will be given the same name as their , and the abelian vectors d d d θ d ijk X b F Finally, there are the additional auxiliary fields of the gravitino multiplet. This multi- Another familiar 4D, There are many more dimension-1 auxiliary fields since most 4D, The definition of the dimension-1/2 gravitini was ambiguous because there is a (Here plet is reviewed in detail in appendix C of [ only one such representation contain one. The vector multipletsare all 4 contain such a real real3-form auxiliary prepotentials: field we the denote conformal by graviton This field is presentcomponent in field, any it forms Lagrangian a theorydefined Lagrange below. of multiplier the pair Because with of gravitinoproportional a this, multiplet to dimension-3/2 and spinor the as we will see explicitly in section no confusion since, throughoutexplicitly this state paper, otherwise. we will always mean thedimension-1/2 superfield invariant unless we 2.1.2 Auxiliary superfields We will now defineof superfields their containing theleading auxiliary component components to as avoid the a leading further term proliferation of new notation. This should cause But this condition isinvariant ambiguous: spinor as superfield wecomponents will explain defined presently above. (cf.them. eq. The ( Curiously, ones they we alsothat have turn all defined out the here to dimension-1/2 be torsions are vanish, the as as simplest we we ones will originally in see derived the in sense section that they imply so that their curls JHEP07(2021)032 – 21 how (2.8i) (2.9c) (2.9a) (2.8g) (2.9b) (2.9d) (2.8h) 3 , which α i ψ . α lmnp does appear ¯ D G y i , ) and to prove . αi abcd ) 1.1 α = 12 G Ψ h , ijklmnp . α ε  . αα i β a abcd ( 1 ε 12 Γ U ∂ 2 ]. i 2 4! ) of the eleven-dimensional − i · D . + 32 α 6 1.6 , . ¯ α lmn D ) + 31 α 1 2 f [ U X i + . In fact, . d αi β ) ( abcd 1 3 G ).) α j ijk lmn α i D G = 0 ) . X α d D i ] = α G ¯  ) D a β 14 . D.3b i ( α i π U = ( 8 D i∂ Γ a + 3 . . β α ( . i∂ − − α ¯ D ¯ a i ) The supersymmetry transformations of the grav- ¯ – 14 – D mnp D + i β ˜ ¯ G β Φ lmn i ( l ¯ Ψ + 3 G [ ∂ f α . D . (Note, however, that the real part of α 2 ( + h a i ¯ i ¯ D D D y lmn ∂ 2 β ) cf. also eq. ( k ( i 1 ajkl 4 ¯ 12 D ψ i D ijklmnp G β 1.6 − +  h b jk ( H i i i D 1 3 4 ) = ijkl ϕ 1 4 D β βj a ψ − − a i Ψ P − 36 ¯ i σ 1 6 2 ψ ) α ( ¯ β ¯ D D lmn := + i i α D 2 i ) + b 6  W 2 F i α λ a i 2 α . ¯ ρ + α ( D ¯ y = lmn D ¯ a i 4 1 D D f S − 2 1 X ( − − 2 a i j 2 ¯ i D y ¯ = h D  := := 1 6 ijk lmn i i i 1 . 2 α − − G − ), there are two dimension-1 components i α αβi y = t := := := := 2.8a j a i i R The second is a complex linear field The auxiliary fields defined in this section are important because they are part of a ijk Γ Z ]. In the 3-form formulation of the theory employed here, the imaginary part of the m explains why it is this combinationtry that transformations. features Expanding prominently in it thepart out gravitino of explicitly, supersymme- we the see gravitino auxiliary thatin field it the is transformation related of to the imaginary The first of these isthe complex the auxiliary chiral field, scalar the field spin-1/223 part strength of of the old-minimal gravitino, and supergravityauxiliary the containing field curvature scalar is [ dualized to the 4-form component that appear naturally in the differential algebra are that they do notthey contribute propagating appear degrees in of the freedom, off-shell we dynamics. examine in section Useful alternative composite fields. itino are mixtures of the auxiliary fields and 4-form flux. The gauge-invariant combinations completing the set of auxiliary fields with a dimension-3/2linear spinor. representation of the subalgebrathem generated appear by explicitly 4 in of the supersymmetry thegravitino. transformation 32 ( supersymmetries. To compare Many of this rule to the known on-shell transformation ( comprising a complex vector and a 2-form, and the top component of field ( JHEP07(2021)032 14 + (3.1c) (2.10) (3.1a) (3.1b) to the in the 7 . 7 + ¯ a D 1 is evident. U + + 1 . 14 and i + 27 . ). (The choice . + α ) by a real field i a 7 D = 27 ¯ ψ i qrs + 2.6b of that action with 2.8f b ∂ ¯ = F 49 ] by including them into 28 − 29 is not gauge invariant). 35 . α = i mnp qrs ¯ ) and ( ψ 14 δS/δφ a 49 G is real. This agrees with the h ∂ h 2 2.6a := . ¯ α D 14 α is invariant. It is somewhat less φ ) i a ] (and to low order in a gravitino E b σ 5 H 16 ijklmnp α ε 12( D 1 2 12 − ¯ D − βi ] all appear in the connections alongside ] structure, with the real ) and the imaginary b i 2 jkl ψ V Φ a G i [ that is a singlet under the 4D part of the Lorentz [ – 15 – B.11 ∂ ∂ and β α a ijk ··· are not constrained except for reality and chirality α i ijkl = 4 ) Ψ + 3 G G ab symbol defines a chiral 4-form k σ j mnp ijk 7 jkl ¯ are complex, whereas the ϕ Z Defining the variation m ϕ ...i γ α of auxiliary fields appears in the supersymmetry transfor- 1 ) and curls of the gravitini ( 1 6 i 1 7 + 2( δ 18 ε 6 j , we obtain the supercurrents i + αi )). Equivalently, this final invariant is the component of the φ + ]). 