Even and odd geometries on supermanifolds(2)

LetΓbeasymmetricconnectionofasymplecticsupermanifold(M,ω).Thecorrespondingco-variantderivative hastoverifythecompatibilityconditionω =0withthesymplecticstructureω.Ineachlocalcoordinatesystem{xi}thecompatibilityconditioncanbeexpressedas

ωij k=ωij,k Γijk+Γjik( 1) i j=0,ωij= ( 1) i jωji(36)

intermsofthecomponentsΓijk( i)ofthesymplecticconnection ,whereweusethenotation

Γijk=ωinΓnjk, (Γijk)= (ω)+ i+ j+ k.(37)

Asymplecticsupermanifold(M,ω)equippedwithasymmetricsymplecticconnectionΓiscalledaFedosovsupermanifold(M,ω,Γ).

LetusconsidernowcurvaturetensorRijklofasymplecticconnection

Rijkl=ωinRnjkl, (Rijkl)= (ω)+ i+ j+ k+ l,(38)

whereRnjklisde nedin(28).Thistensorhasthefollowingsymmetryproperties

Rijkl= ( 1) k lRijlk,

andsatis estheidentity

Rijkl+( 1) l( i+ k+ j)Rlijk+( 1)( k+ l)( i+ j)Rklij+( 1) i( j+ l+ k)Rjkli=0.

5(40)Rijkl=( 1) i jRjikl(39)

We analyze from a general perspective all possible supersymmetric generalizations of symplectic and metric structures on smooth manifolds. There are two different types of structures according to the even/odd character of the corresponding quadratic tensor

ThelaststatementcanbederivedfromtheJacobiidentity

( 1) j lRijkl+( 1) l kRiljk+( 1) k jRiklj=0.(41)

togetherwithacyclicchangeofindices[17].Theidentity(40)involvesdi erentcomponentsofthecurvaturetensorwithcyclicpermutationofallindices,butthesignfactorsdependontheGrassmannparitiesoftheindicesanddonotfollowacyclicpermutationrule,similartothatofJacobiidentity,butarede nedbythepermutationoftheindicesthatmapsagivensetintotheoriginalone.

Fromthecurvaturetensor,Rijkl,andtheinversetensor eldωijofthesymplecticstructureωij,onecanconstructtheonlycanonicaltensor eldoftype(0,2),

Kij=ωknRnikj( 1) i k+( (ω)+1)( k+ n)=Rkikj( 1) k( i+1),

ThistensorKijistheRiccitensorandsatis estherelations[18]

[1+( 1) (ω)](Kij ( 1) i jKji)=0.(43) (Kij)= i+ j.(42)

IntheevencaseKijissymmetricwhereasintheoddcasetherearenotrestrictionsonits(generalized)symmetryproperties.

Now,onecande nethescalarcurvaturetensorKbytheformula

K=ωjiKij( 1) i+ j=ωjiωknRnikj( 1) i+ j+ i k+( (ω)+1)( k+ n).

FromthesymmetrypropertiesofRijkl,itfollowsthat

[1+( 1) (ω)]K=0,(45)(44)

whichprovesthatasinthecaseofFedosovmanifolds[4]thescalarcurvatureKvanishes.

However,foroddFedosovsupermanifoldsKis,ingeneral,non-vanishing.ThisfactwasquiterecentlyusedinRef.[13]togeneralizetheBVformalism[8].

LetusconsidertheBianchiidentity(32)intheform

Rnmij;k Rnmik;j( 1) k j+Rnmjk;i( 1) i( j+ k)≡0.

Contractingindicesiandnwiththehelpof(42)weobtain

Kmj;k Kmk;j( 1) k j+Rnmjk;n( 1) n( m+ j+ k+1)≡0.

Nowusingtherelations

Kij=ωikKkj( 1) k,Kij;m=ωikKkj;m( 1) k

Kij;i( 1) i( j+1)=ωikKkj;i( 1) k+ i( j+1),

itfollowsthat

K,i=[1 ( 1) (ω)]Kji;j( 1) j( i+1).

