G(3)-supergeometry and a supersymmetric extension of the Hilbert–Cartan equation

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G(3)-supergeometry and a supersymmetric extension of the Hilbert–Cartan equation

Boris Kruglikov^a,∗, Andrea Santi^b, Dennis The^a

aDepartmentofMathematicsandStatistics,UiTTheArcticUniversityofNorway, Tromsø90-37,Norway

bDepartmentofMathematics“TullioLevi-Civita”,UniversityofPadova,35121 Padova,Italy

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received19September2019 Receivedinrevisedform31July 2020

Accepted21September2020 Availableonline23October2020 CommunicatedbyRoman Bezrukavnikov

Keywords:

SimpleLiesuperalgebra Vectordistribution Spencercohomology Tanakaprolongation ExceptionalLiealgebraG2 Submaximalsymmetry

WerealizethesimpleLiesuperalgebraG(3) assupersymme- tryofvariousgeometricstructures,mostimportantlysuper- versionsoftheHilbert–Cartanequation(SHC)andCartan’s involutivePDEsystemthatexhibitG(2) symmetry.Wepro- vide the symmetries explicitly and compute, via the first Spencer cohomology groups, the Tanaka–Weisfeiler prolon- gationofthenegativelygradedLiesuperalgebrasassociated with two particular choices of parabolics. We discuss non- holonomicsuperdistributionswithgrowthvector(2|4,1|2,2|0) deforming the flat model SHC, and prove that the second Spencer cohomology group gives a binary quadratic form, thereby indicating a “square-root” of Cartan’s classical bi- nary quartic invariant for generic rank 2 distributions ina 5-dimensional space. Finally,we obtain super-extensions of Cartan’sclassicalsubmaximallysymmetricmodels,compute theirsymmetriesandobserveasupersymmetrydimensiongap phenomenon.

©2020TheAuthor(s).PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

* Correspondingauthor.

E-mailaddresses:[email protected](B. Kruglikov),[email protected](A. Santi), [email protected](D. The).

https://doi.org/10.1016/j.aim.2020.107420

0001-8708/©2020TheAuthor(s). PublishedbyElsevierInc. ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

Contents

1. Introductionandthemainresults . . . . 2

1.1. History:realizationsofG(2) assymmetry . . . . 3

1.2. Newresults:realizationsofG(3) assupersymmetry . . . . 6

1.3. SpencercohomologyofG(3) andcurvedsupergeometries . . . . 8

1.4. Structureofthepaperandfuturedirections . . . 10

2. AlgebraicaspectsandparabolicsubalgebrasofG(3) . . . 11

2.1. RootsystemsandDynkindiagramsofG(3) . . . 11

2.2. ParabolicsubalgebrasandamapofG(3)-supergeometries . . . 12

2.3. SHCandcontactgradings . . . 13

2.4. StructuresassociatedwiththecontactgradingofG(3) . . . 17

2.5. LagrangiansubspacesalongV . . . 25

3. ComputationoftheSpencercohomology . . . 27

3.1. Hochschild–Serrespectralsequence . . . 27

3.2. Anexactsequenceincohomology . . . 28

3.3. Spencercohomologyforp^IV₁ ⊂G(3) . . . 30

3.4. Spencercohomologyforp^IV₂ ⊂G(3) . . . 37

4. G(3) asthesupersymmetryofdifferentialequations . . . 48

4.1. Superjet-spacesandequationsupermanifolds . . . 48

4.2. TheG(3)-contactsuper-PDE . . . 51

4.3. SymmetriesoftheG(3)-contactsuper-PDE . . . 54

4.4. CauchycharacteristicreductionandthesuperHilbert–Cartanequation . . . 59

5. CurvedsupergeometriesfromG(3)-symmetricmodels . . . 61

5.1. Rigidityofthesymbol . . . 61

5.2. Rank(2|4) distributionsina(5|6)-dimensionalsuperspace . . . 66

5.3. IntegralsubmanifoldsoftheSHCdistribution . . . 71

5.4. Super-deformationandsubmaximallysupersymmetricmodels . . . 74

Acknowledgments . . . 84

Appendix A. ParabolicG(3)-supergeometriesandequivalences . . . 85

Appendix B. VanishingofthegroupsH^d,2(m¯1,g)¯0ford3 . . . 87

Appendix C. InternalsymmetriesoftheSHCequation . . . 93

References . . . 96

1. Introductionandthemainresults

IntheearlydaysofLietheory,W.KillingfoundallfiveexceptionalsimpleLiealgebras, yetwithoutconcretegeometricrealizations.Thesimplestofthese,the14-dimensionalLie algebraG(2), discoveredin1887,wasrealizedas thesymmetryalgebraoftwodifferent Klein geometries in 1893 by E. Cartan and F. Engel, in two successive papers in the sameissueofComptesRendus[2,11].

Supersymmetry was brought to lifeinthe context ofquantum fieldtheory and it is based on the theory of Lie superalgebras. The firstsimple (real) Lie superalgebra was computed by J.Wessand B.Zumino in1974 as thesymmetry superalgebraof AdS^5|8 [38],asuperizationoftheantideSitterspaceplayingaspecialroleingeneralrelativity.

ThiswasoneoftheclassicalLiesuperalgebrassu(2,2|1)=osp(4,4|2;R)∩sl(4|1;C).The classificationof simplecomplexLiesuperalgebraswas achievedbyV. Kacin1977[22], and thesimplest exceptionalone,inthelistof Liesuperalgebras withareductiveeven part, is G(3) of dimension (17|14). This Lie superalgebra is traditionally described by introducingthebracketsonitsevenandoddpartsandnotasthesymmetrysuperalgebra

ofsomesimplealgebraicorgeometricstructure.(Arguably,onereasonisthatthesmallest non-trivialrepresentationofG(3) istheadjointrepresentation[32].)