2.8i i Z = 4 ρ ijklmnp ( and α i a d ε 15 1.6 3 d λ α i = 1 γk 1 12 i 4 ρ i − − αj + − b H = = = T α := i 27 a α i )). In the form presented, the decomposition D V U Ψ 2 = (cf. eq. ( ijkl E 4.9 E ¯ E D E i βj 28 4 ] where a Wess-Zumino gauge analysis was carried out to identify the spectrum: i 3 ψ 16 Finally, we mention for completeness that additional invariants are possible that look The last combination The third is a modification of the auxiliary field of the scalars ( Contraction with the constant The gravitino and Kaluza-Klein field were treated non-linearly in reference [ 5 6 the superspace geometry. In this approach The prepotential superfields ofconditions. table Therefore, we can writeequations. an action This in terms was ofsuperfield done them at expansion to determine the in the linearized dynamical [ respect level to in the prepotential [ be in contradiction withthose the fields. Wess-Zumino analysis since we already accounted3 for all of Dynamics defined auxiliary field of the dimension-3/2definition spinor of depends the on auxiliary component.)results how of This we [ conclusion order was guaranteed theany by new derivatives the invariant component would have to correspond to a physical component field, but this would is easy to checktrivial that to the show dimension-3/2 that spinor it satisfies We conclude that this invariant is not new, since it is a combination of the previously- Of these the counting of the first-order invariants ofcorresponding the to the fouranalogous torsion derivatives forms of ( the gauge 3-form (the derivative in the new, but are actually linear combinations of the invariants defined above. For example, it strength. It is an interesting alternative that is chiral. mation of dimension-1 torsion group (see eq. ( JHEP07(2021)032 (3.3) (3.2f) (3.1f) (3.2c) (3.2e) (3.1e) (3.2a) (3.1g) (3.2b) (3.2d) (3.1d) This is 7 . i . α i Ψ X lmnp E E α ijkl , these equations G D ijk G . α ϕ ¯ 8 3 D ijkl i R ψ ijk ijklmnp − ijkl ) − ε ijk 1 48 Φ G ϕ α kl lmn i E 6 χ ) j ijkl ) + 3 ) + Φ − ∂ + ψ . E Φ jkl 1 lmn i 2 96 lmnp E − ¯ f ( lmnp G iψ are those of the gravitino superfield. Φ + G ) + ijk E } ϕ ( + 2 α i lmnp ijk i 2 i lmn ρ ψ ) for the auxiliary fields to find ¯ , f m lmn − ( + ijklmnp ϕ bcd i αβ i 3.1 + ε t X a G , ijk U E a i i i . ijk abcd – 16 – ϕ 576 bcd y E V i ijk lmn + 3 a G { R m 4 we will continue to work off-shell E G ( − − iε ijk i + ijkl − ) abcd ijk ijk that vanish on-shell. − ϕ β ϕ ϕ lmn G iε a ijk ∗ ijk i Ψ ( ¯ 12 f ijkl 1 1 i α Φ E G 72 96 E ijkl a jkl ijk G Ψ + kl α E ψ ( i α i ijk + − ϕ G E V h jkl i λ ϕ 1 D ijk 96 X X 2 4 jkl ijk lmn E ϕ i 1 1 d d ¯ ¯ − 2 36 18 1 6 1 − − D G m + 48 ij contribute nothing new on-shell. Similarly, up to equations of ψ 2 3 2 3 h d i i i i ======t 16 48 48 − − α i Ψ ) a i α αβjk ij X a d = = 2 = = = = d E λ d d 3 ¯ lmn G σ a i α and ij 1 2 X ¯ f ¯ σ jk D ijk Σ i V ( ¯ E y Φ D ϕ E i E 1 2 12 1 E 2 − − − ijk lmn is a sum of three curls, each of which is separately an invariant of dimension 3/2 G = = = ρ a i α i y ρ αβi t Explicitly, we can solve these equations ( This overloading of notion should not be confused with the compensating component fields, which are 7 It follows that motion, — that is — they are components of theunrelated gravitino and curl. have played no role since the earliest part of section The auxiliary fields notHowever, appearing it here is not difficult to show that to indicate that setting thema to useful zero distinction defines because, theare although superspace needed mass-shell to condition. provefreedom the on-shell: claim that the auxiliary auxiliarystrength fields only fields by do differ the not from superfields contribute components any of new the degrees physical of 4-form field These are in bold font because they are gauge-invariant, and we have colored them red JHEP07(2021)032 ]). 35 (4.1) (4.2) (4.3) (4.4) (3.4a) (3.4b) ), up to 1.5 off-shell component . 8 = 0 / D = 0 , = 1 , C c ) ]. The obstruction theory for this jkl ab cd N cd f 34 = 0 M , (Γ ]. cd 33 ijkl cd 36 ψ ,T i AB 6 AB αβjk R αβ R For eleven-dimensional supergravity in − 2 1 ) G 1 2 c 8 . The latter can be rewritten in terms jk i + ijk + d structures [ (Γ λ i C permutations 2 iϕ G D ijk D ] ∇ − ϕ C ) and | C ∼ – 17 – + 2 ]] + E = | i + 4 C c AB T ). These are subject to the Bianchi identities that 2.8g i αβ T ∇ E t d 10 ] αβ , i i ] B ] = AB [ = 4 = 6 B ∇ T ,..., , ) . 0 i α ∇ i β A + 0 ,T , λ λ = α A ∇ D α ] ( [[ a ∇ D = 0 | → [ D ∗ AB γ E T is the covariant superspace derivative of eleven-dimensional super- and αβ C [ T a 32 ∇ ∇ , α ,..., ∇ 1 = = (i.e. its supersymmetry transformation) generate gravitino curls by ( α A ∇ C d These identities become non-trivial when covariant conditions are imposed on vari- Finally, we note that the action of the superspace derivatives on the field strength This can be formalized in terms of first-order-flat = 8 Others follow from the deformationsupersymmetry of constraints algebra defining representations (most of notably, the flat chiral). space Thereis is known as no Spencer known cohomologyThe (cf. Spencer analog e.g. cohomology Schwachhöfer’s of for article 11D “Holonomy this Groups supergravity and was type Algebras” considered in in [ [ convention that follows fromnections the (most possibility importantly, of the shifting spinsuperspace, connection). the these definitions are of the various con- ous components of the torsion and curvature. Some of these conditions are a matter of gravity ( follow from the nested commutator identity The non-trivial part says that approach defines the superspace torsion and curvature through the commutator where are imposed by taking 4 Toward eleven-dimensional off-shell supergeometry