Instanton operators and symmetry enhancement in 5d supersymmetric gauge theories

0.21 MB·English

by  Yuji Tachikawa

IPMU-15-0002
UT-15-01
Instanton operators and symmetry enhancement
5
1 in 5d supersymmetric gauge theories
0
2

b
e
F
YujiTachikawa

3
2

] DepartmentofPhysics,FacultyofScience,
h
t UniversityofTokyo,Bunkyo-ku,Tokyo133-0033,Japan,and
-
p
InstituteforthePhysicsandMathematicsoftheUniverse,
e
h UniversityofTokyo,Kashiwa,Chiba277-8583,Japan
[

3
v
Abstract
1
3
0
1 Supersymmetricgaugetheoriesinfivedimensionsoftenexhibitlesssymmetrythantheultra-
0
violet fixed pointsfrom which they flow. The fixed points mighthave larger flavor symmetry
.
1
or they mighteven be secretly six-dimensionaltheories onS1. Here we provide a simplecri-
0
5 terionwhensuchsymmetryenhancementintheultravioletshouldoccur,byadirectstudyof
1
thefermioniczeromodesaroundone-instantonoperators.
:
v
i
X
r
a
Contents
1 Introduction 1
2 Preliminaries 3
2.1 Supermultipletofbrokencurrents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Instantonoperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 SU(2) 5
3.1 PureN=1theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Withfundamentalflavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3 N=2theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 SU(N) 8
4.1 PureN=1theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Withfundamentalflavors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.3 N=2theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
5 Quivers 10
5.1 SU(2)2theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.2 SU(N )×SU(N )theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1 2
5.3 Somespecialtwo-nodequivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.4 Multi-nodequivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6 Conclusions 15
1 Introduction
Infourdimensionsandinlowerdimensions,weoftenstartfromaLagrangiangaugetheoryin
theultraviolet(UV)andstudythebehaviorofthesystemintheinfrared(IR).Infivedimensions
(5d),Lagrangiangaugetheoriesarealwaysnon-renormalizable,andoneinsteadneedstolook
fornontrivialultravioletsuperconformalfieldtheories(SCFTs)fromwhichtheyflowout.With
supersymmetry,suchastudyisindeedpossible,asdemonstratedin[1–4],bycombiningfield-
theoreticalanalysesandembeddingintostringtheory.
Thereandinotherworks,itwasfoundthattheUVfixedpointcanhaveenhancedsymme-
try: theinstantonnumbersymmetrysometimesenhancestoanon-Abelianflavorsymmetry,
andsometimesenhancestotheKaluza-Kleinmodenumberofasix-dimensionaltheoryonS1.
Forexample,theUVSCFTforN=1SU(2)theorywithN ≤7flavorshas E flavorsymme-
f Nf+1
try; with N = 8 flavors, the UV fixed point is instead a 6d N=(1,0) theorywith E symmetry
f 8
compactified onS1 [5]. For SU(N) theory, the flavor symmetry enhancement only occurs for
somespecificchoiceofthenumberoftheflavorsandoftheChern-Simonslevels.N=2theory
with gaugegroupG in 5d, in contrast, comes from some6d N=(2,0) theorycompactified on
S1,asoriginallyfoundinthecontextofstringduality[6,7].
1
These results were soon extended to include more models, using webs of five-branes, in
e.g.[8–10]. Morerecently,varioussophisticatedtechniquessuchassupersymmetriclocaliza-
tions, Nekrasov partition functions, and refined topological strings have been applied to the
analysis of these 5d systems. The symmetry enhancement of the models mentioned above
has been successfully confirmed by these methods, and even morediverse models are being
explored,seee.g.[11–27].
These resultsare impressive, but thetechniques used are ratherunwieldy.1 In this paper,
wedescribeasimplermethodtoidentifythesymmetryenhancementofagiven5dgaugethe-
ory,assumingthatithasanultravioletcompletioneitherin5dor6d. Althoughheuristic,this
methodtellsuswhatwillbetheenhancedflavorsymmetryandwhethertheultravioletcom-
pletionisa5dSCFTora6dSCFTonS1.
