Non-perturbative gauge couplings from holography

0.1 MB·English

by  Marco Billo

Non-perturbative gauge couplings from holography
MarcoBillo`1∗,MarialuisaFrau1,LucaGiacone1,andAlbertoLerda2
1 DipartimentodiFisica,Universita`diTorino
andI.N.F.N.-sezionediTorino,ViaP.Giuria1,I-10125Torino,Italy
2 DipartimentodiScienzeeInnovazioniTecnologiche,Universita`delPiemonteOrientale
andI.N.F.N.-GruppoCollegatodiAlessandria,VialeT.Michel11,I-15121Alessandria,Italy
2
1 Keywords F-theory,gravitydual,non-perturbativecorrections.
0
WeshowhowD-instantoncorrectionsmodifythedilaton-axionprofileemittedbyanO7/D7systemturning
2
itintothenon-singularF-theorybackgroundwhichcorrespondstotheeffectivecouplingonaD3probe.

n
a
J

0
2
1 Introduction

]
h
WeconsiderthelocallimitofTypeI′stringtheorywithN D7branesclosetoanO7planeandstudythe
t f
-
profileofthecorrespondingdilaton-axionfieldτ. Attheperturbativelevel,τ isnon-trivialandpossesses
p
e logarithmicsingularitiesattheorientifoldandbranepositions. Thesesingularitiesareincompatiblewith
h its roˆle as string couplingconstant, and mustthereforebe resolvedbynon-perturbativeeffects, resulting
[
intoanon-singularF-theorybackground. TheNf = 4casewasconsideredlongagobyA.Sen[1]who,
1 basedonthesymmetriesandmonodromypropertiesoftheTypeI′ configuration,suggestedthattheexact
v dilaton-axionprofilebe givenbythe effectivecouplingofthe 4dN = 2SYM theorywith gaugegroup
1 SU(2)andN = 4flavors, asencodedin thecorrespondingSeiberg-Wittencurve[2]. Thisisthegauge
f
3 theorysupportedbyaprobeD3braneinthelocalTypeI′model,andthustheF-theorybackgroundcanbe
2
interpretedasthegravitydualoftheeffectivegaugecouplingoftheD3braneworld-volumetheory[3].
4
. Here we provide a microscopic description of how the exact F-theory background arises when D-
1
instantonsareintroducedintheD7/O7systemandshowhowtheymodifythesourcetermsintheτ field
0
2 equation[4]. ThecomputationrequiresintegratingovertheD-instantonmodulispaceandthisisdonevia
1 localization techniques [5,6] that allow to obtain explicit results even when all instanton numbers con-
: tribute. Inthiswaywedemonstratehowthenon-perturbativecorrectionstotheeffectivegaugecoupling
v
i are incorporatedin the dualgravitationalsolution. Theagreementoftheexactdilaton-axionprofilethus
X
obtainedwith the couplingconstantofthe D3 branegaugetheorypersists allthe way downto N = 0,
f
r whichamountstosaythatanO7planeplusitsD-instantoncorrectionsrepresentsthegravitationalsource
a
forthegaugecouplingofthepureSU(2)N =2SYMtheoryin4d.
The local configuration we consider contains N D7 branes; the massless excitations of the D7/D7
f
open strings describe a gauge theory in eight dimensions and the orientifold projection implies that its
gaugegroupisSO(N ). Thesedegreesoffreedomcanbeassembledintoanadjointchiralsuperfield
f
1
M =m+θψ+ θγµνθF +··· . (1)
µν
2
A D3-brane(plusits orientifoldimage)in thisbackgroundsupportsa four-dimensionalSp(1) ∼ SU(2)
gauge theory with N hypermultiplets, arising from the D3/D7 strings, and flavor group SO(N ). For
f f
N = 4 this theory has vanishing β-function, and we will mostly consider this case, except in the last
f
section.
∗ Correspondingauthor E-mail:[email protected],Phone:[phone],Fax:[phone].
2 M.Billo`,M.Frau,L.Giacone,andA.Lerda:Holographiccouplings
The transverse space to the O7 plane and the D7 branes (parametrized by a complex coordinate z)
correspondsto the Coulomb branchof the modulispace of the theory: placing the probeD3 brane in z
(and its image in −z) amounts to give a vacuum expectation value φ = (a,−a), with a = z , to
cl 2πα′
the SU(2) complex adjoint scalar. On the other hand, displacing the D7 branes in z (i = 1,...,N )
i f
correspondstogivingavacuumexpectationvalue
z
m =(m ,...,m ,−m ,...,−m ), with m = i , (2)
cl 1 Nf 1 Nf i 2πα′
totheSO(N )complexadjointscalarmofeq.(1).IntheD3braneeffectiveaction,them ’srepresentthe
f i
massesofthehypermultiplets,whiletheroˆleofthecomplexifiedgaugecouplingisplayedbythedilaton-