The analysis that leadson to the the supergeometry results needed of to the solve previous the sections superspace also Bianchi implies identities. the Such constraints a top-down implying that supersymmetricnew derivatives of invariants. the This concludes dimension-1spectrum the reduces field demonstration correctly strength to that the generate the physical no spectrum if the component equations of motion G supersymmetry spacetime derivatives of auxiliary fields and field strengths by ( JHEP07(2021)032 and ) are (4.6) (4.5) . The ) with k j 4.4 δ 4.3 αβ Ansatz iε ) is a tensor, 2 − 4.4 = k ], that these must be α βj T can appear through the 16 ∗ = 0 αβj k T . β α ijk ,T ϕ . β α ) α i ) c ¯ λ b σ Let us now specialize to the supergeometry ( σ i over-constrain the torsions leading to trivial ( 2 1 6 − − – 18 – = = c . β Ansätze α ). Starting with these, then, it must be possible to αbjk 2.1 G ,T = 0 in this case. c supergeometry. . 2 2 / 8 αβ / 3 / T 1 ). The remaining dimension-0 constraints of relevance from ( < < = 1 4.3 . Similarly, we keep only those parts of the Bianchi identity ( N } i ∇ , a is manifest, we retain from the superspace covariant derivatives above only ∇ 8 , . / α ¯ ∇ , Less obvious — but not difficult to verify — is that there are also no dimension-1/2 But now we notice that we already know the solution to the linearized constraints, even Historically, appropriate torsion constraints were often found by making an = 1 α torsions with our choice ofappears gravitino components: only in in the this dimension-1/2 form, the component gravitino auxiliary field of the super-4-form field strength. together with their conjugates. Thisants is with consistent dimension with our finding that there are no invari- {∇ the implied free indices. However,so the any on-shell non-vanishing eleven-dimensional component torsion ofimplication ( it of must this be is retainedquadratic that including terms in torsion ( components of the form constraints can then betransformation gleaned of from the the connections. spectrum of invariantsOff-shell and 11D, the supersymmetry appropriate to theN eleven-dimensional superspace considered in this paper. Since only things) invariants and relations between them.strengths, torsions, The curvatures, invariants are and the theira aforementioned derivatives, superspace field since by these definition. are Theidentities. the relations only In between invariants this them of paper, willantizing we then the have be auxiliary taken the fields. a (reduced) The shortcut Bianchi remaining by invariants building are the the curls connections of and the covari- connections. The We already know fromequivalent the to derivation the of prepotentialsderive the the ( constraints correct from spectrum thisrepeatedly [ solution on directly. the In prepotentialserate principle, with a this can graded superspace be space derivatives done of in by superfields. all acting possible In ways this to large space, gen- we will find (among many other representation constraints. Ofthe course, lack guessing of the symmetry. torsion constraints is complicatedthough by we do notometry, know the the solution to constraints! thefor superspace This Bianchi the is identities field because culminates strengths, in in explicit torsions, the expressions and case of curvatures in off-shell superge- terms of unconstrained prepotentials. multiplets with spins solving the Bianchi identities.solutions Poor or on-shell geometries.ture group We can so circumvent that this there problem are by fewer torsion reducing constraints the and struc- more possibilities for deformed of constraint in eleven-dimensional supergravity, presumably because there are no matter JHEP07(2021)032 , ]. 4 23 > – (4.9) (4.7) (4.8) of the as the 21 λ ¯ D dimension , 2.1.1 ] ∼ ≤ . part k 2 a j y ¯ λ γ , / defines the 11D b ¯ Γ 1 T . γ . R . α γ Dσ superspace [ ) α = 0 a . ) ). 2.1 + b c σ . β ( k σ ) with = 1 . a ab γ l ( . λ i γ 2.9d T ¯ λ 1 2 12 b N 2.8 T ¯ α σ . γ = = ¯ D  D [ . . γ γ kl γ α − j δ , and the dimension-2 curvature αb , αj ϕ β 1 8 γ T T a 1 6 ), we find for the γ kcd ab -expansion. We see that for D − t auxiliary fields that contain the 4D θ − T . T γ D k b γ  α  Γ -valued spinor index decomposes as R torsion from ( ) al γ e α 7 2 δ ¯ σ Γ , i 8 G ∼ − kl k and j β bcde + with a ϕ a ab j i , d 1 6 ck – 19 – G γk − ¯ , Γ of the − abj ] T d 14 DE c t ) j γ . ) (or appendix i σ γ π d α a i ] γ α ) bc Z γ ¯ ) α Γ η 1.6 ∼ − bc α ) a lmn ab ) σ (4 β ab σ bc G i a 6 ( σ ( σ ( E k i i , and  j + 8( + 12 δ h + ( k + bc  + ab k d ∼ η k . t bc . γ , the dimension-3/2 torsion j γ j γ α η , α β α d δ i γ ) α Z ) a γ α δ a d b [5 γ α ) to zero. δ [ ψ abcd σ σ δ  i i 2 2 i ( ( δ 1 2 12 G i i 3.1 ======γ γ . . γk γk γk γk αb αj αb αb αj αj T T T T T T . These are all superfields with the implied component field strengths as their leading , which are reduced dynamically to the aforementioned physical spectrum by setting 2 The remaining torsion component was defined already at the end of section The analogous dimension-1 torsion We could now begin to solve the superspace Bianchi identities subject to these condi- This leaves the dimension-1 conventional constraints. Since there is no bosonic eleven- cd / 3 ab dimension 1 4-form R Taylor coefficient. In addition, we≤ find the auxiliary superfields ( the supercurrents ( and the superspace covariantization spacetime