We do this by identifying the supermultiplet of broken symmetry currents2 by studying
instanton operators, that introduce non-zero instanton number on a smallS4 surroundinga
point.3 When the instanton number is one, the structure of the moduli space and the zero
modesisparticularlysimple,allowingustofindthebrokensymmetrycurrentsratherdirectly.
Thespiritoftheanalysiswillbeclosetotheanalysisofmonopoleoperatorsof3dsupersym-
metricgaugetheoriesin[30],exceptthatweneedtoidentifyanoperatorintheIRthatwould
comefromaUVcurrentoperator,ratherthanviceversa.
Therestofthepaperisorganizedasfollows. InSec.2,wecollectbasicfactsonthesuper-
symmetryandoninstantonsonS4thatweuse.InSec.3,weanalyzeN=1SU(2)gaugetheories
withN ≤8flavorsandN=2SU(2)theory.Theeffectofthediscretethetaanglewillalsobedis-
f
cussed. In Sec. 4, we extend thediscussionto N=1 SU(N) gaugetheorieswith fundamentals
andChern-Simonsterms,andto N=2SU(N) theory. Weusetheresultsthatwillbeobtained
up to this point in Sec. 5 to study symmetry enhancement in quiver gauge theories made of
SUgaugegroupsandbifundamentals.Weassumethattheeffectivenumberofflavorsofeach
SU(N)nodeis2N andthattheChern-Simonslevelsareallzero. Weconcludewithadiscussion
inSec.6.
1Oneofthetrickiestaspectsistheneedtoremovespuriouscontributionsfromtheso-called“U(1)parts”in
theinstantoncountingmethodandfromtheparallellegsinthetoricdiagraminthetopologicalstringmethod.
2Theimportanceandthespecialpropertyofthecurrentsupermultipletsin5dSCFTswereemphasizedin[13].
Theanalysispresentedhereisstronglyinfluencedbythatpaper.
3Recentlythereappearedapaper[28]whereinstantonoperatorsof5dN=2gaugetheorieswerealsostudied.
Theretheemphasiswasonthemulti-pointfunctionsofinstantonoperators.Inthispaperwejustconsiderasin-
gleinstantonoperator,andthemultipletitformsundertheflavorsymmetryandthesupersymmetry. Inanother
recentpaper[29],adifferenttypeofinstantonoperatorswasstudied,whereanexternalSp(1) backgroundwas
R
introducedonS4tohavemanifestsupersymmetryattheclassicallevel.Herewedonotintroducesuchadditional
backgrounds.
2
2 Preliminaries
2.1 Supermultipletofbrokencurrents
Supposethatthe5dsupersymmetricgaugetheoryisobtainedbyamassdeformationofa 5d
superconformal theory. In this setup, supersymmetric mass deformations are always asso-
ciated to the Cartan part of a possibly non-Abelian conserved current supermultiplet,4 that
containstheconformalprimaries
µa , ψa , Ja, Ma (2.1)
(ij) iα µ
with scalingdimension3, 3.5, 4, 4 respectively. Here a is theadjointindexof theflavor sym-
metry,i =1,2istheindexofSp(1) andαisthespinorindexofSO(5);thesymplecticMajorana
R
conditionisimposedonψa .
iα
ThemassdeformationisdonebyaddingδL=h Ma totheLagrangian,thusbreakingsome
a
oftheflavorsymmetry:
∂ Ja ∝ fab h Mc. (2.2)
µ µ c b
where fab isthestructureconstant. Thetheoryisnolongersuperconformal,butwecanstill
c
consider the supermultipletsof operators underthe 5d N=1 supersymmetry, modulospace-
timederivatives∂ . Nocomprehensiveanalysisofthestructureofsuchsupermultipletsin5d
µ
isavailableintheliterature,totheknowledgeoftheauthor.Wecanstillseethatthesupermul-
tipletcontainingµa ofa broken generator is short,because thesuperderivativeofµa only
(ij) (ij)
containsψa whichisadoubletunderSp(1) .Thisfactallowsustoidentifythebrokencurrent
iα R
supermultipletsclearlyin theinfrared gaugetheory: theyare supermultipletscontainingthe
following:
µ , ψ , J (2.3)
(ij) iα µ
whereµ isanSp(1) tripletscalarµ ,ψ isasymplecticMajoranaSp(1) doubletfermion
(ij) R (ij) iα R
ψ , and J is a vector. The component M in (2.1) is a divergence of J , and therefore is a
iα µ µ
descendant.