ThegoalofthispaperistorealizeG(3) asthesymmetryofa(Klein)supergeometry, andthenstudy theinvariantsanddeformationsofthissupergeometry.Weremarkthat G(2) isasubalgebraofG(3),andweextensivelymakeuseofthisimportantfact.Indeed, wewillestablishsuper-analogsofthecelebrateddifferentialequationsassociatedtoG(2).

For simplicity, inthis paper we restrict to Lie algebras and superalgebras over C, the straightforwardversionoverRcorrespondsto thesplit(normal)form.

Letusbrieflyrecalltheclassicalresultsbeforewe describethesuper-models.

1.1. History:realizations of G(2) assymmetry

InCartan’s realization, G(2) is the symmetry of arank 2distribution ina5-space.

Thisdistributionisassociatedto theunderdeterminedordinarydifferentialequation

z =¹₂(u)², (1.1)

forthefunctionsu=u(x),z=z(x).Equivalently,itisthe5-manifoldΣ={z₁= ¹₂(u₂)²} in the mixed jet-space J^2,1(C,C²) = C⁶(x,u,u₁,u₂,z,z₁) equipped with the Pfaffian system

du−u1dx, du1−u2dx, dz−¹₂u²₂dx, (1.2) i.e.,thepullbacktoΣ oftheCartansysteminJ^2,1(C,C²).Symmetriesof(1.2) areoften referredto asinternal symmetriesof(1.1).Inmodernterms,thisPfaffiansystemisthe Kleingeometry encodedasarank2distributionontheflag varietyG(2)/P1, whereP1

istheparabolicsubgroupwith markedDynkindiagram × ofG(2).¹

In Engel’s realization, G(2) is the symmetry of a contact distribution C on a 5- dimensionalspaceMequippedwithafieldofrationalnormalcurvesofdegree3 (twisted cubics)V⊂P(C),inmodernterms,aparaconformalorGL(2)-structure.Werecallthat a twisted cubic is given by λ → [λ³ : λ² : λ : 1] in some (projective) frame of the distribution,moduloprojectivereparametrizationsofλ.

In 1910, E. Cartan [3] realized G(2) as the contact (or external) symmetry of an overdeterminedsystemofdifferentialequations

u_xx=¹₃λ³, u_xy= ¹₂λ², u_yy=λ. (1.3) Uponeliminationoftheparameterλ,oneobtainsaninvolutivePDEsystemthatwecall theG(2)-contactPDEsystem,namely

1 WewillabusethenotationG(2) fortheLiegroupandthecorrespondingLiealgebra,andsimilarlyfor G(3) inthesuper-settinglateron.Forparabolics,PwilldenoteaLie(super-)groupwithLie(super-)algebra p,sometimeswithextraornamentation.

uxx= ¹₃u³_yy, uxy= ¹₂u²_yy. (1.4) It shouldbe mentioned thatD. Hilbertin1912 [20] showed that(1.1) (resp. (1.3)) do notallowintegralcurves(resp.surfaces)tobeexpressedinclosedform,thatis,without quadratures, aphenomenonexplainedbyE. Cartaninamoregeneralcontext in1914.

Henceforth (1.1) is called the Hilbert–Cartan equation. (Variations like the factor ¹₂ in equation (1.1) areinessential. Laterwewill alsohavesimilar differencesinnotation for thesuper-versions.)

SincethenmanymethodstocomputesymmetriesoftheHCequation(1.1) havebeen developed, in particular relating internal to generalized symmetries. For us the most importantapproachwill bethatofN. Tanaka[34] and B.Weisfeiler[37].This givesan upper bound on the symmetry algebra from the algebraic prolongation of the symbol algebra(alsoknownastheCarnotalgebrainoptimalcontrol).Werecallthatassociated to anydistributionDonamanifold,thereistheweakderivedflag

D¹=D⊂D²⊂ · · · ⊂D^k ⊂ · · · , D^k+1= [D,D^k]. (1.5) A distributionD is called regular if theranks ofD^k|x are constantinxfor all k > 0, thatis,theD^k aredistributionsforallk >0.Settingg_−i(x)=Dⁱ|x/Dⁱ⁻¹|x,thesymbol algebra at x ∈ M is mx =

k<0gk(x). If we assume that D is bracket-generating of depthμ(D^μ isthefulltangentbundleand D^μ−1D^μ)and stronglyregularoftypem (allmxareisomorphictoafixednegatively-gradedLiealgebram),thengk= 0 precisely for all k <−μ. Themaximal prolongationof m=

−μk<0gk is then definedas the unique(possibly infinite-dimensional)Z-gradedLiealgebra

pr(m) = +∞

k=−μ

gk

thatextendsm,istransitive(forallk0,ifX ∈gkisanelementsuchthat[X,g₋1]= 0, then X = 0)and is maximal with these properties.In particular g0 =dergr(m) is the Lie algebraofgrade-preserving derivationsofm.

If M = (M_o,AM) is a supermanifold, with underlying topological space M_o and sheaf of superfunctions AM, then a distributionis definedas a(graded) AM-subsheaf D of the tangent sheaf TM = Der(AM) of M that is locally a direct factor, see e.g.

[36, §4.7].(We will often use “superdistribution”as ashorteningof “distributionon a supermanifold”.)AnysuchDinduces avectorbundleD|Mo =∪x∈MoD|xonM_o(where D|x is the evaluation of D at x ∈ M_o), but we note that this bundle does not fully determine D. Theweakderivedflag associated toD isdefinedas in(1.5).Similarly,D is called regular if D^k are superdistributions for all k > 0, and bracket-generating of depthμifD^μ=TM.Hencewe obtainvector bundlesD^k|Mo =∪x∈MoD^k|x onMo and asymbolmxat anyx∈Mo thatisa(finite-dimensional)Liesuperalgebra.