curl of the eleven-dimensionalwere gravitino. also The dimension-2 defined invariants in (curvature) strengths that of section. the eleven-dimensional supergeometry In with summary, 4 we off-shell have supersymmetries demonstrated are the that the only field The new structures include thegravitino real auxiliary (which and vanish imaginary on-shell), parts various of projections the of descendant the 4-form field strength, The first line is theThey same as are those essentially of the textbookpart definitions treatments of of of the 4D, the Ricci tensorthese and are scalar extended curvature by in their components can be deduced directly from the linearized gravitino transformations Collecting the relevant parts from ( dimensional torsion, we have chosenspin the connection constraint in accordingly: terms of the frame as wetions. have chosen This it would be in useful section beyond as their a direct linearized method approximation. of deriving Instead, the we field observe strengths, perhaps that even the dimension-1 torsion JHEP07(2021)032 × 1) , SO(3 . What was un- ]. Acting repeat- 16 terms constructed from et cetera 4 R ]. There the invariant is of the 37 – 20 – . These correspond to our collection of superspace ). These invariants still satisfy a large set of iden- 4.9 ) and ( ). 4.8 2.1 ) correctly describes eleven-dimensional supergravity [ 2.1 ) in superspace. 4 reduced field strength superfields R ∧ To check that such an approach is feasible, we have carried out the analogous calcu- Alternatively, we can proceed to construct higher-derivative deformations of the ac- With the supergeometry torsion constraints so clarified, we could now “start over” with The miraculous-looking series of fortunate events required in this calculation was guar- The calculations are quite cumbersome, but the results are easy to understand: by C invariants of the former that are closed as differential forms (as needed for terms like . This gives a superfield for each physical component field of eleven-dimensional super- R 2 4 metries which are, in turn,R related by the non-manifest∼ ones. We proceed by constructing lation in a five-dimensional version of this scenario in [ tion directly by makingrelated appropriate to combinations the of approach above, themost but tensors direct the derived methods interest herein. used to inderivative us (This action. the are construction is For are the M-theory, different.) these higher-derivativethe are Of invariants curvature needed the 2-form and ones to those containing deformIn terms our the related hybrid two- to superspace, them these by are supersymmetry divided transformations. into groups related by the manifest supersym- the usual superspace Bianchi identity analysis.direct The advantage derivation of doing of this would the beused a super-torsion, to more define super-curvature, higher-derivative and invariantsof directly super-4-form. that the can supergeometry. be These used can to be study deformations or an inconsistent constraint,to but their reduced was field guaranteedclear strengths, to from their construct the reduced outset, invariants Bianchi wasparticular, corresponding identities, the the expression torsion of constraints the needed torsions to in begin terms a of top-down prepotentials, approach to and, the in problem. unconstrained superfields ( anteed to have this outcome, becausesuperfields we had ( previously shown that theedly set of with unconstrained superspace derivatives could, therefore, never generate an equation of motion and solving the formerso-called by the usual methodinvariants in as a expressed top-down in approach, ( givestities a expressing, collection for of example,these superspace identities results derivatives in of explicit one expressions in for the terms reduced of field another. strengths in Solving terms of the manifest. reducing the structure supergroup ofarate eleven-dimensional the supergravity, torsion it constraints is of possibleand that to a supergeometry sep- remainder into a that part is that responsible can be for solved putting off-shell the theory on-shell. Ignoring the latter, We have built up the linearized tensorsunconstrained of prepotentials eleven-dimensional of supergravity directly this from theoryG the with the structure groupgravity. reduced In to addition, theneeded for construction a provides multiplet furnishing superfields a embedding linear representation the with auxiliary four off-shell fields supersymmetries 5 Epilogue JHEP07(2021)032 ) 3 in ]). , 2 4 , | 39 1 4 , (A.1) (A.2) (A.3) 0 (A.4e) (A.4c) (A.4a) (A.4d) (A.4b) R between = a which may v . a,b α structures is . α α α α ) ) ) a D a ˙ C with (anti-)Weyl α σ α ].) A comparison σ , with . ¯ ( β D D . . α 38 a β α ab . ε := ( α η i∂ D ε SL(2 . 4 2 α α αβ − − v ε 2 = − ] = := } ˙ α ˙ α a ) = ¯ v β ¯ D D , , ab Ψ 2 ˙ 2 α α η α 7→ . ( and conjugate D ¯ β D D D [ a a β ˙ D { α α ) v . i∂ αβ α b ¯ D D ε 2 ¯ σ can be extended to 11D, as we intend to D 2 ( − β . α 2 + 2 − α D = , R ) a a ˙ a α o i∂ i∂ σ ˙ ¯ + 2 α ( D 2 2 – 21 – ¯ α α , D − 2 α , ˙ α Ψ D α . ( ˙ D α α D ¯ D . ⇔ 2 α ¯ D ¯ a D D = ¯ ¯ n 2 2 D D -identities in flat space for Minkowski superspace i∂ a . − − − D β D . 