Supposeinsteadthatthe5dtheoryisobtainedbyputtinganN=(1,0)6dSCFTonS1,pos-
sibly with a nontrivial holonomy for the flavor symmetry; we consider N=(2,0) theories as
specialcasesofN=(1,0)theories.Theconservedcurrentsupermultipletin6dcontains
µ , ψ , J , (2.4)
(ij) iα A
where A is the 6d vector index and we omitted the adjoint index for brevity. OnS1, we have
Kaluza-Klein (KK) modes J(n) and M(n) := J(n) where n is theKK mode number. Then the 6d
µ 6
conservationgives
n
∂ J(n)∝ M(n) (2.5)
µ µ L
in5d,whereL isthecircumferenceofS1. Therefore,weagainfindthesamesupermultipletof
brokencurrents.
4Forthedetailsofthesuperconformalmultipletsin5dandin6d,seee.g.[31]andreferencestherein.
3
2.2 Instantonoperators
Supposenowthatwe aregivena 5dgaugetheory,andfurtherassumethatitisa massdefor-
mationofa5dSCFTora6dSCFTonS1.Pickapointp inR5andinsertthereacurrentoperator
thatis broken bythemass deformationin theformer case, and a current operatorwith non-
zeroKKmodenumberinthelattercase. Surroundp byanS4,onwhichwehaveacertainstate.
LetthesizeofS4 besufficientlylargerthanthecharacteristiclengthscaleofthesystemsetby
themassdeformationortheinverseradiusofS1. ThenthestateonS4 can beanalyzedusing
thegaugetheory.WhentheinstantonnumberofthegaugeconfigurationonS4 isnonzero,we
calltheoriginaloperatorinsertedatp aninstantonoperator.
Itshouldbepossibletostudythesupermultipletstructureofinstantonoperatorsindetail
using supersymmetric 5d Lagrangians on S4 times R or S1, using the results in e.g. [11,32–
34]. Hereweonlyprovidearatherimpressionisticanalysisofone-instantonoperators,i.e.the
instantonoperatorswhentheinstantonnumberonS4 isone.
Intherestofthissectionwegatherknownfactsonone-instantonmodulispacesandfermion
zeromodes. Wewillbebrief;acomprehensiveaccountonSU(N)instantonscanbefoundin
e.g.[35]. Forinstantonsofgeneralgaugegroups,seee.g.[36,37].
WhenthegaugegroupisSU(2): AnyinstantonconfigurationonS4 canbeobtainedbycon-
formaltransformationsfromoneonaflatR4.WhenthegaugegroupisSU(2),theone-instanton
modulispaceonR4hastheform
R4×R ×S3/Z , (2.6)
>0 2
whereR4parametrizestheposition,R thesize,andS3/Z thegaugerotationatinfinity.When
>0 2
the instanton is mapped toS4, the first two factors R4 and R combine to form the ball B5
>0
withastandardhyperbolicmetric.5 TheasymptoticinfinityofR4 ismappedtoapointonS4,
thereforethegaugesymmetrythereshouldreallybegauged. Thereforewelosethelastfactor
S3/Z ,butweshouldrememberthatthegaugegroupSU(2)isbrokentoZ .
2 2
ApointinB5veryclosetoapointx inS4describesanalmostpoint-likeinstantonlocalized
at x. A point at the center of B5 corresponds to the largest possible instantonconfiguration,
thatisinfactSO(5)invariant. Thisinvariantconfigurationcanbeidentifiedwiththepositive-
chiralityspinorbundleontheroundS4. ThisconfigurationisalsoknownasYang’smonopole
[42].
AWeylfermionofthecorrectchiralityinthedoubletofSU(2)gaugegrouphasjustonezero
modeonaone-instantonconfiguration.OnS4,thisisasingletofSO(5)rotationalsymmetry.