WealsoconsiderthestalkD^kxofD^k atx∈Moasamoduleoverthelocalring(AM)x

andsetgr(TxM)=

k>0gr(TxM)₋k,wheregr(TxM)₋k =D^kx/D^kx⁻¹.Thisisnaturally

× ×

× ×

Fig. 1.G(2)-twistor correspondence as a relation between PDEs and ODEs.

agraded Liesuperalgebrafreeover (AM)x.Sincesupervectorfieldsare notdetermined bytheir valuesat thepoints ofMo, the correctgeneralization ofthe conceptof strong regularityisgiven intermsofthestalks.

Definition1.1.LetD bearegulardistributiononasupermanifoldM = (Mo,AM) that isbracket-generatingofdepthμ.ThenDisstronglyregularifthereexistsanegatively- graded Lie superalgebra m=

0<kμm_−k such that gr(TxM) ∼= (AM)_x⊗m at any x∈M_o,as gradedLiesuperalgebrasover(AM)_x.

Concretely, a strongly regular superdistribution admits alocal basis of supervector fields adapted to the weakderived flag and whose brackets are given by the structure constantsofm,aftertheappropriatequotientshavebeentaken.Thesuperdistributions consideredinthispaperwill alltacitlybe assumedstronglyregular.

Theproof oftheexistenceanduniquenessofpr(m) givenin[34] extendsverbatimto theLiesuperalgebracaseandthemildgeneralizationpr(m,g0) thatincludesareduction g0 ⊂ dergr(m) of the structure Lie superalgebra is straightforward. In the context of gradedLiesuperalgebraswewill callittheTanaka–Weisfeiler prolongation.

Letuscomebacktotheclassicalcase.ThethreerealizationsofG(2) discussedabove areconvenientlyrelatedbythediagramofFig.1.Ontheleftisthe5-dimensionalspace G(2)/P1equippedwitharank2distributionhavinggrowthvector(2,1,2).(Hereandin thefollowing,thegrowthvector isthelistofdimensionsofthegradedcomponentsofthe symbolalgebra. Anotherconventionisto list dimensionsofthefilteredcomponents, in whichcaseitiscalled a(2,3,5) distribution.)Ontherightisthe5-dimensionalcontact manifoldG(2)/P₂ withareductionofthestructuregrouptoGL(2)⊂CSp(4).

Finally,onthe topisthe6-dimensional spaceG(2)/P₁₂equipped with arank2dis- tributionDhaving growthvector(2,1,1,1,1) andthisgeometryisindeedderivedfrom thePDEmodel(1.4).Namely,asa6-spaceE⊂J²(C²,C),itinherits:

(i) theCartandistributionD²ofrank3,whichcoincideswiththefirstderivedofD.The associated Cauchy characteristicspace Ch(D²) consists of(internal)symmetries of D² thatlieinsideD² itselfandisgenerated by

C₁=D_x−λD_y, where

D_x=∂_x+u_x∂_u+λ³

3 ∂_ux+λ²

2 ∂_uy , D_y=∂_y+u_y∂_u+λ²

2∂_ux+λ∂_uy (1.6)

aretruncatedtotalderivatives.

(ii) 1-dimensionalfibres forE→J¹(C²,C),with verticalbundle spanned byC2 =∂λ. ThelattergeneratesCh(D⁴).

Therank2distributionD canbe recoveredfrom (i)-(ii)as thespanofC1 andC2. The left arrow inFig. 1is the quotient E →Σ ∼=E/C1 of E by C1 and Σ inherits the rank2 distributionD²/C1. Therightarrow isthequotient E→M ∼=E/C2 byC2

and M inherits the contact distribution C = D⁴/C2 equipped with the twisted cubic obtainedas (theZariski-closureof)thepush-forwardofC1 throughtheprojection.

Both quotients exist in generalonly locally and only forG(2)-invariant (flat) struc- tures. In fact,the second arrow doesnot exist for generalcurved parabolic geometries (the field of curves determined by the push-forward of a vector field in Ch(D²) may notbe arationalnormalcurve, thereforeitdoesnotgiveriseto therequiredreduction GL(2) ⊂CSp(4)), while thefirstquotient exists universally,as the geometriesare just determined by thetypeof theassociated distributions.The lattergivesabijection be- tweeninvolutivePDEsystemsofthesecondorderforu=u(x,y) and(byatheoremof E. Goursat)Mongeequations, i.e.,underdeterminedODEsofthetype

z =f(x, u, u, u, z),

for u=u(x),z =z(x).Cartan [3] showedthat (1.4) hasmaximal (contact)symmetry dimensionamongstallsecondorderinvolutivePDEsystemsand(1.1) maximal(internal) symmetry dimensionamongstallMongeequations.

GeneralizationsoftheG(2)/P12→G(2)/P1 fibrationtotheotherexceptionalsimple Lie groupswerefirst investigatedby K.Yamaguchi [39].Recently, workof D.The[35]

gavethefirstexplicitgeometricgeneralizationsoftheCartan–EngelG(2)-modelstothe exceptionals.TheideaissimplyillustratedintheG(2)-case:thetwistedcubicV⊂P(C) from Engel’sG(2)/P₂-pictureisaLegendrianprojectivevariety,i.e.,itsosculationsgive a familyV ofLagrangian subspaces,and thefibres of E→M are modelledon V. The Lagrange–Grassmann bundleM=LG(C)→M(whosefibresareLagrangiansubspaces of the contact distribution C on M) is locally isomorphic to J²(C²,C), so E ⊂ M corresponds toaPDE.Thedifferenceofperspectivein[35] isto viewthePDE(1.4) as anequivalent descriptionofG(2)/P₂-geometry, usingthefactthatV providesthesame reductionto GL(2)⊂CSp(4) asVdoes.

