An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry PDF
Preview An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry
Lecture Notes in Physics 951 Ilarion V. Melnikov An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry Lecture Notes in Physics Volume 951 FoundingEditors W.Beiglböck J.Ehlers K.Hepp H.Weidenmüller EditorialBoard M.Bartelmann,Heidelberg,Germany P.Hänggi,Augsburg,Germany M.Hjorth-Jensen,Oslo,Norway R.A.L.Jones,Sheffield,UK M.Lewenstein,Barcelona,Spain H.vonLöhneysen,Karlsruhe,Germany A.Rubio,Hamburg,Germany M.Salmhofer,Heidelberg,Germany W.Schleich,Ulm,Germany S.Theisen,Potsdam,Germany D.Vollhardt,Augsburg,Germany J.D.Wells,AnnArbor,USA G.P.Zank,Huntsville,USA The Lecture Notes in Physics The series Lecture Notes in Physics (LNP), founded in 1969, reports new devel- opmentsin physicsresearch and teaching-quicklyand informally,but with a high qualityand the explicitaim to summarizeand communicatecurrentknowledgein anaccessibleway.Bookspublishedinthisseriesareconceivedasbridgingmaterial between advanced graduate textbooks and the forefront of research and to serve threepurposes: (cid:129) to be a compact and modern up-to-date source of reference on a well-defined topic (cid:129) to serve as an accessible introduction to the field to postgraduate students and nonspecialistresearchersfromrelatedareas (cid:129) to be a source of advanced teaching material for specialized seminars, courses andschools Bothmonographsandmulti-authorvolumeswillbeconsideredforpublication. 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Melnikov An Introduction to Two-Dimensional Quantum Field Theory with (0,2) Supersymmetry 123 IlarionV.Melnikov DepartmentofPhysicsandAstronomy JamesMadisonUniversity Harrisonburg,VA,USA ISSN0075-8450 ISSN1616-6361 (electronic) LectureNotesinPhysics ISBN978-3-030-05083-2 ISBN978-3-030-05085-6 (eBook) https://doi.org/10.1007/978-3-030-05085-6 LibraryofCongressControlNumber:2018964707 ©SpringerNatureSwitzerlandAG2019 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland To VictorMelnikov Preface Oneofthemostremarkable discoveriesinelementaryparticle physicshasbeenthatoftheexistence ofthecomplexplane. TheanalyticS-matrix Eden,Landshoff,Olive,& Polkinghorne Complex geometry has played a key role in most developments of the last 30 yearsinquantumfieldtheoryandstringtheory.Thishascomeaboutnotonlyviathe analytic S-matrix but more generally through the beautiful interrelations between supersymmetry and complex geometry. Both structures introduce a great deal of rigidity compared to the more general categories of non-supersymmetric theories andrealdifferentialgeometry,andthisrigidityallowsforgeneralconceptualresults anddetailedquantitativepredictions.Amongthe highlightsin these developments we might recall the web of dualities between ten-dimensional string theories, the Seiberg-WittensolutionofthelowenergydynamicsofN=2supersymmetricgauge theories,mirrorsymmetryandSeibergduality,and,morerecently,theconstruction and investigationof a large class of non-Lagrangianfield theoriesassociated with the(2,0)superconformaltheoryinsixdimensionsanditscompactifications. Oncloserexamination,wefindthatmostoftheseremarkableresultshaveused relations between theories with four or more supercharges and Kähler geometry. While Kähler complex manifolds constitute the most familiar class of complex manifolds, a generic complex manifold does not admit a Kähler metric but still hasagreatdealmorerigiditycomparedtoagenericrealmanifold. Whilegreatprogresshasbeenmadeinunderstandingtheorieswithfourormore supercharges,itisimportanttoextendthesesuccessesasfaraspossibletotheories with less supersymmetry. In that sense, two-dimensional quantum field theories with (0,2) supersymmetry are the “ultimate frontier” where we can still use tools fromcomplexgeometryto constrainthe kinematicsanddynamicsof a non-trivial theory. Explorations of this frontier have been playing a growing role in modern mathematicalphysics. vii viii Preface The (0,2) theories were introduced relatively early on in string theory. Pertur- bative heterotic string theory involves the least supersymmetric theory of them all: a (0,1) two-dimensional supergravity theory which describes the propagation of a string in a ten-dimensional background. The resulting spacetime theories are chiral and propagate non-abelian gauge fields and have been studied at great lengthwiththegoalofprovidingunifiedmodelsforelementaryparticlephysics.A criticalheteroticstringbackgroundwithminimalfour-dimensionalsuper-Poincaré invariance requires the internal theory to be a (0,2) superconformal field theory. This realization led to an intensive study of such theories. A number of general techniques were constructed for obtaining a large class of models of this sort. Therelationthroughcompactificationtosupersymmetricfour-dimensionaltheories leads to powerful constraints on properties of the two-dimensional theories, and insights gained from this point of view have proven useful in more general (0,2) quantumfieldtheories. Morerecently,(0,2)theorieshavecometoprominenceasdescriptionsofsurface defects and low energy dynamics of solitonic strings in four-dimensional super- symmetric theories, where they providesome of the probesof a four-dimensional theorybeyondperturbationtheory.In addition,such theoriesnaturallyarise in the context of holography, as well as compactifications of the (2,0) six-dimensional superconformaltheoriesonfour-manifolds. There is another conceptualreason for interest in (0,2) quantum field theories: they may be considered as models for N=1 field theories in four dimensions: some(0,2)theoriesexhibitconfinementandsupersymmetrybreaking,whileothers have a rich IR dynamics controlled by superconformal theories with chiral sym- metries, marginal deformations, and accidental symmetries. So, one can develop useful analogies with four-dimensional dynamics in the context of simpler two- dimensional theories. The analogy becomes a concrete relation through