Intheoddcasethisimpliesthat

K,i=2Kji;j( 1) j( i+1).(51)(50)(48)(49)(47)(46)

IntheevencaseK,i=0butinthatcasetherelation(50)doesnotprovidesanynewinformationbecauseinthiscaseK=0.

6

We analyze from a general perspective all possible supersymmetric generalizations of symplectic and metric structures on smooth manifolds. There are two different types of structures according to the even/odd character of the corresponding quadratic tensor

5Riemanniansupermanifolds

g=gijdxjdxi,gij=( 1) i jgji, (gij)= (g)+ i+ j,(52)LetMbeasupermanifoldequippedbothwithametricstructureg

andasymmetricconnection withacovariantderivative compatiblewiththesuper-Riemannianmetricg

m i jgij k=gij,k gim m=0.jk gjm ik( 1)(53)

ItiseasytoshowthatasinthecaseofRiemanniangeometrythereexiststheuniquesymmetricconnection ijkwhichiscompatiblewithagivenmetricstructure.Indeed,proceedinginthesamewayasintheusualRiemanniangeometryoneobtainsthegeneralizationofcelebratedChristo elformulafortheconnectioninsupersymmetriccase[12]

lki=1

We analyze from a general perspective all possible supersymmetric generalizations of symplectic and metric structures on smooth manifolds. There are two different types of structures according to the even/odd character of the corresponding quadratic tensor

Intheevencasewehave

R,i=2Rji;j( 1) j( i+1),(62)

whichisasupersymmetricgeneralizationofthewellknownrelationofRiemanniangeometry[19].IntheoddcaseR,i=0andtherelation(61)impliesthatR=const.

Therefore,oddRiemannsupermanifoldscanonlyhaveconstantscalarcurvatureR=const.

ItiswellknownthatspecialtypesofRiemannianmanifoldsplayanimportantroleinmodernquantum eldtheory.Inparticular,aconsistentformulationofhigherspin eldtheoriesispossibleonAdSspace(see,forexample[20]).Inthiscasethecurvature,Ricciandscalarcurvaturetensorshavetheform

Rijkl=R(gikgjl gilgjk),Rij=(N 1)Rgij,R=N(N 1)R,(63)

whereNisthedimensionoftheRiemannianmanifoldMwithametrictensorgijandRisconstant.LetusanalyzethestructureofsupersymmetricextensionsofAdSspaces(63).Ifgijisthegradedmetrictensor(52)oftheAdSspaceonecande nethefollowingcombinationofmetrictensors

Tijkl=gikgjl( 1) (g)( i+ k)+ k j

whichtransformsasatensor eld.Thereforeanaturalgeneralizationof(63)satis esthat

Rijkl=R(gikgjl( 1) (g)( i+ k)+ k j gilgjk( 1) (g)( i+ l)+ l j+ l k)=

=(gikRgjl( 1) k j gilRgjk( 1) l j+ l k)( 1) (g),

whereR( (R)= (g))isaconstant.TheRiccitensorsatis es

Rij=gklRlikj( 1)( (g)+1)( k+ l)+ i k=R(N 1)gij( 1) (g)

andthescalarcurvaturetensorveri esthat

R=RN(N 1),

wherewedenote

iN=δi( 1) i(64)(65)(66)(67)(68)

andNisnothingbutthedi erencebetweenthenumberofbosonicandfermionicdimensionsofthesupermanifold.

TheaboveRiemanniantensorsobeythefollowingsymmetryproperties

Rijkl= ( 1) k lRijlk,Rijkl= ( 1) (g)+ i jRijkl,

Rijkl=( 1)( i+ j)( k+ l)Rklij+[1 ( 1) (g)]gilRgjk( 1) (g)+ l( j+ k),

Rij=( 1) i jRji.

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