A Weyl fermion of the correct chirality in the adjoint of SU(2) gauge group has four zero
modes. Two are obtained by applying supertranslations and the other two by applying spe-
cialsuperconformaltransformationstotheoriginalbosonicconfiguration.Whenconformally
mapped to S4, these four modes transform in the spinor representation of SO(5). They are
5ThehyperbolicballB isalsoknownastheEuclideanAdS . Thisfactwasusedeffectivelyinaseriesofearly
5 5
worksrelatingAdS/CFTandinstantons,seee.g.[38–41].
4
exactlythemodesobtainedbyapplyingthe5dsupersymmetrytothebosonicinstantoncon-
figuration. Inthissensetheinstantonconfigurationbreaksallthesupersymmetryclassically,
butthisdoesnotmeanthattheinstantonoperatorisinageneric,longmultipletofsupersym-
metry,aswewillsoonsee.
Whenthegaugegroupis general: When the gauge group is a general simplegroupG, any
one-instantonconfigurationisobtainedbyembeddinganSU(2)one-instantonconfiguration
bya homomorphismϕ :SU(2)→G determinedbya longroot. Therefore, theone-instanton
modulispaceonR4isoftheform
R4×R ×G/H (2.7)
>0
where H is the part ofG that is unbroken by the SU(2) embedded by ϕ. A further quotient
G/(SU(2)×H)isknownastheWolfspaceoftypeG.TheformofH iswell-known;hereweonly
notethatforG =SU(N)wehaveH =U(1)×SU(N −2).
TheconformaltransformationtoS4 againcombinesR4andR totheball B5,andwelose
>0
G/H asbefore. TheremainingeffectisthatwehaveH astheunbrokengaugesymmetry.Ana-
lyzingthefermioniczeromodesofanarbitraryrepresentationRofG aroundthisconfiguration
isnotanyharderthanforSU(2),sincetheactualgaugeconfigurationisstillessentiallythatof
SU(2). WeonlyhavetodecomposeR underSU(2)×H,andtoutilizeourknowledgeforSU(2).
3 SU(2)
N
3.1 Pure =1theory
Afterthesepreparations,letusfirstconsiderone-instantonoperatorsofpureN=1SU(2)gauge
theory.Asrecalledintheprevioussection,theone-instantonmodulispaceonS4isjusttheball
B5. WeconsiderastatecorrespondingtothelowestSO(5)-invariantwavefunction.
Now we need to takefermioniczero modes intoaccount. Thegauginois a spinorfield in
fivedimensions,thatgivestwoWeylfermionswiththecorrectchiralityintheadjointofSU(2)
gauge group and in the doublet of Sp(1) when restricted onS4. As recalled in the previous
R
section,thezeromodesareinthedoubletofSp(1) andinthespinorofSO(5)rotationalsym-
R
metry. Letusdenotethembyλ ,wherei =1,2isforSp(1) andα=1,2,3,4isforSO(5). The
iα R
symplectic Majorana condition in 5d means that we need to quantizethese zero modes into
‘gammamatrices’satisfying
{λ ,λ }=ε J . (3.1)
iα jβ ij αβ
Thestatesonwhichthesezeromodesact,then,canbefoundbydecomposingtheDiracspinor
representationofSO(8)intermsofitssubgroupSp(1) ×SO(5)suchthatthevectorrepresen-
R
tationofSO(8)becomesthedoublettimesthequartet.Wefindthefollowingsixteenstates:
(cid:12)µ+(ij)E, (cid:12)ψ+iα¶, (cid:12)Jµ+E, (3.2)
(cid:12) (cid:12)
(cid:12)
(cid:12) (cid:12)
5
that form exactly the broken current supermultiplet(2.3) recalled in the last section. We put
theplussignsassuperscriptstoremindusthattheyareone-instantonoperators.
LetusassumethatthisgaugetheoryisamassdeformationofaUV5dSCFT.ThentheUV
SCFTshouldsimultaneouslyhaveboththemultipletthatcontains J+ inthegaugetheoryand
µ
themultipletthatbecomestheinstantonnumbercurrent
J0∝ε trF F . (3.3)
µ µνρσυ νρ συ
Now, J+hascharge1under J0,because J+isaone-instantonoperator.Therefore,theyshould
µ µ µ
formanSU(2)flavorsymmetrycurrent. Thisconclusionagreeswiththeoriginalstringyanal-
ysis[1].