4 and find α ] (which, for most practical calculations, agrees with [ ε = = = i with Minkowski metric (components ˙ ] = 4 . ) = α 23 } β α αβ α α ¯ α 4 ε ¯ D Ψ structure (components D ). These are related by the Pauli matrices 2 , D ) β R 2 ) ,D . , α α ,D α − 2 1 D 2 C ¯ D is the gravi-photon. (Such an analysis was possible because enough ) D ¯ D , ¯ D = h . α D [ α = A { b ¯ Here we recall some formulae which all follow from the fundamental D . a D + 2 β . := α η . . α, α SL(2 a ¯ D ¯ + 2 D 4 . α α,β, α where s we have + 2 ¯ D D 2 . ( α D α R α ¯ D D ∧ algebra. D ( α A D ( R = 1 With three definition First we define qualification. Usingexpressed it, by the the Fierz compatibility identity of the Minkowski and N and a compatible spinor indices be interpreted as thevectors components and of hermitian isomorphisms matricessuperspace acting calculations on that Weyl spinors. we use The this operation is special so notation pervasive throughout in the paper without A 4D, N=1 superspaceIn identities this appendix, we rederive the the conventions of reference [ The bosonic subspace is Acknowledgments It is ailluminating pleasure discussions. to This thank work isgrant Jim partially PHY-1820921. supported Gates, by Konstantinos National Science Koutrolikos, Foundation and Warren Siegel for of the 5D hybridof supergeometry the had 11D been and workedimplies out that 5D the previously geometries results in reveals ofdemonstrate [ the that explicitly 5D in the analysis future latter for work. is embedded in the former. This form JHEP07(2021)032 2 → G ) (B.1) (B.2) maps (A.5) (A.6) (A.7) (A.8) (A.9) → Y (A.10) ( γ 3 D 2 :Λ ¯ D s GL(7) is said to be γ . ϕ D ) . β a jmn γ ϕ 4 . 4 a . α ) ( i∂ ) ˙ 356 β γ klmn ˙ ) 4 β e ˙ γ β ˜ γ ω ) − i∂ ˙ ¯ γ 4 β − D 2 4 ikl 2 α ¯ = D α D ( ˙ − ϕ ( ˙ 2 2 γ D i 347 γ ¯ 2 1 α ∂ e = D 24 D ]. ∂ ( ˙ ¯ γ ] ) α D is definite, and we may assume it ¯ b . − αβ ( ˙ − D β , 44 αβ D ) 2 ¯ – ¯ ˙ 4 iε D β D iε = αβ ϕ a ˙ 2 2 257 α D [ ( ε 40 7 αβ e ε h ∂ 2 k ij D − ε b + 4 6 . s α − αβ ∂ k γ reduces the structure group ( − γ ) ˙ + 5 ε 8 ) ˙ ) ¯ k β ) D β Y β − 246 , β ω 4 e 4 + 2 4 D ˙ γ D k  ˙ = 2 γ b 3 2 α + , if necessary). When this is true, we may take α ¯ ∂ ( a , D ¯ ( jk – 22 – a D ∂ ϕ . α ∂ . γ ˙ 4 ∂ ϕ α β ( ˙ ) 167 + 16 β ( ˙ γ 2 ij 8 α β ˙ e α k D δ γ D 1 ˙ D linear algebra [ − iε β ˙ 4 → − iε + 2 β ˙ D . 2 ik α = γ ˙ 2 4 α 2 ¯ = ε α D ϕ ϕ ε − G ( ¯ − ij 7 γ 145 D

) k . The imaginary operator b be a fixed constant 3-form. Define the map e γ = = i∂ ) D = 2 = ϕ ··· 2 ] ) ( 4 1 D + b  V s k , b − ( Y ε a 4 ( ab 4 3 a 123 = 2 = [ 4 e , 1 F e Λ )  a 144 o i∂ β 2 4 = ∈ 4 7→ − ¯ D  D − 2 ϕ ϕ , V ¯ 2 := − D (up to factors). Finally we point out that the dimension-3 operator α D ij ( ) -structure manifold. In this case, ) n 2 D V . ω ( ) ( G s and let is a symmetric bilinear form. When this form is invertible, ab A ( F 7 ] ij is a ab ) R miscellanea to the component Maxwell equation. In this sense the operator 45 F 7→ ϕ is normalized so that b Y ( 2 from 3-forms to symmetric tensors by = ∂ s V V s . A stable 3-form on the tangent spaces of ) G Y Y 7→ ( a 2 to be positive-definite (by flipping it to be [ Then Then stable so that In this appendix, we review some B.1 Linear algebra Set S maps A B maps This operator maps There are other useful identities involvingpart commutators. of First the of product all, there is the symmetric and the anti-symmetric part and and usual ones with four JHEP07(2021)032 (B.6) (B.7) (B.3f) (B.4c) (B.4e) (B.3c) (B.3e) (B.5a) (B.3a) (B.4a) (B.4d) (B.5b) (B.3b) (B.3d) (B.4b) , , or ij decomposes , , , ω 14 . mnp such that jk Y η l 0 . ω ijk k ] Y ω 0 q i ϕ j δ 4 = 2 ψ on k ijkl lmnp − η kl ~ω . ψ − ψ mnp denote the projection [ ] 1 2 = 14 0 ij ijk . k ϕ k η ω ijkl ϕ − 2 and δ i 2 14 0 ) and 3-form kl , ψ η π j ~η ] j [ = = ij jklm η 1 δ = 4! ijk 24 jk ψ ψ ] ij := ω + 2 0 ijk η − i k 7 lm 2 0 7 = 2 ω i π j ) ω = 7 . π i 0 ( -dimensional space of 3-forms on and ψ 0 0 k ω π η 0 1 6 0 k k i i 7 0 0 [ [ ij jklmnpq 35 ij j j π δ  7 0 0 , ϕ , let = i i 24( ϕ 9 η 0 .) These tensors satisfy the algebraic l l ω ω ω 2 4 − = 2 δ ) ijkl ijk − i   − ψ ψ 0 0 , = ω ~ω k k = id 0 0 ( ι ijk ⇔ ⇔ = j j ij 2 = 24 0 0 lm ϕ -form i i , ϕ 0 ∗ kl 0 η 2 14 p k ϕ ϕ 7 0 kl 9 η and 0 j η and l 0 , ijkl 0 i klm -dimensional space of 2-forms on – 23 – ijk ijk and kl k ) ψ ϕ ϕ ϕ ϕ + 2 , ij η ψ , we are defining the vectors jk 21 ] . Similarly, the , 7 ) 2 2 1 1 ψ q ij i g − jkl ijk 24 56 ] ijkl ) π i 2 7 δ Y ] l ( ω ϕ 14 ( k − ij k η jk . (Recall + − = 3 ] δ ] 1 6 η 4 0 0 0 ] ] gδ . For any 0 0 ϕ ⊕ Λ j l l k k 7 , the k k ϕ j j ijkl jk − ∗ , lmnp δ ~η 2 , ψ δ δ √ = π δ ϕ [ 7 ∈ 0 η ι 0 0 ψ 27 0 0 0 ( i j j k i i G ψ , k [ k i [ [ [ k 0 0 = kl kl ω 48 6 1 i i 1 = := δ δ δ δ ijk ⊕ η η 0 12 − ij ij k kl ϕ [ [ → 0 7 ψ   = η j 1 6 = 21 and ψ ψ 0 = 8 = 6 = 6 = = 10 = 6 kl kl 2 i ij ⊕ ) 8 3 8 3 ) l l 0 0 η ij ij ω i l 0 0 = 0 0 Y jkl 1 j k η − + ψ ψ ( 0 k krs 0 ijk i 0 0 0 ( ω 2 j 1 6 1 6 jk ijkl j 0 ijk ϕ k k k k GL(7) = 7 ϕ 0 ϕ i Λ η i δ δ ψ ψ π 0 0 − + acts on 2-forms as 7 ψ ∈ ϕ jklmnpq j j jpq j j l l ijk j j π 35 ijkl  δ δ δ δ η ϕ ψ ϕ 0 0 ijkl ijkl k k i i ijk ijkl i i i i ψ ijk ijk ψ δ δ δ δ ψ ϕ ψ ϕ 4 1 3 2 3 1 4 3 ϕ imn 1 1 1 6 12 42 ϕ     = = = = = := := representations as i i ij ij -dimensional representation. Explicitly, for any 2-form ijk ijk ijk η η η i ω 2 -projections of 2- and 3-forms play an important role in the gravitino analysis. We 7 ω ω ω lmnpqrs 7 14  G 7 1 π Define the Hodge dual That is, for any 27 π π π decomposes as 9 π Note that this implies that there are conversion factors in squares The dual 4-form The define for such projections the vectors fields Under the reduction into Y to the identities JHEP07(2021)032 (B.8) (B.9c) ), it is (B.12) (B.10) (B.9a) (B.9b) acts as (B.11a) (B.11b) B.4e . Since the mnp ϕ )–( 27 on a smooth 7- + ] need not vanish; ϕ p may be expressed B.4c is proportional to ] 14 ijklmnp ] j kl i ijk  ϕ ] T + 1 ϕ ]. 