Wenow turntoourresultsinthesuper-setting.

1.2. Newresults: realizationsof G(3)as supersymmetry

Wewill demonstratethatasuper-extensionoftheHCequation (1.1) isgiven bythe following system of partial differential equations that we call SHC (for super Hilbert- Cartan):

zx= ¹₂u²_xx+uxνuxτ, zν =uxxuxν, zτ =uxxuxτ, uντ =−uxx, (1.7)

where u = u(x,ν,τ) and z = z(x,ν,τ). It is a submanifold Σ of codimension (2|2) inthemixedjet-superspaceJ^2,1(C^1|2,C^2|0), equipped withthe pullbackof theCartan system.Unlike(1.1),whichhasgeneralsolutiondependingononearbitraryfunctionof onevariable,thespaceofsolutionsof(1.7) dependsonlyonfivearbitraryconstants,see Section5.3.

From the internalperspective, this corresponds to asuperdistribution of rank(2|4) ina(5|6)-dimensionalsupermanifoldwithgrowthvector(2|4,1|2,2|0),andtheequation canbedirectlyproducedbyintegratingthegradednilpotentLiesuperalgebraassociated totheparabolicp^IV₂ ⊂G(3) (seeSections2.2and2.3fornotations)viathesuper-version oftheBaker–Campbell–Hausdorffformulaandthenlocallyrectifyingthecorresponding Pfaffian system on G(3)/P₂^IV. This is how we obtained (1.7) initially; however inthis paperwe presentamethodclosertothatused in[35].Wethen provethattheinternal symmetrysuperalgebraof(1.7) isG(3) inTheorem 4.13.

Specifically, we begin with acontact grading. From Table 9, we note that there are two contact gradings on G(3): theone associated to p^I₁ which corresponds to a purely odddistribution(i.e.,whichgivesrisetoa“consistent”Z-gradinginKac’sterminology), andtheotherassociatedtop^IV₁ correspondingtoadistributionofmixedparity(i.e.,an

“inconsistent”Z-grading).Bothhavepurelyevennormalbundleandwewillexplorethe secondoption.

The homogeneous superspace G(3)/P₁^IV has dimension (5|4) and it comes with a contact superdistribution C of rank (4|4). We first determine an invariant cone field init, characterizing thereduction of the structure groupto COSp(3|2) ⊂CSpO(4|4);

note theorder of letters.In other words,at any fixed topological point x∈G(3)/P₁^IV, the projectivization of C|x contains a distinguished subvariety V|x of dimension (1|2).

ThissupervarietyisisomorphictotheuniqueirreducibleflagmanifoldofthesimpleLie supergroupOSp(3|2),namely

V|x∼=OSp(3|2)/P₁^II,

wherep^II₁ istheparabolicsubalgebra × .Wecallitthe(1|2)-twistedcubic,because itsunderlyingclassical manifoldis arationalnormal curveof degree3and it issuper- deformedin2odddimensions.

Lagrangiansubspacesobtainedasosculationsofthefieldof(1|2)-twisted cubicsV⊂ P(C) determineasubmanifoldEinthesuperspaceof2-jetsJ²(C^2|2,C^1|0) andweobtain thefollowingextensionof(1.4),whichwecall G(3)-contactsuper-PDE system:

uxx= ¹₃u³_yy+ 2uyyuyνuyτ, uxy= ¹₂u²_yy+uyνuyτ,

u_xν=u_yyu_yν, u_xτ =u_yyu_yτ, u_ντ =−u_yy, (1.8) where u=u(x,y,ν,τ). SeeTheorem 4.7. Furthermore, we showthatthe contact sym- metryalgebraofthis super-PDEsystemisexactlyG(3), seeTheorem4.10.

× ×

× ×

Fig. 2.G(3)-twistor correspondence considered in this paper.

ThemodelsweobtainarerelatedbythediagramofFig.2.Ontheleftweseethe(5|4)- dimensionalsuperspaceG(3)/P₁^IVwithafieldof(1|2)-twistedcubicsintheprojectivized contact distribution, that is, with a reduction of the structure group CSpO(4|4) to COSp(3|2).On the right there is the(5|6)-dimensionalsuperspace G(3)/P₂^IV endowed withasuperdistributionofgrowth(2|4,1|2,2|0),whichcorrespondstotheSHCequation (1.7).

Finally,onthetopweseethe(6|6)-dimensionalsuperspaceG(3)/P₁₂^IV equippedwith thesuperdistributionD ofgrowth(2|2,1|2,1|2,1|0,1|0).Thisisderivedfromthesuper- PDE model (1.8). As in the G(2)-case, the superdistribution D is not the Cartan distribution of the super-PDE modelconsidered as a submanifold E of J²(C^2|2,C^1|0).

Infact,theCartandistributionisthefirst derivedD²ofD andhasrank(3|4);wealso note thatD³ hasrank(4|6),whileD⁴ hasrank(5|6).

ThesuperdistributionD²hasanon-trivialCauchycharacteristicspaceCh(D²),which we compute inSection4.4 (we remark thatD² is simply denotedby the symbol H in thatsection).Itisspanned bytheevensupervectorfield

C=D_x−λD_y−θD_ν−φD_τ, (1.9)

where Dx,Dy,Dτ,Dν are truncated total derivatives (explicit formulae are given in Section 4.4) and λ = u_yy, θ = −u_yτ, φ = u_yν. On the other hand D⁴ has a (1|2)- dimensionalCauchycharacteristicspaceCh(D⁴)=∂_λ|∂_θ,∂_φ.

