Table Of ContentLecture 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.
Editedvolumesshould,however,consistofaverylimitednumberofcontributions
only.ProceedingswillnotbeconsideredforLNP.
VolumespublishedinLNParedisseminatedbothinprintandinelectronicfor-
mats,theelectronicarchivebeingavailableatspringerlink.com.Theseriescontent
isindexed,abstractedandreferencedbymanyabstractingandinformationservices,
bibliographicnetworks,subscriptionagencies,librarynetworks,andconsortia.
Proposalsshouldbe sent to a memberof the EditorialBoard, ordirectly to the
managingeditoratSpringer:
ChristianCaron
SpringerHeidelberg
PhysicsEditorialDepartmentI
Tiergartenstrasse17
69121Heidelberg/Germany
[email protected]
Moreinformationaboutthisseriesathttp://www.springer.com/series/5304
Ilarion V. 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