3! n j ijkl 7 m δ mn ψ 43 ] ] [ = . m i ∇ m [ + [ ij l [ l δ 43 ∂ ψ 1 T k l -dimensional representations. ijkl [ -dimensional representations, [ torsion δ = ψ ijkl ijk 2 + 18 14 3 4 27 ψ ϕ , G 3 7 49 3 + k structure τ l mnp ϕ ∗ + T 2 l ϕ . There is also a natural (indefinite) l - and l ] ∧ m lmn + G 7 -, and Y T 2 ijk ϕ jk . Explicitly [ 7 ijkl ϕ τ 3 ψ ϕ ψ In terms of projectors ( , ) ijk ijkl ∧ 1 3 -, + i 1 6 2 [ ϕ and, therefore, a unique compatible torsion- ψ 1 1 l , ψ τ ] + 10 1 ) 1 T + j 36 = , k ∧ ϕ ] k ( j + – 24 – + 3 1 T mnp + δ g l τ ijk k i ] ] mnp = 0 ϕ k [ ψ l i k ϕ n in the ij [ [ i 0 T mnp k δ µ δ τ i ( ϕ T m torsion of the components of the = 4 ijkl k 3 j m 4 3 ψ 1 6 4 3 3 2 ε , T ∇ δ = 1 i As we have reviewed, a stable 3-form 1 2 l − − 4 7 [ [ − − − 49 , dψ δ = = = dϕ 1 2 = = = -dimensional part drops out.) Similarly, the differential of . Its partial contractions satisfy i ) = = 0 ij ) 0 27 14 ) ijk 1 := τ π µ ) 2 τ ( 3 τ ( -forms with ( τ + jkl mnp jkl mnp qrs mnp ( lmn µ 7 G G G π ijk structure. ijk ijkl − G The co-calibration with raised indices matrix called the ϕ 2 action, this torsion decomposes into ψ 1 . The exterior derivative of the calibration 7 ijklqrs π 2 G ∇ ε 4 3 -form with ϕ G × µ − 7 defines a Riemannian metric . It follows that the = ] is a Y G ijk µ ϕ τ m [ Under the ∂ This definition differs from that of our previous papers by a burdensome factor of 18. 10 ijk ϕ instead in terms of four the co-calibration can be expanded as in terms of a 1-formThe and a 1-forms 2-form in corresponding to these expansions are the same because connection is torsion-free, antisymmetrizingexterior all derivative of indices the reduces calibration. the This is left-hand a side generic to 4-form the which may bewhere expanded as respectively. (Evidently the without loss of generality, it can be parameterized as in terms of a Torsion of the manifold free connection to which we willgiven refer by as the Hitchin metric. a(n indefinite) metric onmetric the on the space space of ofwith 3-forms 2-forms resulting respect on from to the second variation of the Hitchin functional Hitchin metric. JHEP07(2021)032 l - k ] . is ) j ] jk 14 δ 7 ) to ω ϕ k k , but l i 6 i ) ( [ (B.16) (B.17) (B.18) (B.13) (B.14) (B.15) k B.3 5 ω T η ) found k ∧ ω = 2.1 4 component j l k ] 3 ϕ 7 jk jk ∧ ϕ l ϕ ], a non-abelian i i 2 [ ϕ k ) 20 , so the symmetric ( 1 so its dimension is η k ik . 7 ∗ k . 2 π k ϕ 1 6 ) S ( ]. considered as a 2-form M j ikl + − ) δ ij ⊕ 16 T superconformal transfor- 3 δ k 7 i τ 2 1 6 k ( ). The significance of it is ( 6 Λ k ) = − M 5 = 1 jkl k ω k projects out the ) ' ( ϕ B.16 j ϕ N s 1 4 ij δ ∗ 1 4 ) = does not survive, and it is easily k s k Λ i 3 acting on ( − l M . ] jk ( j M ⊗ i t M ω jk kl ) is 14 ) 1 ) 1 3 2 ◦ . ) that 2 ϕ π j k Λ l τ 6 1 − 0 i ω h ( [ B.1 ik -dimensional. This suggests that the kl 1 2 ≡ i M ϕ ( B.4d ], local 4D, ) = ) 35 3 2 − ϕ + ω s.t. 19 k 1 6 from vector-valued 1-forms to 3-forms by ( M 7 ) ( := 7 k 1 3 – 25 – t 6 = π τ ◦ k kl Λ ( representation by ( 5 ) and ( ◦ ijk s s ij k ) jk The map ( ) s kl i → 7 ϕ δ ω 4 ϕ M ∗ 1 ( k ( ij B.3d s 3 t Λ δ 1 3 jk ) + ⊗ 0 ϕ with the projectors above. Similarly ) to give alternative forms for it. For example, when 1 , we find τ 2 − . This matrix space ( s k 0 j ∗ 1 i 1 ij :Λ δ -dimensional one since the trace of the symmetric part B.3 s Λ ik ≡ t 4 1 ω l representation since it is ] [ ⊗ 27 7 = jk 1 := 3 k 14 j ϕ Λ i ··· l ij i 1 T [ h k ∈ ) ε η M is in the 14 whereas the image is at most 144 1 π 3 · ( is in the τ 3 28 ], and extensions thereof (gravitino transformations) [ 2 does not come back to itself even up to a coefficient. This suggests we define a τ − 21 ⊕ M = , and 21 k ]. They contain abelian transformations of the M-theory 3-form [ ] ij j Next, define the map ) . δ = 16 ω ω k i ( [ s C Prepotential transformations andIn partial this invariants section, wein collect [ the linearized transformationsgauging thereof of under the 7D prepotentials diffeomorphismsmations ( [ [ Note that this mapthat projects it out defines the the graviton as a metric fluctuation from the linearized calibration. dimensional projection of thechecked anti-symmetric that part of as required. Composing with part of new map for any matrix 49 In this form, itof is also clear from ( This is the entire kernel, as we conclude by counting dimensions. This is the same map, becauseto the the first and second third term termshow together (with that are either a symmetric and form factor evaluate reduces of to 2). the more Going economical the other direction, we can use ( we can use theobtained identities by ( expanding a stableexpression 3-form around the calibration, we obtain the symmetrical T is projected out. Conversely, Linearized Riemannian metric. Note that JHEP07(2021)032 ). (C.3) (C.4) (C.5) C.1 (C.2e) (C.1c) (C.2c) (C.1a) (C.2a) (C.1b) (C.1d) (C.2b) (C.2d) superconformal ]. The prepoten- 21 [ ] = 1 α 20 L N . ] ] j X i u jk ∂ i [  = 0 V i . ∂ α The 4D, α [ i − ]. At the linearized level, this field ] . 2 α α j ∂ τ ¯ Υ Υ  . ¯ i 3 . 