compact- ificationofN=1d=4theoriestotwodimensionsonanappropriatebackground. Thepurposeoftheselecturenotesistointroducethereadertothesefascinating theories. The audience is assumed to have some basic background in conformal theory,quantumfieldtheory,andgeneralrelativity/differentialgeometryatthelevel ofVolumeIofPolchinski’sstringtheorytextandsomeexposuretosupersymmetry. A major theme will be to point out and utilize the relations between structures from complex geometry and field theory. To that end, we will need to introduce a numberof mathematicalconcepts. Our treatment of these will not be complete, but we will strive to explain the essential results and ideas, as well as to provide referencesfor further study. Some of these are given in the appendix,while other notionsaredevelopedinthemainbodyofthetext.Throughoutthetextthereader will find a number of exercises. Rather than being afterthoughts tacked on to the endofasection,theexercisesareanintegralpartofthetextanddevelopresultsor point out subtleties that are used in subsequent developments. There has been no deliberateattemptatobfuscationbutalsonoattempttohaveagrandunifiedtheory of notation: the reader should not be dismayed if the same symbol is sometimes usedfordifferentpurposes. Preface ix Wewillbeginwithathoroughexaminationofthebasicstructuresof(0,2)quan- tumfieldtheoryandconformalfieldtheory.Whilesettingdownthefundamentals, thiswillalsohelpustoestablishasetofconventionsandnotationthatwewillusein whatfollows.Next,wewillturntoasimpleclassofLagrangiantheories—the(0,2) Landau-Ginzburgmodels—anddiscuss the resulting renormalizationgroupflows, dynamics,andsymmetries.Wewillalsomakecontactwiththemorefamiliar(2,2) theoriesandcompareandcontrastthe(0,2)and(2,2)theories.Havinggottensome experience with this simplest class of models, we will examine (0,2) non-linear sigmamodels.Thesetheorieshavearichgeometricstructureandyieldanimportant generalizationoffamiliarKählergeometry.Theyarealsomoredelicateandexhibit anomalies that break global symmetries or even invalidate a particular theory but are particularly fascinating because of a direct connection with compactification of the heterotic string. After developing these structures, we will be in a position to appreciate the many simplifications offered by the (0,2) linear sigma model approach,whichprovidesaunifiedframeworkfortreatingnon-linearsigmamodels and Landau-Ginzburg theories. Here we will touch on the rich subject of mirror symmetry, mainly developed in the context of (2,2) theories and only recently generalizedtoclassesof(0,2)models. There are several glaring omissions in these notes and the following three deserve special mention. First, this is by no means a complete catalogue of every (0,2) application or result. Although we will meet many concrete examples, each illustrating either a general feature or a particular subtlety, there are many more to be found in the literature. Second, our point of view will be very much two- dimensional, so we will not discuss the many ways to obtain (0,2) theories from higherdimensionalconstructions.Finally,wedonotpresentthemodernlocalization toolsthathavebeenandcanbeappliedwithgreatsuccesstothesetheories. Acknowledgements I would like to first thank my collaborators who taught me most of what I know of this beautifulsubject: it has been fun, and I look forwardto more of it in your brilliantcompany!IamespeciallygratefultoR.Minasian,R.Plesser,S.Sethi,and S. Theisen for their continued encouragement to undertake and finish this book. Thanksalsotomyfamilywhoheardmore(0,2)complaintsthantheydeserve.This workwascompletedwithsupportfromtheCollegeofScienceandMathematicsat JamesMadisonUniversity,the4-VAInitiativegrant“Frontiersinstringgeometry,” the KITP Scholar program and the National Science Foundation under Grant No. NSFPHY-1748958,andtheMaxPlanckInstituteforGravitationalPhysics. Harrisonburg,VA,USA IlarionV.Melnikov Contents 1 (0,2)Fundamentals .......................................................... 1 1.1 TheLorentzGroupandLight-ConeCoordinates..................... 1 1.1.1 Light-ConeConventions...................................... 2 1.1.2 SpinorsinTwoDimensions................................... 2 1.2 The(0,2)SupersymmetryAlgebra .................................... 3 1.3 Minkowski(0,2)Superspace........................................... 4 1.4 SuperspaceDerivativesandMultiplets................................ 5 1.5 SupersymmetricActionsandFermiMultiplets....................... 6 1.6 (0,2)YukawaModels................................................... 8 1.6.1 SuperspaceEquationsofMotion ............................. 10 1.7 TheSupercurrentAlgebraviaSuperspace............................ 10 1.7.1 AssumptionsontheCurrentAlgebra......................... 11 1.7.2 TheS andRMultiplets....................................... 12 1.7.3 ASuperconformalTheory .................................... 13 1.7.4 AFewCommentsontheSupermultiplets.................... 14 1.7.5 TheSupercurrentAlgebraoftheYukawaModels........... 15 1.8 EuclideanWorldsheet.................................................. 15 1.8.1 EuclideanFermions ........................................... 16 1.8.2 Superspace..................................................... 17 1.8.3 EuclideanYukawaTheory .................................... 18 2 Conformalities................................................................ 21 2.1 TheBasics .............................................................. 21 2.1.1 TheConformalGroupinTwoDimensions................... 21 2.1.2 UnitaryCompactCFTs........................................ 22 2.1.3 TheEnergy-MomentumTensor............................... 23 2.1.4 WardIdentitiesforT .......................................... 25 2.1.5 Operator-StateCorrespondence............................... 28 2.1.6 SomeKeyProperties.......................................... 29 2.1.7 MinimalModels............................................... 30 2.1.8 DecomposableCFTs .......................................... 31 xi
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