Effectofthediscretethetaangle. Intheanalysissofar,weneglectedtheeffectofthediscrete
thetaangleoftheSU(2)theory. Thethetaangleisassociatedtoπ (SU(2))=Z ,andtherefore
4 2
only takes the values θ = 0 or π. On a five-manifold of the form M ×S1 such that the SU(2)
configurationonM hasinstantonnumber1andthereisanontrivialZ holonomyaroundS1,
2
thethetaangleθ =πassignsanadditionalsignfactor−1inthepathintegral.6 OnR4×S1,this
has an effect that makes the wavefunctions on the one-instantonmodulispace sections of a
nontriviallinebundlewithholonomy−1onS3/Z .
2
Inoursetup,thesupermultiplet(3.2)iskeptwhenθ =0,butisprojectedoutwhenθ =π.
Therefore,thereisanenhancementoftheinstantonnumbersymmetrytoSU(2)whenθ =0,
butweseenoenhancementwhenθ =π. Thiseffectofthediscretethetaanglematcheswhat
wasfoundoriginallyin[2]usingstringyanalysis.
A caveat here is that in our crude analysis, we can only say that the states considered so
far do not give any broken current supermultiplet. It is logically possible that exciting non-
zero modesintheone-instantonsectororconsideringoperatorswithinstantonnumber2or
largergivesrisetobrokencurrentsupermultiplet,enhancingtheinstantonnumbersymmetry
toSU(2)evenwithθ =π. Acarefulstudyonthispointisdefinitelyworthwhile,butisoutside
ofthescopeofthispaper. Thesamecaveatisalsoapplicabletotherestofthearticle,butwe
willnotrepeatit.
3.2 Withfundamentalflavors
Next, let us consider N=1 SU(2) theory with N flavors in the doublet. Stated differently, we
f
add 2N half-hypermultipletsin the doublet of SU(2). At the classical Lagrangian level, they
f
transformunderSO(2N )symmetry.
f
Inaone-instantonbackground,theygive2N fermioniczeromodes,thatneedtobequan-
f
tizedasgammamatrices:
{Γa,Γb}=δab (3.4)
wherea =1,...,2N istheindexofthevectorrepresentationofSO(2N ).TheyactontheDirac
f f
spinorrepresentationS ⊕S ofSO(2N ).Tensoringwiththeresult(3.2)ofthequantizationof
+ − f
6Foraderivation,seee.g.theappendixAof[43].
6
theadjointzeromodes,wefindthattheone-instantonoperatorisabrokencurrentsupermul-
tipletintheDiracspinorofSO(2N ).
f
NowweneedtotaketheunbrokenZ gaugesymmetryintoaccount. Thegeneratorofthis
2
Z symmetryactsby−1onthedoubletrepresentation,andthereforeby−1onthegammama-
2
tricesin(3.4). Therefore,thisZ actsasthechiralityoperatoronS ⊕S . Therefore,depending
2 + −
onthevalueofthediscretethetaanglebeing0orπ,wekeeponlyone-instantonoperatorsin
S orS . ThesetwochoicesarerelatedbytheparityoperationofflavorO(2N )andtherefore
+ − f
equivalent.Inconclusion,wefoundthefollowingcurrentmultiplets:
• theconservedSO(2N )currents J[ab],
f µ
• theconservedinstantonnumbercurrent J0,
µ
• the broken currents coming from one-instanton operators J+,A where A is the chiral
µ
spinorindexofSO(2N ).
f
Take 1 ≤N ≤7. If we assume thatthe gaugetheoryis a mass deformationof a UV fixed
f
point,weseethatthecurrentslistedaboveneedtocombinetogivetheflavorsymmetryE .
Nf+1
This can be seen by attaching an additional node, representing a Cartan element for J0, to
µ
the node of the Dynkin diagram of SO(2N ) that gives the chiral spinor representation. For
f
example,wehave
◦
|
•−◦−◦−◦−◦−◦−◦ (3.5)
when N =7,wheretheblacknodeis fortheinstantonnumbercurrent.7 Thisenhancement
f
patternagreeswithwhatwasfoundoriginallyin[1].