The left arrow is the quotient by Ch(D⁴) and the contact superdistribution C on G(3)/P₁^IVat thebottomleftisD⁴/Ch(D⁴).TherightarrowisthequotientbyCh(D²) andthesuperdistributionofrank(2|4) onG(3)/P₂^IVatthebottomrightisD²/Ch(D²).

Note thattheleftandright arrowsareswappedw.r.t.theclassicalpicturepresentedin Fig.1;thisisduetothereversalofthetriplearrowintheDynkindiagramofG(3) w.r.t.

thatofG(2).

The local quotient given by theleft arrow exists ingeneral onlyfor special (in par- ticularflat)structures.However thelocalquotientto therighthasnoobstructions, and there isanequivalenceofcategoriesforthecorresponding geometries,seeAppendixA.

1.3. Spencercohomology ofG(3)and curvedsupergeometries

Our proof that g = G(3) is the internal symmetry superalgebra of (1.7), resp. the contactsymmetrysuperalgebraof(1.8),isbasedontheexplicit determinationofallthe supersymmetriesand onthecomputationof theTanaka–Weisfeilerprolongationof the

symbol algebra associated to the relevant distribution, resp. with a furtherorthosym- plectic reduction cosp(3|2) ⊂ dergr(m). We emphasize that in [35], the proof that the givenPDEshavesymmetryrealizedbytheexceptionalLiealgebras wasgivenindepen- dentlyfrom theexplicitsymmetrycomputation.Thatresultwasbasedonthetheoryof parabolicgeometries, whichisnotavailableinthesuper-setting.

Theorem 3.16, resp. Theorem 3.9, computes the first Spencer cohomology group H¹(m,g) (see(3.4) foritsdecompositionintohomogeneouscomponents)ofthenegatively gradedLiesuperalgebram⊂gcorrespondingtotheZ-gradinginducedbyp^IV₂ ,resp.p^IV₁ . Vanishinginnonnegative,resp.positive,degreesamountsexactlytopr(m)∼=G(3),resp.

pr(m,g₀)∼=G(3), thatis, G(3) is themaximal prolongation.

In addition, we compute the second Spencer cohomology groups H²(m,g) that are classicallyidentifiedwith thespacesof(fundamental) curvaturesor structurefunctions [18,19,27]. Traditionally, this hastwo motivations. First, Spencer cohomology consists ofcompatibilityconstraintsof theLieequation onsymmetry andcanbe expressedvia curvatures[25]. Second,inparabolic geometryH²(m,g) containsthecompleteobstruc- tionsto flatnessofthe(regular,normal)Cartanconnection,i.e.,theso-calledharmonic curvatures.Inthis paperwe takeinsteadthedeformationapproach,inwhichH²(m,g) classifiesfiltereddeformationsofgradedsubalgebras[5];note thatthesymmetrybreak- ing mechanism of [26] is also based on this deformation idea and we apply it in the contextofsupergeometry.

TherelevantLiealgebracohomologiesarecomputedintheclassicalcaseviaKostant’s versionoftheBott–Borel–Weiltheorem[23],aresultwhichdoesnotholdinthegeneral super-case.Cohomologygroupshavebeenknownforsomeirreducible(i.e., depthμ= 1) supergeometries [27] and some distinguished (i.e., corresponding to aDynkin diagram with justoneoddroot)Borel subalgebras[7], butnotfor theparabolic subalgebras of depthμ= 2,3 thatweconsiderinthispaper.(NotethatG(3) doesnothave|1|-gradings.) Our proofs use various techniques, such as the Hochschild–Serre spectral sequence for p^IV₁ and a combination of different exact sequences with the representation theory of osp(1|2) forp^IV₂ .

In the latter case, it is intriguing that H^2,2(m,g) ∼= S²C², which yields a “square root” of Cartan’sclassical binary quartic invariant for(2,3,5)-distributions, see Theo- rem3.20.A similarphenomenonhasbeen observedinthecontextof supergravity [12].

Forthesecond cohomologygroupofthegraded Liesuperalgebra associatedto p^IV₁ ,see Theorem3.9.

Notethatcomparing(super)dimensions(p|q) hasdifferentmeaningsintheliterature:

sometimesthemaximal dimensionis understoodinthe evensense (maxp),sometimes inthe odd sense (max q) or inthe total sense (max p+q). In this paper, we use the strongernotion of partial order: (p|q)(p|q) iff p pand q q. Inthis sense, we proveinTheorem4.9that(17|14) isthemaximalsupersymmetrydimensionfor(locally transitive)G(3)-contactsupergeometries.

Finally, we discuss curved geometries modelled on the homogeneous superspace G(3)/P₂^IV. We consider distributions in (5|6)-dimensional superspaces with growth

(2|4,1|2,2|0) and prove that the SHC symbol is rigid, i.e., given this growth (plus somemildnon-degeneracyconditions),thegradedLiesuperalgebrastructure isunique.

See Theorem 5.1 and Corollary 5.3. Contrary to the case of rank2 distributions in 5- dimensionalspaces,thegenericrank(2|4) distributionsin(5|6)-superspaceshavedepth μ = 2, so the distributionswith the indicated growth are not generic. Weaddress the restrictionsthisputs ontheirevenpart.

We investigate integral submanifolds of general superdistributions with the growth vector(2|4,1|2,2|0) inTheorem5.10andnoticeadifferenceitmakeswiththeevencase.

Then we observe a supersymmetry gap phenomenon in Theorem 5.12: The maxi- mal supersymmetry dimension of (locally transitive) distributions with growth vector (2|4,1|2,2|0) of SHC type is (17|14), and among all such distributions any symmetry superalgebra differentfromG(3) hasdimensionatmost(10|8).

