Superanalogs of symplectic and contact geometry an

arXiv:hep-th/9406120v117Jun1994Superanalogsofsymplecticandcontactgeometryandtheirapplicationstoquantumfieldtheory.AlbertSchwarz∗DepartmentofMathematics,UniversityofCalifornia,Davis,[e-mail]insashortreviewofthetheoryofsymplecticandcontactmanifoldsandofthegeneralizationofthistheorytothecaseofsupermanifolds.Itisshownthatthisgeneralizationcanbeusedtoobtainsomeimportantresultsinquantumfieldtheory.Inparticular,regardingN-superconformalgeometryasparticularcaseofcontactcomplexgeome-try,onecanbetterunderstandN=2superconformalfieldtheoryanditsconnectiontotopologicalconformalfieldtheory.Theoddsymplecticge-ometryconstitutesamathematicalbasisofBatalin-Vilkoviskyprocedureofquantizationofgaugetheories.Theexpositionisbasedmostlyonpublishedpapers.However,thepapercontainsalsoareviewofsomeunpublishedresults(inthesectiondevotedtotheaxiomaticsofN=2superconformaltheoryandtopologicalquantumfieldtheory).ThepaperwillbepublishedinBerezinmemorialvolume.Introduction.Itisagreatpleasureformetopublishmypaperinthisvolume.F.Berezinmadeextremelyimportantcontributiontomathematicalphysicsinmanydiffer-entdirections.However,themostsignificantpartofhisheritageisrelatedtotheideathatthetheoryoffermionsbecomesverysimilartothetheoryofbosons,iftheusualfunctions(”functionsofcommutingvariables”)arereplacedbythe”functionsofanticommutingvariables”(elementsofaGrassmannalgebra).Hewasthefirstwhorealizedthatalongwithstandardalgebra,analysisandge-ometryonecanconstructalgebraandanalysisoffunctionsdependingnotonlyoncommuting,butalsoonanticommutingvariables,anddevelopgeometryofmanifoldswithcommutingandanticommutingcoordinates.Theseideasfoundveryimportantapplicationstophysics.Theywereusedtoanalyzeanewkind∗ResearchsupportedinpartbyNSFgrantNo.DMS-92013661ofsymmetry.Thissymmetry,mixingbosonsandfermions,iscalledsupersym-metry;thereforecorrespondingmathematicalconceptsarealsoprovidedwiththeprefix”super”.Thepresentpaperisbasedcompletelyonsuchconcepts.Itbeginswithabriefintroductiontotheideasofsupergeometry.Themainpartofthepapercontainsashortreviewofthetheoryofsymplecticandcontactmanifoldsandofthegeneralizationofthistheorytothecaseofsupermanifolds.Itisshownthatthisgeneralizationcanbeusedtoobtainsomeimportantresultsinquantumfieldtheory.Inparticular,regardingN-superconformalgeometryasaparticularcaseofcontactcomplexgeometry,onecanbetterunderstandN=2superconformalfieldtheoryanditsconnectiontotopologicalconformalfieldtheory.TheoddsymplecticgeometryconstitutesamathematicalbasisoftheBatalin-Vilkoviskyprocedureofquantizationofgaugetheories.Ourexpositionisbasedmostlyonthepapers[1]-[7].However,thepapercontainsalsoareviewofsomeunpublishedresults(inthesectiondevotedtotheaxiomaticsofN=2superconformaltheoryandtopologicalquantumfieldtheory).Supergeometry.Asmoothm-dimensionalmanifoldcanbedefinedasanobjectobtainedfromdomainsinRmpastedtogetherbymeansofsmoothtransformations.Thisdefinitioncanbeformulatedinapurelyalgebraicway.Namely,onecanidentifyadomainU⊂RmwiththealgebraC∞(U)ofallsmoothfunctionsonUandasmoothmapofUintoVwithahomomorphismofC∞(V)intoC∞(U).Suchanalgebraicconstructioncanbegeneralizedasfollows.Bydefinition,weidentifyan(m|n)-dimensionalsuperdomainUnwiththeZ2-gradedalgebraC∞(U)⊗ΛnwhereUisadomaininRmandΛnisaGrassmannalgebrawithngeneratorsξ1,...,ξn.InparticularifU=RmweobtainasuperdomaindenotedbyRm|n.OnesaysthatRm|nisan(m|n)-dimensionallinearsuperspace.AmapofUnintoVn′,whereUisadomaininRm,VisadomaininRm′,isdefinedasanevenhomomorphismofC∞(V)⊗Λn′intoC∞(U)⊗Λn.(WesaythatanoperatoractingonZ2-gradedspacesisevenifitisparitypreservingandoddifitisparityreversing.)ElementsofthealgebraC∞(U)⊗ΛncanbewrittenasformallinearcombinationsF=XkXi1,...,ikfi1,...,ik(x)ξi1...ξik,(1)wherefi1,...,ik(x)aresmoothfunctionsonUandξiξj=−ξjξi.Withoutlossofgeneralityweassumethatthecoefficientsfi1,...ikareantisymmetricwithre-specttoapermutationofi1,...,ik.ItisconvenienttoconsidertheelementsofC∞(U)⊗Λnasfunctionsdependingoncommutingvariables(x1,...,xm)∈Uandanticommutingvariablesξ1,...,ξn.(InotherwordselementsC∞(U)⊗ΛnareconsideredasfunctionsonthesuperdomainUnwithmcommutingcoor-dinatesx1,...,xmandnanticommutingcoordinatesξ1,...,ξn).Amapofa2superdomainUnwithcoordinatesx1,...,xm,ξ1,...,ξnintoasuperdomainVn′withcoordinates˜x1,...˜xm′,˜ξ1,...,˜ξn′canbespecifiedbytheformulas˜xi=ai(x1,...xm,ξ1,...,ξn)˜ξj=αj(x1,...xm,ξ1,...,ξn)whereai,1≤i≤m′,andαj,1≤j≤n′arecorrespondinglyevenandoddelementsofC∞(U)⊗Λn.(Itiseasytocheckthatthesubstitutionofaiandαjintothefunctionsofvariables˜xi,˜ξjdeterminesahomomorphismfromC∞(V)⊗Λn′intoC∞(U)⊗Λnifthefunctionsai(x1,...,xm,0,...,0),1≤i≤m′determineamapofthedomainU⊂RmintothedomainV⊂Rm′.)Nowonecandefinean(m|n)-dimensionalsupermanifoldasanobjectpastedtogetherfrom(m|n)-dimensionalsuperdomainsbymeansofinvertiblemaps.Thedefinitionofsuperdomainandsupermanifoldgivenaboveiscompletelyalgebraic.Inthisdefinitionasupermanifold”hasnopoints.”However,ifMisasupermanifoldandΛisanarbitraryGrassmannalgebra,onecanconstructthesetMΛofΛ-pointsofM.IntheparticularcasewhenMisasuperdomainUnwedefineMΛasthesetofallrows