19 α L Σ . α − i α i i ∂ α ¯ [ ¯ D ¯ − V D Σ  ∂ . are real. It is almost manifest that these D + α α ij  i − ¯ ¯ D Λ D − F u + 2 2 α ijk α α − ¯ − ¯ Φ ij Υ D ] L D ): α V 1 4 . ij and ] i 2 αi α jk − α ¯ τ Λ D ¯ ¯ − Σ Λ D D – 26 – jkl H i   X D α [ ijk 4 1 − 2 2 Φ i i = ∂ i 1 1 i D Φ ¯ ¯ := [ 2 2 − D D i α τ   ∂ is chiral. Together they define an eleven-dimensional . ( 1 4 1 4 α i i = 3 = = = i α 1 1 W α carrying a 7D vector index, and transforming as 1 2 2 − − 2 α i ij i Υ X is by definition the contraction of the vector index by ijk δU V Σ = = = = 4 = V = Φ i a Under abelian 2-form symmetry only the prepotentials cor- i ath The abelian tensor hierarchy was coupled to the non-abelian α ij G ath ath δ ijk U V H δ δ ijkl ath 7 W δ F . δ E αα ) a σ are chiral and that is real, and E i u := (¯ . α α , and U W , = is chiral, G a ij containing the conformal part of the 4D frame suffers the pregauge transformation U Λ a U where the Pauli matrices (thisdiacritical invertible mark operation directly on is the the index instead same of as the field). Feynman’s slash In calculations, but it with is useful the to note is invariant and chiral. Extended 4D superconformaltransformations transformations. are parameterized by a spinortial parameter superfield Its field strength combinations are invariant under the linearized non-abelianKaluza-Klein gauge gauge transformations field. ( Kaluza-Klein field gauging the 7D diffeomorphismsis in described [ by a real superfield Note that where 2-form. The field strengths of the abelian tensor hierarchy are responding to the components of the 3-form transform. Explicitly, [ Abelian tensor hierarchy. JHEP07(2021)032 ] (see (C.6) (C.8f) (C.7c) (C.7e) (C.8c) (C.8e) (C.8a) (C.7a) (C.7b) (C.7d) (C.8b) (C.8d) 47 , 21 ) l  . ¯ α Ω ¯ L − . α l ¯ D (Ω ] + k α ∂ L ij [ . l α ] α j ) α ϕ ] where it was shown that under D k i Ξ . L . α 2  i i ¯ 3 [ Ω l 16 i ¯ L ) ∂ ¯ i∂ i . ∂ Ω α − + 2 2 ¯ Ω ¯ k D  − ¯ − l D + 2 +  ) ¯ + i Ω (Ω . i α i k 2 in the tensor hierarchy carrying 4D polar- ijkl Ω α . ¯ ¯ Ξ Ω ¯ ijk α D ψ (Ω . α i L α m ϕ i 2 2 1 ) X α − Ξ ¯ D i i + ¯ has a gauge-for-gauge ambiguity corresponding D Ω 1 2 – 27 – k l D − − − 2 ¯ Ω + α − ¯ Ω D (Ω L α i = = = = = 2 ] + α l αi α i i α i ∂ D L ij Ξ X V D ijk  α V Σ α = Ξ (Ω 2 sc ijk sc Φ [ sc α sc D δ ¯ δ α i D D δ δ 2 sc m ijkl D  δ Ψ ¯ 2 ψ D i ijk δ iψ itself, so 1 ¯ 2 1 4 1 4 2 ϕ D ]. This implies that it must transform in a specific way under L i 8 1 8 − − − − 32 ======requiring the compensation. We do this step-wise, first defining is complex and completely unconstrained. These two parameters i i α ], this parameter must also enter the transformation of the gravitino α ij These transformations can be removed by combining them with the G ] for a review of this multiplet). ijk α i H ijkl Ω sc W 46 W F 16 Ψ sc δ E ]. For the field strengths of the non-abelian tensor hierarchy, this gives sc δ sc sc δ sc δ δ δ 21 and a U ], it was observed that the field appears instead of 18 is chiral and part of the superconformal transformations. It was also conjectured there that all . This claim was demonstrated explicitly in [ Ξ α ¯ 3 DL In [ As explained in [ L Partial invariants. superfields Crucially, many of thesepensating transformations fields are [ Stückelberg shifts — the hallmark of com- superconformal transformations, old-minimal supergravity [ the the fields of thetions. tensor hierarchy This suffer is similartable compensating needed superconformal for transforma- the spectrum of component fields to match as summarized in Here are the conformal supersymmetry parameters ofalso the appendix matter C gravitino multiplet of [ [ izations of the 3-form, was also coupling exactly as the chiral conformal compensator of to a shift by a chiral spinor field. superfield: that JHEP07(2021)032 α i Ξ is a that J (C.9c) (C.9e) (C.9a) (C.9b) (C.9d) G and (C.10e) (C.10c) (C.11a) (C.12a) (C.10a) (C.11b) (C.12b) (C.10b) (C.10d) α L The second : the KK-ino, 11 2 . a G by the derivative . U ¯ L Ψ ¯ D ) . ) α . − ¯ α ε parameter. . ¯ ε α ]). i . ] α ) ¯ j τ parameter. Next, l DL D ¯ 50 ¯ D τ Ω – i Ξ + chiral compensator α αjkl l − and 48 α − χ D V . transformation in this way, because α α ] ε i l ¯ [ a Ψ k α ε α α ∂ i ∂ ijkl α D U L = 1 (Ω U ] D ]  i ij ψ . i ( H [ D − α k ∂ l ( ¯ ∂ i 3 2 ¯ Ω N D i Ψ 12 . ) + 2 a 1 − α or the derivative of the compensator iϕ ij 2 ¯ α [ − is the complex combination of scalars k D ) + l 3 ¯ ) U . + D ¯ α i . D Ω  a α i b Ψ [  α i F iϕ − ¯ i ∂ Ψ ¯ 1 4 Ψ ) = ) = ¯ l − . Ω D α ¯ . . αjk Ω α α α b k 2 ¯ H + 3 − ] D ¯ ¯ − L L + ) + D c α j ¯ W l ) 2 2 D i i (Ω ¯ − G Ψ Ω ijkl ¯ + α -dimensional representation of G Ω D D i ijk with respect to the 2 − – 28 – [ . . 2Ω ψ 7 α α α − α i αi ϕ i D ∂ +  i D 2 ¯ ¯ 2 i 2 D D D Ψ Ψ Ω G W α . . ¯ 2 α α α G ( D − + 2 ( ijkl + − − i ¯ D ¯ transforming into D D 1 D D ) covariantized by the derivative of the KK prepotential 2 1 2 2 ψ ijk α α α ij i α ( (   ijk ¯ 1 2 i i X ϕ D L L i i 1 1 2 2 1 1 W W F 2 2 i := := 2 8 1 4 2 − 2.5b ¯ ¯ ( D D and S P = = = = = := := := := := α α a i i . . i α α i α α α ijk α ij D D α ij U b b α i α α i ( ( ijk ijk χ H J J H i b c δ b 3 8 8 c L δ W W F F X X δ δ δ = = S P δ δ under only is just the covariantization of . Alternatively, it is the partial covariantization of the derivative of c a W Additionally, we define combinations The other two combinations cannot be covariantized under the U . Note that it is the only combination transforming under the . 