Non-Compact WZW Conformal Field Theories

We discuss non-compact WZW sigma models, especially the ones with symmetric space $H^{\bf C}/H$ as the target, for $H$ a compact Lie group. They offer examples of non-rational conformal field theories. We remind their relation to the compact WZW models but

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aNON-COMPACTWZWCONFORMALFIELDTHEORIES1,2KrzysztofGaw¸edzkiC.N.R.S.,I.H.E.S.91440Bures-sur-Yvette,FranceABSTRACTWediscussnon-compactWZWsigmamodels,especiallytheoneswithsymmetricspaceHC/Hasthetarget,forHacompactLiegroup.Theyo erexamplesofnon-rationalconformal eldtheories.WeremindtheirrelationtothecompactWZWmodelsbutstresstheirdistinctivefeatureslikethecontinuousspectrumofconformalweights,divergingpartitionfunctionsandthepresenceoftwotypesofoperatorsanalogoustothelocalandnon-localinsertionsrecentlydiscussedintheLiouvilletheory.Gaugingnon-compactabeliansubgroupsofHCleadstonon-rationalcosettheories.Inpartic-ular,gaugingone-parameterboostsintheSL(2,C)/SU(2)modelgivesanalternative,explicitlystableconstructionofaconformalsigmamodelwiththeeuclidean2Dblackholetarget.Wecomputethe(regularized)toroidalpartitionfunctionanddiscussthespectrumofthetheory.AcomparisonismadewithmorestandardapproachbasedontheU(1)cosetoftheSU(1,1)WZWtheorywherestabilityisnotevidentbutwhereunitaritybecomesmoretransparent.1.INTRODUCTION

ThefouryearswhichpassedsincethepreviousCarg`eseInstituteoftheserieshavebroughtamarkedprogressintheunderstandingofrationalConformalFieldTheories(CFT’s),aclassof2Dmasslessquantum eldmodels,seee.g.[1].Thesimplestofthosetheoriesisthefree eldwithvaluesinacircleofrationalradius,morecomplicatedexamplesareprovidedbytheWess-Zumino-Witten(WZW)sigmamodelswithageneralcompactLiegroupGasthetarget[2],[3][4]orbythecosettheoriesobtainedbygaugingasubgroupofGintheWZWtheory[5],[6].ThecharacteristicpropertyoftherationalCFT’sis

1.decompositionoftheeuclideanGreenfunctionsintoa nitesumofproductsof

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