Even and odd geometries on supermanifolds

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

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aEvenandoddgeometriesonsupermanifoldsM.Asoreya vrovb aFacultadDepartamentodedeF´ sicaTe´orica,50009CienciasZaragoza,UniversidadSpaindeZaragoza,bDepartmentTomskofMathematicalTomskStatePedagogical634041,RussiaUniversity,Analysis,Weanalyzefromageneralperspectiveallpossiblesupersymmetricgeneralizationsofsymplecticandmetricstructuresonsmoothmanifolds.Therearetwodi erenttypesofstructuresaccordingtotheeven/oddcharacterofthecorrespondingquadratictensors.Ingeneralwecanhaveeven/oddsymplecticsupermanifolds,FedosovsupermanifoldsandRiemanniansupermanifolds.ThegeometryofevenFedosovsupermanifoldsisstronglyconstrainedandhastobe at.Intheoddcase,thescalarcurvatureisonlyconstrainedbyBianchiidentities.However,weshowthatoddRiemanniansupermanifoldscanonlyhaveconstantscalarcurvature.WealsopointoutthatthesupersymmetricgeneralizationsofAdSspacedonotexistintheoddcase.1IntroductionThetwomainquadraticgeometricalstructuresofsmoothmanifoldswhichplayasigni cantroleinclassicalandquantumphysicsareRiemannianmetricsandsymplecticforms.Riemanniangeometryisnotonlybasicfortheformulationofgeneralrelativitybutalsofortheveryformulationofgauge eldtheories.Thesymplecticstructureprovidesthegeometricalframeworkforclassicalmechanics(see,e.g.[1])and eldtheories[2].TheFedosovmethodofquantizationbydeformation[3]isalsoformulatedin

termsofsymplecticstructuresandsymplecticconnections(theso-calledFedosovmanifolds[4]).TheintroductionoftheconceptofsupermanifoldbyBerezin[5](seealso[6,7])openednewperspectivesforgeometricalapproachesofsupergravityandquantizationofgaugetheories[8,9,10].Insummary,thegeometryofmanifoldsandsupermanifoldspercolatesallfundamentalphysicaltheories.

Inthisnoteweaddresstheclassi cationofpossibleextensionsofsymplecticandmetricstruc-turestosupermanifoldsintermsofgradedsymmetricandantisymmetricsecond-ordertensor elds.ThecasesofevenandoddsymplecticandRiemanniansupermanifoldsareanalyzedinsomedetail.Gradednon-degeneratePoissonsupermanifoldsaredescribedbysymplecticsupermanifoldsthatifequippedwithasymmetricsymplecticconnectionbecomegradedFedosovsupermanifolds.TheevencasecorrespondstoastraightforwardgeneralizationofFedosovmanifold[4]wherethescalarcurvaturevanishesasforstandardFedosovmanifolds.GradedmetricsupermanifoldsequippedwiththeuniquecompatiblesymmetricconnectionalsocorrespondtogradedRiemanniansupermanifold.Thescalar

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

curvatureisnontrivial,ingeneral,foroddRiemannianandFedosovsupermanifolds,butinthe rstcaseitmustalwaysbeconstant.ThereisasupersymmetricgeneralizationofAdSspacebutitistrivialintheoddcase.

Thepaperisorganizedasfollows.InSect.2,weconsiderscalarstructureswhichcanbeusedfortheconstructionofsymplecticandmetricsupermanifolds.Thepropertiesofsymmetrica neconnectionsonsupermanifoldsandtheircurvaturetensorsareanalyzedinSect.3.InSect.4,weintroducetheconceptsofevenandoddFedosovsupermanifoldsandevenandoddRiemanniansupermanifoldsareanalyzedinSect.5.Finally,weconveythemainresultsinSect.6.WeusethecondensednotationsuggestedbyDeWitt[11]andde nitionsandnotationsadoptedin[12].

2ScalarFields

LetMbeasupermanifoldwithadimensiondimM=Nand{xi}, (xi)= ialocalsystemofcoordinatesoninthevicinityofapointp∈M.Letusconsidernowthemostgeneralscalarstructuresonsupermanifoldswhichcanbede nedintermsofgradedsecond-ranksymmetricandantisymmetrictensor elds.

Ingeneral,thereexisteighttypesofsecondranktensor eldswiththerequiredsymmetryproperties

ωij= ( 1) i jωji,

ij=( 1) i j ji,

Eij= ( 1) i jEji,

gij=( 1) i jgji, (ωij)= (ω)+ i+ j, ( ij)= ( )+ i+ j, (Eij)= (E)+ i+ j,

(gij)= (g)+ i+ j.(1)(2)(3)(4)

Usingthesetensor elds(1)-(4)itisnotdi culttobuilteightscalarstructuresonasupermanifold:

{A,B}=

(A,B)= rA rA xj

xj,, ({A,B})= (ω)+ (A)+ (B), ((A,B))= ( )+ (A)+ (B),

(Eijdxj∧dxi)= (E),

(gijdxjdxi)= (g),(5)(6)(7)(8)E=Eijdxj∧dxi,g=gijdxjdxi,

whereAandBarearbitrarysuperfunctions.

Thebilinearoperation{A,B}(5)obeysthefollowingsymmetryproperty

{A,B}= ( 1) (ω)+( (A)+ (ω))( (B)+ (ω)){B,A}

whichintheevencase( (ω)=0)reducesto

{A,B}= ( 1)( (A) (B){B,A}

andintheoddcase( (ω)=1)to

{A,B}=( 1)( (A)+1))( (B)+1){B,A}.

Ontheotherhand,thebilinearoperation(A,B)(6)hasthesymmetryproperty

(A,B)=( 1) (ω)+( (A)+ (ω))( (B)+ (ω))(B,A)

whichintheevencase( (ω)=0)reducesto

(A,B)=( 1) (A) (B)(B,A)

2(13)(12)(11)(10)(9)

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

andintheoddcase( (ω)=1)to

(A,B)= ( 1)( (A)+1))( (B)+1)(B,A).(14)

Onecaneasilycheckthatintheevencase( (ω)=0)thebilinearoperation{A,B}satis estheJacobiidentity

{A,{B,C}}( 1) (A)( (C)+{C,{A,B}}( 1) (C)( (B)+{B,{C,A}}( 1) (B)( (A)≡0(15)ifandonlyifωsatis es

kl

ωij ωk xj( 1) l +ωkj ωli

i( l+1)

xj( 1) + lj ik

xj( 1) k( i+1)≡0.(17)

Therefore,becauseoftheidentities(16)and(17),onecanidentify{A,B}( ({A,B})= (A)+ (B))and(A,B)( ((A,B))= (A)+ (B)+1)withthePoissonbracketan …… 此处隐藏：5699字，全部文档内容请下载后查看。喜欢就下载吧 ……