Finally,weshow inTheorem 5.13thatthefollowingdeformationoftheSHC z_x=f(u_xx) +u_xνu_xτ, z_ν =f(u_xx)u_xν, z_τ =f(u_xx)u_xτ, u_ντ =−f(u_xx), givesarealizationoftheabovedimensionboundwheneverthefunctionf ofone(even) variable is f(s) =

s^kds with k = −2,−²₃,−¹₃,0,1. These non-flat models can be considered as super-extensions of Cartan’s classical submaximally symmetric G(2)/P1

structures.

1.4. Structureof thepaperandfuture directions

In Section2 we recall the basics of G(3), its parabolic subalgebras and Z-gradings.

Fig. 4 gives all the 19 generalized flag varieties of G(3) and twistor correspondences, further discussed in Appendix A. (The diagram is complete in the flatcase, while for curved geometries some arrows may disappear.) Associated to the G(3)-contact case, i.e., to the flag superspace G(3)/P₁^IV, we compute the supervariety V, its osculations, and super-symmetricforms onanaturallyassociatedJordan superalgebra,leadingtoa collectionV ofLagrangiansubspacesalongV.

Section 3 is devoted to cohomology – this is an important ingredient in the proof thatG(3) isthe symmetrysuperalgebra ofthe two maindifferential equations thatwe will derive inSection 4. Some technicalcohomological computations are postponedto theAppendixB.Wethen explaininSection4therelationbetweenthetwodifferential equations and give theexplicit expression of supersymmetries:we encodethem by the generatingfunctionofthecontactvectorfieldinthecaseofthesuper-PDE(1.8) andwe delegate theformulaeto AppendixCinthecaseoftheSHC equation(1.7).

Finally in Section 5 we discuss curved geometries of type G(3)/P₂^IV: their symbol, genericity and in which respects they differ from SHC. The submaximally symmetric models arederivedat the end of thissection, and their supersymmetriesareexplicitly given.

Throughoutthepaper,wewillworkwithLiesuperalgebrasoverthecomplexfieldand freelyexponentiatetothecorresponding Liesupergroupsandhomogeneoussuperspaces viathefunctorofpoints (seeSection2.4.3and,e.g.,[1] formoredetails).

InforthcomingworkswewilldevelopthetheoryofCartanconnectionsandparabolic geometries in the super-setting. In particular, this will allow us to deal with curved geometries with an intransitive symmetry superalgebra as well investigate the precise geometricrelationshipbetweenourfundamentalbinaryquadraticinvariantandCartan’s classicalbinaryquartic.GeometriesmodelledonothersimpleLiesuperalgebrasarealso ofimportance,e.g.,theLiesuperalgebraF(4) ispopularduetoitsrelationtoconformal fieldtheories.This workproposes G(3) asthesupersymmetryof differentialequations.

TherelationofourconstructiontothetwistorspinorsassociatedtoNurowski’sconformal metricswill bediscussedelsewhere.

2. AlgebraicaspectsandparabolicsubalgebrasofG(3)

2.1. Rootsystemsand Dynkindiagrams ofG(3)

The (complex) Lie superalgebra (LSA) g = G(3) has dimension (17|14), with even andoddparts:

g¯0=G(2)⊕A(1), g¯1=C⁷C². (2.1) Here,weusethenotationG(2) andA(1)∼=sp(2) todenotecomplexsimpleLiealgebras, while g¯1 is the g¯0-representation that is the (external)tensor product of thestandard G(2) andA(1) representations. Thesomewhat unusualnotation sp(2) will be reserved specificallytotheidealA(1)⊂g¯0throughout thewholepaper,to avoidconfusion with othersl(2)-subalgebras.

ACartansubalgebrahofgisbydefinitionaCartansubalgebraforg¯0.Allareconju- gate,sowefixonesuchchoice.Weadopttherootconventionsin[14,§2.19].Fixvectors δ, 1, 2, 3inh^∗with 1+ 2+ 3= 0 suchthat i, j= 1−3δij,δ,δ= 2 and i,δ= 0.

TheG(3) root systemΔ= Δ¯0∪Δ¯1⊂h^∗\{0}isgivenby:

Δ¯0={±2δ,± i, _i− j}, Δ¯1={±δ,±δ± i},

where1i=j3.Given anyevenrootαonehasα,α= 0,andtheusualreflection Sα(β) =β−2β, α

α, αα (2.2)

onh^∗ preserves eachof Δ¯0 and Δ¯1. TheWeylgroupof gisgenerated byallsuch even reflections. For any fixed simple root system Π and any oddisotropic root α∈ Π, we definetheoddreflection(see[33])

I II III IV

2 4 2

α1 α2 α3

3 4 2

α1 α2 α3 2 2

3

α1 α2

α3 2 3 3

α1 α2 α3

α1=δ+ 3

α2= 1

α3= 2− 1

α1=−δ− 3

α2=δ− 2

α3= 2− 1

α1=−δ+ 2

α2=δ− 1

α3= 1

α1= 2− 1

α2= 1−δ α3=δ

Fig. 3.Inequivalent simple root systems forG(3).

S_α(β) =

⎧⎪

⎪⎨

⎪⎪

⎩

β+α, α, β = 0;

β, α, β= 0, β=α;

−α, β =α;

(2.3)

forany β∈Π.

Up to Weyl group equivalence, there are four inequivalent simple systems Π = {α₁,α₂,α₃}, and each leads to a Cartan matrix and corresponding Dynkin diagram –seee.g.[4, §3] fordetails.These diagramsaregiveninFig.3.

• Nodesarewhite,black,orgreyaccordingtowhetherthecorresponding simpleroot αi iseven,oddwithαi,αi= 0,oroddwithαi,αi= 0;

• Dynkinlabelsm1,m2,m3 inscribedaboveeachnodecorrespond tothehighestroot α_high=m₁α₁+m₂α₂+m₃α₃.

