Polyakov conjecture for hyperbolic singularities

arXiv:hep-th/0308131v2 30 Sep 2003LPTOrsayAugust2003hep-th/0308131

Polyakovconjectureforhyperbolicsingularities

LeszekHadasz1LaboratoiredePhysiqueTh`eorique,Bˆat.210,Universit´eParis-Sud,91405Orsay,FranceandM.SmoluchowskiInstituteofPhysics,JagiellonianUniversity,Reymonta4,30-059Krak´ow,Poland

ZbigniewJask´olski2PhysicsInstituteUniversityofZielonaG´oraul.Szafrana4a,65-069ZielonaG´ora,Poland

AbstractWeproposetheformoftheLiouvilleactionsatisfyingPolyakovconjectureontheaccessoryparametersforthehyperbolicsingularitiesontheRiemannsphere.1IntroductionLetusconsidertheFuchsianequation∂2ψ(z)+1

(z−zj)2+cj

(1−|w|2)2,w(z)=χ1(z)2eϕ(5)forallthatzforwhichw(z)iswelldefined.ItisconvenienttousenormalizedsolutionswithWronskiansatisfying:∂χ1(z)χ2(z)−χ1(z)∂χ2(z)=1,(6)

sothattherelation(4)canbewritteninasimpleform

e−ϕ(z,¯z)2󰀄χ1(z)χ1(z)󰀅.(7)Notethatϕ(z,¯z)isrealbyconstruction.Ifinadditionχ1(z),χ2(z)satisfytheSU(1,1)monodromyconditionthenϕ(z,¯z)issingle-valued.Undersomerestrictionsonconformalweightstherelationabovecanbemademorepre-cise.ThecaseofallparabolicsingularitieswasanalyzedbyPoincar´einthecontextoftheuniformizationproblem[7].HeshowedthattheLiouvilleequation(5)hasauniquereal-valuedregularonXsolutionwiththefollowingbehavioratthepunctures:

ϕ(z,¯z)=󰀔−2log|z−zj|−2log|log|z−zj||+O(1)asz→zj,

−4log|z−zj|+O(1)asz→∞.

2Thissolutiondefinesametricds2=eϕ|dz|2whichiscompleteonX.Ithasconstantnegativecurvature−1andparabolicsingularitiesateachzj.Theenergy–momentumtensorofthesolutionϕ,Tϕ(z)=−12∂2e−ϕ

2,j=1,...,n.Itfollowsfrom(8)thatthereexistsapairofsolutionsχ1,χ2totheFuchsianequation(1)relatedtoϕby(4)[1–3,5,6].Sinceϕisrealandsingle-valuedthissolvestheSU(1,1)Riemann-Hilbertproblem.TheexistenceandtheuniquenessofthesolutiontotheLiouvilleequationwiththeellipticsingularities,

ϕ(z,¯z)=󰀔−2󰀄1−θi

2−12π󰀇2

,j=1,...,n.

AsintheparaboliccaseonecanshowthatthereexistsasolutiontothecorrespondingSU(1,1)monodromyproblem[1–6].ThePolyakovconjecturestatesthatthe(properlydefinedandnormalized)LiouvilleactionfunctionalS[φ]evaluatedontheclassicalsolutionϕ(z,¯z)isageneratingfunctionfortheaccessoryparametersofthemonodromyproblemdescribedabove:

cj=−∂S[ϕ]basedonadirectcalculationoftheregularizedLiouvilleactionforparabolicandgeneralellipticsingularities,wasproposedbyTakhtajanandZografin[6].TheaimofthisLetteristofindtheactionwhichsatisfies(9)forthesingularitiesofthehyperbolictype.TheSU(1,1)monodromyproblemiswellposedinthiscase,butwhetherithasasolutionforarbitraryconformalweights∆j>1

2,λj∈R.Thenthefundamentalsystemdefinedby󰀖ψ1ψ2󰀊=󰀓2󰀖11i−i󰀊󰀖χ1χ2󰀊

4isalsonormalizedandhasSL(2,R)monodromyaroundallpunctures.Intermsof{ψ1,ψ2}theformula(7)reads

e−ϕ(z,¯z)2󰀄ψ1(z)ψ1(z)ψ2(z)󰀅.(10)TheadvantageofusingsolutionswithSL(2,R)monodromyisthatanyhyperbolicelementM∈SL(2,R)(tr(M)>2)isSL(2,R)-conjugatetoadiagonalmatrix.ThusforeachsingularityzjthereexistsanelementBj∈SL(2,R)suchthatthesystem󰀖ψj+ψj−󰀊=Bj󰀖ψ1ψ2󰀊

hasadiagonalmonodromyatzj.Itfollowsthatψj±havethecanonicalformψj±(z)=e±iϑj󰀐2uj±(z),(11)whereϑj∈R,anduj±(z)areanalyticfunctionsuj±(z)=∞󰀍l=0uj±,l(z−zj)l,uj±,0=1,onthediscDj={z∈X:|z−zj|T(z)=n󰀍j=1󰀈∆jz−zj󰀉=∞󰀍k=0tjk(z−zj)k−2,onegetstj0=∆j,tj1=cj,tjk=󰀍i,i=j󰀈(k−1)∆i(zi−zj)k−1󰀉fork󰀂2.(12)

TheFuchsianequation(1)thenimpliesuj±,l=−1

f′(z)−3f′(z)󰀇2

andtheFuchsianequation(1)thattheratioAj(z)=ψj+(z)satisfiestherelationT(z)={Aj(z),z}.(14)

Foreachhyperbolicsingularitywedefine

ρj(z)=(Aj(z))1

λj

󰀈z−zj+cj

dz󰀇2

󰀑Tj(ρj(z))+{ρj(z),z},(17)

where󰀑Tj(ρ)=󰀌ρiλj,ρ󰀏=

∆j

2󰀑Tj(ρ)ψ(ρ)=0

onthecomplexρplaneandanormalizedfundamentalsystemofsolutionswiththediagonalmonodromyatρ=0ofthefollowingform

󰀑ψj±(ρ)=(iλj)−

1

2.(19)

ThecorrespondingsolutionoftheLiouvilleequationreads[14]

󰀑ϕj(ρ,¯ρ)=log󰀋λ2j

2)},l∈Z,andinfinitelymanysingularlines:󰀑Sl={ρ∈C:λjlog|ρ|=πl},l∈Z.

Usingthetransformationrule(17)andtheexpansion(16)onecanshowthatonDj⊂Xthemetriceϕd2zcoincideswiththepull-backofthemetrice󰀑ϕjd2ρbythemapρj(z).Asρj(Dj)isanopenneighborhoodof0thereareinfinitelymanygeodesics󰀑Glandsingularlines󰀑Slcontainedinρj(Dj).TheirinverseimagesGl=ρ−1j(󰀑Gl),Sl=ρ−1j(󰀑Sl),areclosedsingular

linesandclosedgeodesicsofthemetriceϕd2zonX.Thisprovidesadetaileddescriptionofthesingularhyperbolicgeometryinasufficientlysmallneighborhoodofthehyperbolic

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