Instanton Strings and HyperKaehler Geometry

arXiv:hep-th/9810210v1 26 Oct 1998October1998

hep-th/9810210

utfa-98/26

spin-98/4

InstantonStringsandHyperK¨ahlerGeometry

RobbertDijkgraaf∗

DepartmentsofMathematicsandPhysics

UniversityofAmsterdam,1018TVAmsterdam

&

SpinozaInstitute

UniversityofUtrecht,3508TDUtrecht

Abstract

Wediscusstwo-dimensionalsigmamodelsonmodulispacesofinstantonsonK3surfaces.

TheseN=(4,4)superconformalfieldtheoriesdescribethenear-horizondynamicsof

theD1-D5-branesystemandaredualtostringtheoryonAdS3.Wederiveaprecise

maprelatingthemodulioftheK3typeIIBstringcompactificationtothemoduliof

theseconformalfieldtheoriesandthecorrespondingclassicalhyperk¨ahlergeometry.We

concludethatintheabsenceofbackgroundgaugefields,themetricontheinstanton

modulispacesdegeneratesexactlytotheorbifoldsymmetricproductofK3.Turningon

aself-dualNSB-fielddeformsthissymmetricproducttoamanifoldthatisdiffeomorphic

totheHilbertscheme.Wealsocommentonthemathematicalapplicationsofstring

dualitytotheglobalissuesofdeformationsofhyperk¨ahlermanifolds.1.Introduction

Recentlyinstringtheorytherehasbeengreatinterestinaparticularclassoftwo-

dimensionalconformalfieldtheorieswithN=(4,4)supersymmetryandcentralcharge

c=6kwithkapositiveinteger.Theseconformalfieldtheoriesarisebyconsideringacer-

tainlow-energylimitofboundstatesofD5-branesandD1-branesintypeIIBstringtheory

andinvariousotherdualincarnations.Theyhavebeeninstrumentalinthemicroscopic

descriptionofthequantumstatesoffive-dimensionalblackholesasintheground-breaking

computationofStromingerandVafa[1].Thesamesigmamodelshaveappearedinthequantizationofthesix-dimensionalworld-volumetheoryoftheNSfivebrane,thatcanbeusedtodescribeblackholesin

matrixtheory[2,3,4,5,6,7].Inthiscontextonesometimesreferstothesemodelsas

second-quantizedmicrostrings,littlestringsorinstantonstrings.

Morerecently,intheworkofMaldacenaondualitiesbetweenspace-timeconformal

fieldtheoriesandsupergravitytheoriesonanti-deSitterspaces,thesemodelswereidenti-

fiedwiththedualdescriptionofsix-dimensionalstringtheoryonAdS3×S3[8].Theholo-

graphiccorrespondencebetweentheseAdSquantumgravitytheoriesandtwo-dimensional

conformalfieldtheorieswiththeirrichalgebraicstructure,seemstobeafertileproving

groundforvariousconceptionalissuesingravityandstringtheory,seee.g.therecentpa-

pers[9,10,11,12,13]andreferencesthereintothealreadyextensiveliterature.Finally,

thereisafascinatingrelationbetweentheworld-sheetCFTofastringmovinginthis

AdS3backgroundandthespace-timec=6kCFTthatwestudyinthispaper[14].

ThisparticularclassofN=(4,4)SCFTwithcentralcharge6kcanbeidentified

withsigmamodelsonmodulispacesofinstantonsonK3orT4[1].Withtheright

characteristicclassesandcompactification,thesemodulispacesaresmooth,compact4k

dimensionalhyperk¨ahlermanifolds.Independentofthedevelopmentsinstringtheory

therehasbeenrecentlymathematicalinterestinthesehigher-dimensionalcompacthy-

perk¨ahler(orholomorphicsymplectic)manifoldsandtheirdeformationspaces,following

theimportantworkofMukaionmodulispacesofsheavesonabelianandK3surfaces

[15,16,17].Seeforexample[18]foragoodreviewoftherecentdevelopments.