OnecannotextendtheWeylgrouptoalargergroupthatincludesreflectionsforisotropic odd roots, since the latter cannot in general be extended to linear transformations of h^∗ thatsendrootsinto roots.Nevertheless,applying(2.3) toany β∈Π transformsone simple root system to another, and dashed arrows in Fig.3indicate such transforma- tions.

2.2. Parabolicsubalgebras andamapofG(3)-supergeometries A Z-grading of g is a decomposition g =

k∈Zgk satisfying [gi,gj] ⊂ gi+j for all i,j ∈Z.Inparticular,g0isaLSAandeachgk isag0-module.SincetheKillingformon G(3) is non-degenerate, then gk = (g₋k)^∗ as g0-modules. The corresponding parabolic subalgebra is p=g0 =

k0gk, and g₋ istheassociated symbol algebra,anilpotent gradedLSA.Lettingdergr(g₋) denotetheLSAof(super-)derivationsofg₋ofzerodegree, wehaveg₀⊂der_gr(g₋).Moreover,forthesegradingsg₋₁isbracket-generating,i.e.,g₋₁ generatesallofg₋ byiterativelybracketingwithg₋₁,so dergr(g₋)→gl(g₋₁).

Such Z-gradingsareobtainedfromachoiceofgradingelementZ∈h.Namely,define gk ={x ∈ g: [Z,x] = kx} for any k ∈ Z. Note thatZ ∈ h⊂ g0 and the root space gα⊂gk forα∈Δ suchthatα(Z)=k.ItfollowsthatZ∈z(g0).Ifμ= max{k:gk= 0}, then μ = αhigh(Z) and g is said to have a |μ|-grading. Given a simple root system

Fig. 4.Map ofG(3)-supergeometries. Green nodes are the focus of this article.

{α₁,α₂,α₃},let{Z₁,Z₂,Z₃}⊂hbeitsdualbasis.ThenZ=

i∈AZ_ispecifiesagrading element foranynonempty subsetA⊂ {1,2,3}, andinturn aparabolicsubalgebra p_A. Allnon-trivialgradingsofG(3) areofthisform,varyingthechoiceofsimplerootsystem (labelledI toIV asinFig.3)[21].

WerefertoM_A=G/P_A asaG(3)-supergeometry, whereGandP_A are(connected) Lie supergroups corresponding to g and p_A respectively. If B ⊂ A, there are natural fibrationsM_A→M_B.ConsideringeachsimplerootsystemfromFig.3leadstothemap ofG(3)-supergeometriesgiveninFig.4,seeAppendixAforfurtherdetails.

ForeachG(3)-supergeometry,itisnaturaltosearchforsomeexplicitgeometricstruc- tureswhose symmetryalgebrais precisely G(3). Ourgoalis to carryoutthis program intwo cases,labelledinFig.4asG(3)-contactandSHC.

2.3. SHC andcontactgradings

We will work with Π = {α1,α2,α3}labelled IVin Fig. 3. This givesan associated positiveΔ⁺ (resp.negativeΔ⁻)systemofroots.Explicitly,

Δ⁺_¯0 : α1, 2α3, α2+α3, α1+α2+α3, α1+ 2α2+ 2α3, α1+ 3α2+ 3α3, 2α1+ 3α2+ 3α3;

Δ⁺_¯1 : α2, α3, α1+α2, α2+ 2α3, α1+α2+ 2α3, α1+ 2α2+α3, α1+ 2α2+ 3α3; withΔ⁻_i =−Δ⁺_i ,i∈Z₂.Whilethere aremanypossibilitiesforZ, wewillfocusontwo choices:

• contactgrading definedbyZ=Z1,whichisa|2|-grading;

• SHC (“superHilbert–Cartan”)grading definedbyZ=Z2,whichisa|3|-grading.

These gradings closely parallel those considered in the classical cases in [35,39], and indeed the even parts (g₋)¯0 give precisely the symbol algebras considered there. The abovechoicesofΠ andZweremotivatedfromthefollowingdistinguishingfeatureinthe classicalcases:

Let g be a complex simple Lie algebra not of type A or C. The adjoint representation of g has highestweight (root) that isa fundamental weightλ_k =α_high=

im_iα_i with mk= 2 and

i∈Nkmi= 3,whereNk aretheneighbourstothekth-node(excluding the kth-node) inthe Dynkindiagram. The element Z=Zk defines acontactgrading, while Z=

i∈NkZ_i defines a|3|-grading generalizingtheoneforclassical (2,3,5)-geometries (in particular dimg₋3= 2 asfortheHilbert–Cartan grading).

For bothgradingsabove,z(g₀)= span{Z}⊂(g₀)¯0.Below aretherootsorganizedby parity and grading (with Δi(k) = {α ∈ Δi : α(Z) = k} for i ∈ Z2), along with the modulestructure forthesemisimplepart(g0)^ss_¯0 of(g0)¯0.

Contact grading:p^IV₁

k Δ¯0(k) Δ¯1(k)

0 ±(α₂+α₃), ±2α₃ ±α₂, ±α₃, ±(α₂+ 2α₃) 1 α₁, α₁+α₂+α₃,

α₁+ 2α₂+ 2α₃, α₁+ 3α₂+ 3α₃

α₁+α₂, α₁+α₂+ 2α₃, α₁+ 2α₂+α₃, α₁+ 2α₂+ 3α₃ 2 2α₁+ 3α₂+ 3α₃

(2.4)

k (gk)¯0 (gk)¯1 dim 0 C⊕sl(2)⊕sp(2) S²C²C² 7|6

−1 S³C²C C²C² 4|4

−2 CC 1|0

(2.5)

Note thatα₂+α₃ = ₁ and2α₃ = 2δ arethe positive rootsof sl(2) andsp(2), respec- tively. ThebracketΛ²g₋1 →g₋2 yieldsag0-invariant conformalsymplectic-orthogonal structure on g₋₁, so we cannaturally viewg0 ⊂cspo(g₋₁) ∼=cspo(4|4).We will make this explicitinSection2.4.1. Also,g0=CZ1⊕f∼=cosp(3|2),where

f= (g₀)^ss_¯0 ⊕(g₀)¯1∼=osp(3|2) (2.6) is thesemisimplepartofg0.