LetusboldlytakeN =8. WenowhavethecombinedDynkindiagram
f
◦
|
•−◦−◦−◦−◦−◦−◦−◦ (3.6)
which is known as E or Eˆ . If we assume that the gauge theory is an outcome of a massive
9 8
deformation of some UV completion, the UV completion needs to have an affine E flavor
8
symmetryasa5dtheory. Stateddifferently,thismeansthattheUVcompletionneedstobea
6dtheorywithE flavorsymmetry. Thisconclusionmatcheswithwhatwasfoundin[5].
8
Weseea problemwhen N >8: thecombinedDynkindiagramdefinesa hyperbolicKac-
f
Moodyalgebra,whosenumberofrootsgrowsexponentially. Itisthereforeunlikelythatthere
isaUVcompletionwhenN >8.Again,thismatchestheoutcomeofadifferentanalysisin[1].
f
Beforeproceeding,wenoteherethatithasbeenwell-knownthattheinstantonparticlesin
thisgaugetheorygiverisetospinorsofSO(2N )flavorsymmetry. Theonlyslightlynewpoint
f
inthissectionisthatwhenstudiedinthecontextofinstantonoperators,theyareindeedpart
ofbrokencurrentmultipletsgivenby(3.2).
7Strictlyspeaking, thisprocedureisnotuniquewhenatheorywithverysmallN isconsideredalone. Asa
f
simplerexampleofthisissue,supposeweknowwehaveanSU(2)currents,aninstantonnumbercurrent,anda
brokencurrentcomingfromone-instantonoperatorsinthedoubletofSU(2). Thenwehavetwochoices,either
anSU(3)correspondingto•−◦oranSp(2)correspondingto•⇐◦. Theycanonlybedistinguishedbystudying
thetwo-instantonoperators. Inourcase,however,wecanjuststudytheN =7caseandthenapplythemass
f
deformationtomakeN smaller,toconcludethattheflavorsymmetryisalwaysE .
f Nf+1
7
N
3.3 =2theory
As a final example of SU(2) gauge theory, let us consider N=2 gauge theory. On the one-
instantonbackgroundonS4,wehavefourWeylfermionsintheadjointofSU(2),andthezero
modes can be denoted by λ where i = 1,2,3,4 is now for Sp(2) and α = 1,2,3,4 is for the
iα R
spacetime SO(5). Again, the symplectic Majorana condition in 5d means that they become
‘gammamatrices’withthecommutationrelation
{λ ,λ }= J J . (3.7)
iα jβ ij αβ
Thesegammamatricesactonthefollowingone-instantonstatesonS4:
(cid:12)µ+(ab)E, (cid:12)ψ+ai α¶, (cid:12)J[+ab]µE, (cid:12)Qi+µαE, (cid:12)T(+µν)E, (cid:12)Xa+[µν]E, (3.8)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
wherea =1,2,3,4,5 isthevectorindexforSp(2) =SO(5) . Thegamma-tracelessnesscondi-
R R
tiononψ+,Q+ andthetracelessnessconditiononµ+,T+ needtobeimposed.
Whenthediscretethetaangleiszero,theseoperatorsareallkept,andthestructureofthe
operators is exactly that of KK modes of the 6d N=2 energy-momentumsupermultiplet. For
example,thecurrents J+ intheadjointofSp(2) suggestthattheSp(2) symmetryofthe5d
[ab]µ R R
gaugetheoryenhances to theaffineSp(2) , and thesymmetrictraceless T+ is theKK mode
R (µν)
of the 6d energy-momentum tensor. This is as it should be, since theS1 compactification of
N=(2,0)theoryoftypeSU(2)onS1 isdescribedbyN=2SU(2)gaugetheoryin5d. Therelation
ofthe5dN=2theoryandthe6dN=(2,0)theoryonS1hasbeenextensivelystudied,seee.g.[44,
45].
When the discrete theta angleis π, theseoperators are all projected out. In this case too,
the gauge theory is the IR description of N=(2,0) theory of type SU(3) on S1 with an outer-
automorphismZ twist around it [43]. We expect that the operators with the same structure
2
toariseinasectorwithhigherinstantonnumber,buttocheckitisoutsideofthescopeofthis
paper.
