On Non-Perturbative Results in Supersymmetric Gaug

arXiv:hep-th/9611152v120Nov1996CERN-TH/96-268hep-th/9611152ONNON-PERTURBATIVERESULTSINSUPERSYMMETRICGAUGETHEORIES–ALECTURE1AmitGiveon2TheoryDivision,CERN,CH-1211,Geneva23,SwitzerlandABSTRACTSomenotionsinnon-perturbativedynamicsofsupersymmetricgaugetheoriesarebeingreviewed.Thisisdonebytouringthroughafewexamples.CERN-TH/96-268September19961LecturepresentedattheWorkshoponGaugeTheories,AppliedSupersymmetryandQuantumGravity,ImperialCollege,London,UK,5-10July,1996.Toappearintheproceedings.2Permanentaddress:RacahInstituteofPhysics,TheHebrewUniversity,Jerusalem91904,Israel;e-mailaddress:[e-mail]1IntroductionInthislecture,wepresentsomenotionsinsupersymmetricYang-Mills(YM)theories.Wedoitbytouringthroughafewexampleswherewefaceavarietyofnon-perturbativephysicseffects–infra-red(IR)dynamicsofgaugetheories.Weshallstartwithageneralreview;someofthepointsweconsiderfollowthebeautifullecturenotesin[1].PhasesofGaugeTheoriesTherearethreeknownphasesofgaugetheories:•CoulombPhase:therearemasslessvectorbosons(masslessphotonsγ;noconfinementofbothelectricandmagneticcharges).ThebehaviorofthepotentialV(R)betweenelectrictestcharges,separatedbyalargedistanceR,isV(R)∼1/R;theelectricchargeatlargedistancebehaveslikeaconstant:e2(R)∼constant.ThepotentialofmagnetictestchargesseparatedbyalargedistancebehaveslikeV(R)∼1/R,andthemagneticchargebehaveslikem2(R)∼constant,e(R)m(R)∼1(theDiraccondition).•HiggsPhase:therearemassivevectorbosons(WbosonsandZbosons),electricchargesarecondensed(screened)andmagneticchargesareconfined(theMeissnereffect).ThepotentialbetweenmagnetictestchargesseparatedbyalargedistanceisV(R)∼ρR(themagneticfluxisconfinedintoathintube,leadingtothislinearpotentialwithastringtensionρ).ThepotentialbetweenelectrictestchargesistheYukawapotential;atlargedistancesRitbehaveslikeaconstant:V(R)∼constant.•ConfiningPhase:magneticchargesarecondensed(screened)andelec-tricchargesareconfined.ThepotentialbetweenelectrictestchargesseparatedbyalargedistanceisV(R)∼σR(theelectricfluxisconfined1intoathintube,leadingtothelinearpotentialwithastringtensionσ).ThepotentialbetweenmagnetictestchargesbehaveslikeaconstantatlargedistanceR.Remarks1.InadditiontothefamiliarAbelianCoulombphase,therearetheorieswhichhaveanon-AbelianCoulombphase[2],namely,atheorywithmasslessinteractingquarksandgluonsexhibitingtheCoulombpoten-tial.Thisphaseoccurswhenthereisanon-trivialIRfixedpointoftherenormalizationgroup.Suchtheoriesarepartofotherpossiblecasesofnon-trivial,interacting4dsuperconformalfieldtheories(SCFTs)[3,4].2.Whentherearematterfieldsinthefundamentalrepresentationofthegaugegroup,virtualpairscanbecreatedfromthevacuumandscreenthesources.Inthissituation,thereisnoinvariantdistinctionbetweentheHiggsandtheconfiningphases[5].Inparticular,thereisnophasewithapotentialbehavingasV(R)∼Ratlargedistance,becausethefluxtubecanbreak.ForlargeVEVsofthefields,aHiggsdescriptionismostnatural,whileforsmallVEVsitismorenaturaltointerpretthetheoryas“confining.”Itispossibletosmoothlyinterpolatefromoneinterpretationtotheother.3.Electric-MagneticDuality:MaxwelltheoryisinvariantunderE→B,B→−E,(1.1)ifweintroducemagneticchargem=2π/eandalsointerchangee→m,m→−e.(1.2)Similarly,Mandelstamand‘tHooftsuggestedthatunderelectric-magneticdualitytheHiggsphaseisinterchangedwithaconfiningphase.Con-finementcanthenbeunderstoodasthedualMeissnereffectassociatedwithacondensateofmonopoles.2DualizingatheoryintheCoulombphase,oneremainsinthesamephase.ForanAbelianCoulombphasewithmasslessphotons,thiselectric-magneticdualityfollowsfromastandarddualitytransforma-tion,andisextendedtoSL(2,Z)S-duality,actingonthecomplexgaugecouplingbyτ→aτ+bcτ+d,τ=θ2π+i4πg2,a,b,c,d∈Z,ad−bc=1.(1.3)Inthenon-AbelianCoulombphasethisdualityisnotobvious.Thesim-plestversionsofsuchS-dualityinnon-trivialinteractingtheoriesareinN=4supersymmetricYMtheory[6],andinfiniteN=2super-symmetricYMtheories[7].Electric-magneticdualitycanbeextended,sometimes,toasymptoticallyfreeN=1theories[1].2TheLow-EnergyEffectiveTheoryWewillconsiderthelow-energyeffectiveactionforthelightfields,Leff(lightfields),integratingoutdegreesoffreedom(suchasmassivevectorbosons,res-onances,etc.)abovesomescale.Leffhasalinearlyrealizedsupersymmetry(aslongasweareabovethescaleofpossiblesupersymmetrybreaking).Supersymmetrycanbemademanifestbyworkingwithsuperfields[8].AllrenormalizableLagrangianscanbeconstructedintermsofchairal(=scalar),anti-chiralandvectorsupermultiplets:•ChiralandAnti-ChiralSuperfields:thelightmatterfieldscanbecom-binedintochiralandanti-chiralsuperfieldsXr=φr+θαψαr+...,X†r=φ†r+¯θ˙αψ†˙αr+....(2.1)Chiral(anti-chiral)superfieldsobey¯DXr=0(DX†r=0).•VectorSupermultiplet:therealsuperfieldV=V†combiningthelightvectorbosonsAμandthegauginosλα,λ†˙α;schematically,V∼θσμ¯θAμ+θ2(¯θλ†)+¯θ2(θλ)+....(2.2)3Byeffective,wemeaninWilsonsense:Z[modespμ]e−S=e−Seff(μ,lightmodes),(2.3)so,inprinciple,Leffdependsonascaleμ.Butduetosupersymmetry,thedependenceonthescaleμdisappear(exceptforthegaugecouplingτwhichhasalogμdependence).Whentherearenointeractingmasslessparticles,theWilsonianeffec-tiveaction=the1PIeffectiveaction;thisisoftenthec