SHC grading:p^IV₂

k Δ¯0(k) Δ¯1(k)

0 ±α₁,±2α₃ ±α₃

1 α2+α3, α1+α2+α3 α2, α1+α2, α2+ 2α3, α1+α2+ 2α3

2 α1+ 2α2+ 2α3 α1+ 2α2+α3, α1+ 2α2+ 3α3

3 α1+ 3α2+ 3α3, 2α1+ 3α2+ 3α3

(2.7) k (g_k)¯0 (g_k)¯1 dim

0 C⊕sl(2)⊕sp(2) CC² 7|2

−1 C²C C²C² 2|4

−2 CC CC² 1|2

−3 C²C 2|0

(2.8)

Notethatα1= 2− 1and2α3= 2δarethepositiverootsofsl(2) andsp(2) respectively, andg0∼=C⊕sl(2)⊕osp(1|2).

Proposition 2.1. For the contact grading of G(3), the subalgebra g0 ⊂ cspo(g₋₁) is a maximalsubalgebra. FortheSHC gradingof G(3),wehaveg0∼=dergr(g₋).

Proof. To establishthefirstclaim, itsuffices toshow thatf=osp(3|2)⊂k:=spo(4|4) isamaximal subalgebra.Thedecompositionsoffandkintoevenandoddpartsare²

f=f¯0⊕f¯1∼= (sl(2)⊕sp(2))⊕(S²C²C²), (2.9) k=k¯0⊕k¯1∼= (sp(4)⊕so(4))⊕(C⁴C⁴), (2.10) andf¯0→k¯0 viatheactionong₋1= (g₋1)¯0⊕(g₋1)¯1∼= (S³C²C)⊕(C²C²).Note thatso(4)∼=sl(2)⊕sp(2), withsp(2)⊂f¯0 embedded purely inthelatter factorsp(2).

Ontheother hand,sl(2)⊂f¯0 isdiagonallyembeddedinsp(4)⊕sl(2) andwedenoteits imagebysl(2)diag.

Step 1.Wefirstclaim thattheuniquesubalgebrahproperlycontainedbetween f¯0 and k¯0ish=sl(2)⊕so(4),wheresl(2)⊂sp(4) viatheirreducibleactiononS³C².

Indeed, if h is a subalgebra such that f¯0 h k¯0 then h = h⊕sp(2) for some subalgebra

sl(2)_diaghsp(4)⊕sl(2),

2 Weusethesymboltodenotetheexternaltensorproductofsl(2) andsp(2) representations.

wheresl(2) isthefirstfactorofso(4).Howeversp(4)⊕sl(2)∼= (sl(2)⊕S⁶C²)⊕sl(2)diag

as ansl(2)diag-module,where sp(4)∼=sl(2)⊕S⁶C².Since S⁶C² isansl(2)-moduleand sp(4) isasimpleLiealgebra,weimmediatelyseethatS⁶C²isnotasubalgebraofsp(4).

Bythepreviousdiscussion h=

sl(2)⊕sl(2)diag=sl(2)⊕sl(2), or S⁶C²⊕sl(2)_diag,

but thesecond casedoes notcorrespond to any subalgebra of sp(4)⊕sl(2), hence the claim.

Step 2.Asf¯0-modules,

k¯1∼= (S³C²C)⊗(C²C²)∼= (S⁴C²C²)⊕(S²C²C²), where thelastisomorphismgivesthedecompositioninto irreducibles.

Assumefis asubalgebra such thatff k, so we get corresponding inclusionsof their even and odd parts. Sincef¯1 is an f¯0-module, thenf¯1 =k¯1 orf¯1 =f¯1 by Schur’s lemma andweconsider thetwo possibilitiesseparately:

(a) f¯1 =k¯1. Inthis casef¯0 =f¯0 orf¯0 =h,thus [k¯1,k¯1]⊂h.However, [k¯1,k¯1]=k¯0 since k=spo(4|4) issimple.

(b) f¯1=f¯1. Heref¯0=k¯0 orf¯0 =hand inboth casesf¯1 ⊂k¯1 ish-invariant. However k¯1

ish-irreducible.

In both cases,we obtainedcontradictions, so f⊂kis maximal, hencethefirst claimis proven.

ThesecondclaimfollowsfromourTheorem3.16(inparticular,H^0,1(g₋,g)= 0).

Despite tensor fields on supermanifolds M = (Mo,AM) not being determined by theirvaluesatpointsx∈Mo,theG-invariantgeometricstructuresonthehomogeneous supermanifold M =G/P correspond bijectivelyto p-invariant data ong/p, cf. [31,16].

ForthecontactandSHC casesabove,wewill describegeometricstructuresgivenby:

• asuperdistribution correspondingto (g₋₁⊕p)/p;

• a reduction of the structure group Autgr(g₋) with associated LSA dergr(g₋) to a connected(super-)subgroupG0 withsubalgebrag0.

Sinceg₊=

k>0g_k actstriviallyon(g₋₁⊕p)/p,itsufficestoconsidertheg₀-actionon g₋₁.

As no reductionis required forSHC,the rest of Section2isdevoted to considering the contact case and giving explicit descriptions of various g0-invariant structures on g₋1.Moreprecisely,wewillintroducethe(1|2)-twisted cubicV|x⊂P(C|x),whichgives