Borcherds symmetries in M-theory

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1、PreprinttypesetinJHEPstyle-HYPERVERSIONLPT-ENS-02/20hep-th/0203070BorcherdssymmetriesinM-theoryDedicatedtoPr.S.Hawkingonhis60thbirthday.PierreHenry-Labord`ere,BernardJuliaandLouisPaulotLaboratoiredePhysiqueTh´eoriquedel’EcoleNormaleSup´erieure∗24rueLhomond,75231ParisCedex05,F

2、rancephenry@lpt.ens.frbernard.julia@lpt.ens.frpaulot@lpt.ens.frAbstract:ItiswellknownbutrathermysteriousthatrootspacesoftheEkLiegroupsappearinthesecondintegralcohomologyofregular,complex,compact,delPezzosurfaces.Thecorrespondinggroupsactonthescalarfields(0-forms)oftoroidalcomp

3、actificationsofMtheory.TheirBorelsubgroupsareactuallysubgroupsofsupergroupsoffinitedimensionovertheGrassmannalgebraofdifferentialformsonspacetimethathavebeenshowntopreservetheself-dualityequationobeyedbyallbosonicform-fieldsofthetheory.Weshowherethatthecorrespondingdualitysuperal

4、gebrasarenothingbutBorcherdssuperalgebrastruncatedbytheabovechoiceofGrassmanncoefficients.ThefullBorcherds’rootlatticesarethesecondintegralcohomologyofthedelPezzosurfaces.OurchoiceofsimplerootsarXiv:hep-th/0203070v27May2002usestheanti-canonicalformanditsknownorthogonalcomplemen

5、t.AnotherresultisthedeterminationofdelPezzosurfacesassociatedtootherstringandfieldtheorymodels.DimensionalreductiononTkcorrespondstoblow-upofkpointsingeneralpositionwithrespecttoeachother.AlltheoriesoftheMagictrianglethatreducetotheEnsigmamodelinthreedimensionscorrespondtosing

6、ulardelPezzosurfaceswithA8−n(normal)singularityatapoint.ThecaseoftypeIandheterotictheoriesifonedropstheirgaugesectorcorrespondstonon-normal(singularalongacurve)delPezzo’s.WecommentonpreviousencounterswithBorcherdsalgebrasattheendofthepaper.∗UMR8549duCentreNationaldelaRecherch

7、eScientifiqueetdel’EcoleNormaleSup´erieure´Contents1.Introduction22.Geometricalprerequisites32.1Divisorsandtheirclasses32.2Intersectiononalgebraicsurfaces42.3Ampledivisors,projectiveembeddingsanddegrees42.4CanonicalclassKX42.5Blowinguppoints53.FromdelPezzosurfacestoBorcherdssu

8、peralgebras54.SmoothdelPezzo’s74.1Mtheory74.2IIAtheory104.3IIBtheory