Abelian varieties and theta functions as invariants for compact Riemannian manifolds constructions

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Abelian varieties and theta functions as invariants for compact Riemannian manifolds; constructions inspired by superstring theory. Chris Peters Institut Fourier, Universite Grenoble I St.-Martin d'Heres, France January 13 2010 1 Introduction In some forms of superstring theory particular theta-functions come up as partition functions. The associated Abelian varieties come either from cer- tain cohomology groups of the underlying universe or, in more recent theories (e.g. [Witten] [Mo-Wi]), are linked to their K-groups. Expressed in mathematical terms, one canonically associates to the co- homology or the K-theory of an even dimensional compact spin manifold a principally polarized Abelian variety. Moreover, if the dimension is 2 mod 8 a particular line bundle is singled out whose first Chern class is the principal polarization. This bundle thus has a non-zero section represented by a theta function which, after suitable normalization, is indeed the partition function of the underlying theory. Let me give some further detail on the physical motivation. There are several types of superstring theories, e.g. type I which is self-dual and types IIA and IIB wich are related via T -duality. The theories start from a space- time Y which in a first approximation can be taken to be Y = X ? T where T is the time-“axis” 1 and X is some compact Riemannian manifold.

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AbelianvarietiesandthetafunctionsasinvariantsforcompactRiemannianmanifolds;constructionsinspiredbysuperstringtheory.ChrisPetersInstitutFourier,Universite´GrenobleISt.-Martind’He`res,FranceJanuary1320101IntroductionInsomeformsofsuperstringtheoryparticulartheta-functionscomeupaspartitionfunctions.TheassociatedAbelianvarietiescomeeitherfromcer-taincohomologygroupsoftheunderlyinguniverseor,inmorerecenttheories(e.g.[Witten][Mo-Wi]),arelinkedtotheirK-groups.Expressedinmathematicalterms,onecanonicallyassociatestotheco-homologyortheK-theoryofanevendimensionalcompactspinmanifoldaprincipallypolarizedAbelianvariety.Moreover,ifthedimensionis2mod8aparticularlinebundleissingledoutwhoseﬁrstChernclassistheprincipalpolarization.Thisbundlethushasanon-zerosectionrepresentedbyathetafunctionwhich,aftersuitablenormalization,isindeedthepartitionfunctionoftheunderlyingtheory.Letmegivesomefurtherdetailonthephysicalmotivation.Thereareseveraltypesofsuperstringtheories,e.g.typeIwhichisself-dualandtypesIIAandIIBwicharerelatedviaT-duality.Thetheoriesstartfromaspace-timeYwhichinaﬁrstapproximationcanbetakentobeY=X×TwhereTisthetime-“axis”1andXissomecompactRiemannianmanifold.InTypeIIAtheorytheRamond-RamondﬁeldisacloseddiﬀerentialformG=G0+G2+∙∙∙onXwithcomponentsofallevendegreeswhileintypeIIBGisanodddegreecloseddiﬀerentialformonX.Moreover,theseformsareintegral(thatistheyhaveintegralperiodsoverintegralhomology1Tcouldbeacircleinphysicaltheoriesbutitcouldevenbeapoint(“absenceofbranes”).1

cycles).ThereasonisthattheyarePoincare´dualtocertainsubmanifoldsofXwhicharethe“world”-partofabraneinY.Suchaﬁeldshouldbethoughtofassomeconﬁgurationinthetheory.Thepartitionfunctionassemblesallpossibleconﬁgurationsinsomegeneratingfunctionwhichcaninturnbeusedtoderivefurtherphysicalpropertiesofthemodel.IntypeIIAtheorythispartitionfunctionisoftheformΘ(0)/ΔwhereΘissomenormalizedtheta-function.WhileΔiscanonicallyassociatedtotheRiemannianmanifoldX,thisisnolongerthecaseforΘ.Instead,assuggestedbyWittenin[Witten]andlaterbyMooreandWittenin[Mo-Wi]oneshouldliftthediscussionuptoK-theoryusingtheCherncharacter.Butthen,inordertomakeacanonicalchoiceforΘonehastoassumethatthemanifoldhasaspinstructureandhasdimension2mod8.Seethediscussionin§2formoredetails.Foralgebraicgeometerstheseconstructionslookabitesotericatﬁrstsight,themoresincetheyarephrasedintermsforeigntothem.Forin-stance,analgebraicgeometermightask:istheconstructionrelatedtotheWeiljacobian?ThiswasthequestionposedtomebyV.Srinivasandmotivatedthepresentnote.Clearly,ananswerentailsacarefulanalysisoftheconstructionproposedin[Mo-Wi].ThispresupposespreciseknowledgeoftopologicalK-theoryrelatedtotheindextheoremsofAtiyah-Singeret.al.,asubjectnottoowell-knownamongalgebraicgeometers.TomakethisnotereadablefortheaveragealgebraicgeometerIhaveplacedthesefactsin2appendices.Hereisanoutlineofthepaper.Thebackgroundfromphysicsiscollectedin§2.Itisnotnecessaryforanunderstandingoftherestofthepaper,butitpurportstoexplainhowphysicistcametotheparticularJacobiansandthenormalizedtheta-functions.Thebasicconstructionimplicitlyusedin[Mo-Wi]isreallysimpleandgivenin§3.ItisapparentlywellknownamongphysicistsbutIcouldnotﬁndareferenceforitinthispreciseform,althoughavariantiswellknowninsymplecticgeometry,cf.[McD-S,Prop.2.48(ii)].Thenin§4.1examplesusingcohomologyaregivenandin§4.2examplesusingK-groupsleadinguptotheexamplein[Mo-Wi].Itshouldbenotedthattheproofforthecrucialunimodularitypropertyismissinginloc.cit.Iexplaininthissectionhowitcanbeviewedasaspecialcaseofanoldresult[AH3]onnormalizedmultipliers.Thenfollowsashortdigressiononnormalizedthetafunctionsandﬁnallyin§5.2thisisappliedtogiveamathematicalformulationofthepertainingresultsof[Mo-Wi,§3].ThisusesinacrucialwaysomeconstructionsfromrealK-theory.2

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