11 i These are essentiallytransforms a nicely slight (as new-minimal rewriting and of virial compensators the [ 4D, there are no fields analogous to transforming appearing in the tensorino V of is an analog ofX this for thecombination scalar of component the of spin-1/2 fieldsthe in abelian the gaugino, and the tensorino. Finally, The first of these is a partial covariantization of the vector component of These combinations are invariant underparameters: the transformations generated by the partially-invariant building blocks. A convenient set is JHEP07(2021)032 = ]. (cf. abci , is a 22 super- , S (C.13f) (C.13e) (C.13c) (C.13a) G (C.13b) (C.13d) 2 2.1 21 ¯ D = 1 1 6 − N = ) R l V a ∂ − l a l a X X ] ( k ] k ∂ ∂ ij l a [ l ij i [ i X l ϕ ] b 3 k ϕ H ] 3 ∂ . . δ − a i . α ij i [ ) − ¯ i l D X k α l m h.c., the 35 1-forms and the 35 0-form field , ) ϕ δ b b ijk 2 3 X H H . b + ¯ ¯ a α D F ). The process that gives rise to this result D m [ − α . ] ∂ β α ( i + ¯ – 29 – − ∂ 2.6 D jkl ijβ ijkl 2 α d i ) and, through those, the torsion and curvature ψ G . ijk ψ αijk is chiral, the chiral superfield D D X β ( 1 2 ϕ + ¯ χ ) 2.6 α ] α 1 2 ]. Continuing to make other dimension-1 invariants in 1 2 2 α − ) + L ¯ D D jkl ab 2 − + ) 32 ijk σ ¯ , i Φ ¯ ijk ¯ + + D b ij R F b ) 1 4 H 2 F 31 + α β ] ] = ( ) and ( − ] . . − α α D D αijk c ( W . ¯ ¯ Taking combinations of superspace derivatives of these partial α R jkl D D h 2.3 χ α ( = , abij , . ¯ ( Φ α D α α h h ¯ D G α D i abcd [ D D abcd i ε 2 1 2 h  [ [ ∂ i i 4 1 2 2 1 2 1 − − = = := 2 := := := := = 3 a i ijk ˜ ijkl G a abc i abcd αβij G G G G G )). Taking the bosonic curl of these components gives an alternative (equivalent) , the 21 2-forms i d ˜ 2.2 G Instead of doing this, we explicitly construct the gauge 3-form in section In the modification of old-minimal supergravity employed by eleven-dimensional su- abcd The manifest supersymmetryfermionic transformations directions. act in Inderivatives this superspace on section, the by we gravitino translationsis present superfields in not the ( the algorithmic: resultgauge of the transformation acting and desired with spin result the connection fermionic is terms, and obtained in only a up form to that is unknown not field-dependent easily parsed dimensional connections ( invariants. The off-shell spectrumthe is connections completed with by supersymmetry auxiliary transformations. fields that arise byD acting on Gravitino descendants  strengths. eq. ( derivation of the dimension-1 4-form invariants. There, we also define the other eleven- It can be checked that the new superfields represent the dual of the seven 3-forms in a Taylor expansion and the four-dimensional curvature scalarpergravity, at the higher prepotential order of [ theof chiral the compensator real is components real. of4-form The the curl effect complex of of a auxiliary this gauge field isthis 3-form becomes that way, [ we (the one find 4D (among Hodge other dual things) of) the the following combinations invariants, we can constructconsideration superfields (covariant at that non-linear aresymmetry order). invariant parameter at For example, thedimension-1 since linearized invariant. the level In 4D, fact, under minimal it supergravity is that the contains linearized the complex chiral auxiliary scalar scalar curvature as invariant the of leading old- component 4-form field strength. JHEP07(2021)032 (D.5) (D.3c) (D.4c) (D.1c) (D.2c) (D.3a) (D.4a) (D.1a) (D.2a) (D.3b) (D.2b) (D.4b) (D.1b) i b k  . Γ j α i ¯ λ on each of the four α + β k D j ¯ D d λ . . − . . β β α δ j α ¯ . D β α αj and λ h . ¯ iδ α Ψ jk α ¯ . D i α D δ  D ϕ + 2  d 1 2 1 6 . δ β m + ) . j α βγ i + γ b t ¯ i H t ε αj j a β α ) + d i β 6 δ i ( ¯ α Ψ b i Γ a i . δ α i 3 jklm − 2 klm Γ 6 + d i 2 5 ¯ b i D ψ j i i i βγ + 12 2 N R G − + i + i − . . ( Z β α β j a ω ∂ − j j δ − i j α βγ N a + β α Γ t a h a = = = – 30 – . . i i β α 2 X ω ∂ iδ jkl kl ) i β β i iδ β i i -dimensional one (Y). − ) 2 F i + ω = = = j P 7  ψ ψ j i i ψ . ) b − β β β β kl kl i + 2 γ β j a a a j ( γβ ¯ − a . ω D D j ijkl α a ω ϕ ψ ψ b j δ D ψ G . h S ψ β β a kl ( γ ¯ Γ 3 1 6 1 j ( + 2 ¯ − ω D D δ + 2 14 ) ϕ j D j b γ = := := iπ ε + 2 N N j 2 i β + 2 i i j ( a α N − ∂ ∂ j b δ N N i 3 N N 2 = = = a a j − ∂ ∂ ) βj βj β i i i = = = ψ ψ ψ . j β β ) γ βj βj ( ¯ β -singlet index (X) or a D a a D a 2 D ψ ψ ψ . G β β γ ( ¯ D D D The fermionic derivatives of the gravitino with polarization XX may be decomposed as We will not need this explicitparameters form, but agree it is across important equations thatcorrect the (and decomposition normalizations is of of the an the shared descendants). important technical The spin aid connections in are determining given the in terms of the In this derivation, the field-dependent Wess-Zumino gauge transformations are Finally, the YY part gives The XY gravitino is messier: The YX descendants are superspace crossword puzzle asgravitino polarizations: irreducible the projections 1-form of may index have may a lie along 4D (X) or 7D (Y), and the spinor into 4-form terms and combinations of auxiliary fields. We present the solution to this JHEP07(2021)032 ]. = 2 DOI (D.7) , (D.6f) (D.6c) (D.6e) (D.6a) (D.6b) (D.6d) , N Nucl. , i ) . α ¯  . β i ( ] b ¯ (1981) 275 a D Class. Quant. Grav. [ β α e , . 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