Inthispaperwewillseehowtheinsightsfromstringtheoryandfromhyperk¨ahler

geometrycombinenicelyandallowustomakeapreciseidentificationbetweenthemoduli

ofthestringcompactificationandthoseofthe(space-time)CFT.Roughlywefindtwo

setofcanonicalidentifications:

(1)betweenthemoduliofaspecial(attractor)familyofworld-sheetK3sigmamodel

andtheclassicalhyperk¨ahlergeometryoftheinstantonmodulispace;

(2)betweenthemoduliofaspecialfamilyofnon-perturbativeIIBstringtheorycom-

pactificationonK3andthe(4,4)superconformalfieldtheoryontheinstantonmoduli

space.

2WewillseeindetailhowboththemetricandB-fieldonK3areusedtodetermine

themetriconthecompactifiedinstantonmodulispace.TheRRfieldsthenencodethe

B-fieldofthesigmamodel.Furthermore,non-perturbativeU-dualitiesoftheIIBstring

compactificationtranslateintoT-dualitiesofthehyperk¨ahlersigmamodel.

Theoutlineofthispaperisasfollows.In§2weintroducetheD5-braneworld-volume

theoryanditsrelationtoasigmamodelontheinstantonmodulispace.In§3wewill

describeindetailthemoduliofK3compactificationsinstringtheory.Thenin§4wewill

usethisexplicitdescriptiontodeterminethevalueofthesemodulionD-branebound

statesthatminimalizetheBPSmassthroughtheso-calledattractormechanism.Wewill

treatindetailtheexamplesoftheD0-D4-branesystemandD2-branesinIIAtheory(or

equivalentlyD1-D5andD3inIIBtheory).In§5wewillreviewsomeofdiscussionof

hyperk¨ahlermanifoldsandtheirdeformationspacesinthemathematicalliterature.Then

in§6wewillusethesemathematicalfactstogiveanexplicitmapbetweentheK3moduli

andthehyperk¨ahlerstructuresontheinstantonmodulispace,amapthatwewillthen

extendin§7tostringtheorymoduliandthemodulioftheN=(4,4)SCFT.Wewill

endwithsomecommentsonglobalissuesofthesemodulispaces.

2.D5-branesandinstantonstrings

2.1.Somemathematicalpreliminaries

Inthispaperwewillbestudyingsigmamodelswithtargetspacethemodulispace

Mofinstantonsonafour-manifoldX.So,mathematicallyspeaking,weareinterested

inthequantumcohomologyormoregenerallytheGromov-WitteninvariantsofM.The

classicalcohomologyofMisessentiallyequivalenttotheDonaldsoninvariantsofX,

sothesigmamodelonMdefinesa“stringy”generalizationoftheseinvariants—we

willrefertothemasinstantonstrings.However,weshouldhastentoaddthatforthe

particularcaseofaK3orT4manifold,onwhichwewillmostlyconcentrateinthis

paper,themodulispaceisahyperk¨ahlermanifold,andthequantumcohomologyreduces

totheordinarycohomology.Sooneshouldturninthiscasetomorerefinedstringtheory

invariantssuchastheellipticgenus[1,19].ThequantumcohomologyofthemodulispaceofvectorbundlesonaRiemannsurface

iswell-knowntoplayanimportantroleinthestudyofinstantonson4-manifoldsthat

canbewrittenastheproductoftwoRiemannsurfacesor,moregenerallyfibrationsof

thisform[20,21].IfXisoftheformΣ×Σ′,intheadiabaticlimitwherethevolume

ofΣ′isverysmallcomparedtoΣ,instantonsonXreducetoholomorphicmapsof

ΣintothemodulispaceofholomorphicvectorbundlesonΣ′.Inasimilarspiritthe

quantumcohomologyofinstantonmodulispacesonafour-manifoldXshouldberelated

3