4 SU(N)
N
4.1 Pure =1theory
Now let us moveon to SU(N) gaugetheories. Our first exampleis thepure SU(N) theory. As
recalledinSec.2,inone-instantonconfigurationsonS4,thegaugefieldstakevaluesinSU(2)⊂
SU(N). TheunbrokensubgroupisU(1)×SU(N −2). WetakethegeneratoroftheU(1)partto
be
diag(N −2,N −2,−2,−2,...,−2). (4.1)
Inthisnormalization,whentheChern-Simonslevelisκ,theone-instantonconfigurationhas
theU(1)charge(N −2)κ.
An adjoint Weyl fermion of SU(N) on this background decomposes into an SU(2) adjoint
Weylfermion,andN−2WeylfermionsinthedoubletofSU(2)inthefundamentalofSU(N−2)
8
and with U(1) charge N, and Weyl fermions that are neutral. The gaugino in 5d gives two
adjoint Weyl fermions of SU(N) onS4. The SU(2) adjointpart gives the same broken current
supermultiplet(3.2), and weneed totakeadditional2×(N −2)zero modescomingfrom the
SU(2)doubletintoaccount.
By quantizing them, we have fermionic creation operators B where i = 1,2 is now for
ia
Sp(1) and a = 1,...,(N −2) is for SU(N −2). The U(1) charge of B is N. The SU(N −2)
R ia
invariantstatesarethen
|0〉, εa1···aN−2B B ···B |0〉, (B )∧2(N−2)|0〉, (4.2)
i1a1 i2a2 iN−2aN−2 ia
withU(1)charge−(N −2)N,0,+(N −2)N,respectively.
When κ is neither 0 nor ±N, all states are projected out, due to the nonzero U(1) gauge
charge. Whenκis±N,onesingletstateiskept,andwehaveabrokencurrentsupermultiplet
(3.2).ThereforeweexpecttheenhancementoftheinstantonnumbersymmetrytoSU(2).This
enhancementwasrecentlydiscussedin[19].
When κ is 0, the one-instanton operators are the tensor product of the broken current
supermultiplet(3.2) times εa1···aN−2Bi1a1Bi2a2···BiN−2aN−2|0〉, which transforms in the N −1 di-
mensional irreducible representation of Sp(1) . This is a short supermultiplet, but does not
R
correspondtoabrokenflavorsymmetry.
4.2 Withfundamentalflavors
Our next example is SU(N) theory with N hypermultiplets in the fundamental representa-
f
tion.Onone-instantonconfigurationsonS4,eachflavordecomposesintoapairofSU(2)dou-
bletsandanumberofneutrals;theyallhaveU(1)chargeN −2. Therefore,wehaveadditional
fermioniccreation operatorsC , s =1,...,N , of U(1) charge N −2. They act on thestates of
s f
theform
C C ···C |0〉 (4.3)
s1 s2 sk
fork =0,...,N ,withU(1)charge(N −2)(k −N /2).
f f
Tensoring (3.2), (4.2) and (4.3) and imposing the U(1) gauge neutrality condition, we see
thatabrokensymmetrysupermultipletsurviveswhenwehave
κ±N +(k −N /2)=0. (4.4)
f
Now, in [4] it was shown that we need |κ| ≤ N −N /2 to have a 5d UV SCFT behind the
f
gaugetheory.Therefore|κ±N|≥N /2.Wealsotriviallyhave|k−N /2|≤N /2.Therefore,the
f f f
equality(4.4)canonlybesatisfiedwhen±κ=N −N /2.
f
When ±κ = N −N /2, the surviving broken current supermultipletcomes from k = 0 or
f
k = N in (4.3). They have instanton number one and baryonic charge ±N /2. Therefore,
f f
when κ = N −N /2, one U(1) enhances to SU(2), and when −κ = N −N /2, another U(1)
f f
enhancestoSU(2). Whenκ=N −N /2=0,thecombinationI ±B/N oftheinstantoncharge
f
I and the baryonic charge B both enhance to SU(2) , making the UV flavor symmetry to be
±
SU(N )×SU(2) ×SU(2) . Thisenhancementpatternwasfoundin[19,27].
f + −
9