axion field τ belonging to the closed string sector. Actually, τ is the first componentof a chiral scalar
superfieldT inwhichallrelevantmasslessclosedstringsdegreesoffreedomcanbeorganizedandwhich
isschematicallygivenby[7]
T =τ +θλ+···+2θ8 ∂4τ¯+··· , (3)
(cid:0) (cid:1)
where∂ standsfor ∂ . BoththeO7planeandtheD7branescoupletoT andproduceanon-trivialprofile
∂z
for it. This fact allows to establish an explicit gauge/gravityrelation: the effective couplingτ(a) of the
SU(2)theoryontheprobeD3braneisthedilaton-axionbackgroundτ(z)producedbytheD7/O7system.
Suchabackgroundisna¨ıvely(i.e. perturbatively)singularbut,aswewillshowinthenextsections,itcan
be promotedto a full-fledgednon-singularF-theorybackgroundby takingintoaccountnon-perturbative
D-instantoncorrections. Onthegaugetheoryside,thisamountstopromotetheperturbativeSU(2)gauge
couplingtotheexactoneencodedinthecorrespondingSeiberg-Wittencurve.
2 The dilaton-axionprofile
Aswementionedabove,theD7branesandtheO7planeactassourcesforτ, localizedinthetransverse
directions. Theclassicalperturbativedilaton-axionprofilecorrespondingtoN = 4D7branesplacedin
f
z isgivenby
i
4 z−z z+z ∞ (2πα′)2ℓ trm2ℓ
2πiτ (z)=2πiτ + log i +log i =2πiτ − cl , (4)
cl 0 Xh z z i 0 X 2ℓ z2ℓ
i=1 ℓ=1
where in the second step we used eq.(2). This profile, which matches the 1-loop running of the gauge
couplingoftheSU(2)SYMtheorywithN =4flavors,canbeobtainedbycomputingthe1-pointfunction
f
oftheτ emissionvertexwiththeboundarystatesoftheD7branesandthecrosscapstateoftheO7plane.
Thedilaton-axion(4)solvestheequationofmotion(cid:3)τ =J δ2(z),wheretheclassicalcurrent
cl cl
∞ (2πα′)2ℓ
J =−2i trm2ℓ ∂2ℓ (5)
cl (2ℓ)! cl
X
ℓ=1
arisesfrominteractionsontheD7world-volumebetweenτ andtheSO(8)adjointscalarmwhenthelatter
isfrozentoitsvacuumexpectationvalue(2). Suchinteractionscanbeobtainedfromasourceactionofthe
form
1
S = − d8xJ τ¯ (6)
source (2π)3(2πα′)4Z cl
where the dimensionful coefficient is the ratio of the D7 brane tension and the gravitational coupling
constant,whichistheappropriatenormalizationforaD7sourceaction[4]. UsingthesuperfieldsM and
Holographiccouplings 3
T ofeq.s(1)and(3),we caneasilyrealizethattheaboveinteractionscanbederivedfromthefollowing
perturbative8dprepotential
(2πα′)2ℓ−4
F (M,T)=2πi tr M2ℓ∂2ℓ−4T . (7)
cl (2ℓ)!
X
ℓ
ComparingthecorrespondingclassicalactionS = 1 d8xd8θ F (M,T)withthedefinition(6)of
cl (2π)4 cl
thesourceaction,wethusobtain R
(2πα′)4 δF (2πα′)4
J =− cl ≡ − δ¯F , (8)
cl 2π δ(θ8τ¯)(cid:12)(cid:12)T=τ0,M=mcl 2π cl
(cid:12)
whereweintroducedthehandynotationδ¯⋆ ≡ δ⋆ .
From this analysis it is clear how one shouldδ(pθ8rτo¯)c(cid:12)eTe=dτt0o,Mo=btmaicnl the complete dilaton-axionsource J.
(cid:12)
Onehasfirsttopromotetheclassicalprepotentialtothefullonebyincludingnon-perturbativecorrections,
i.e. F(M,T)=F (M,T)+F (M,T),andthenwrite,inanalogytoeq.(8),J =−(2πα′)4 δ¯F.
cl n.p. 2π
Thenon-perturbativecontributiontotheprepotentialariseswhenD-instantonsareaddedtotheD7/O7
system; in thiscase newtypesofexcitationsappearcorrespondingtoopenstringswith atleastoneend-
point on the instantonic branes, i.e. D(–1)/D(–1)or D(–1)/D7 strings. Due to the boundaryconditions,
theseexcitationsdonotdescribedynamicaldegreesoffreedombutaccountinsteadfortheinstantonmod-
uli,whichwecollectivelydenotebyM ,wherek istheinstantonnumber. Amongthem,onefindsthe
(k)
coordinatesofthecenterofmassandtheirfermionicpartners,whichcanbeidentifiedwiththe8dsuper-
spacecoordinatesxandθ, respectively. Theinteractionsamongthemoduliareencodedintheinstanton
actionandcanbecomputedsystematicallybystringdiagramsasdescribedinRef.s[8,9]. Inthecaseat
hand,theinstantonactioncanbewrittenas
S (M ,M,T)=S(M )+S(M ,M)+S(M ,T) (9)
inst (k) (k) (k) (k)
whereS(M )isthepuremoduliaction,whichcorrespondstotheADHMmeasureonthemodulispace,
(k)
S(M ,M)isthemixedmoduli/gaugefieldsactionandfinallyS(M ,T)isthemixedmoduli/gravity
(k) (k)
action.Herewefocusonthemostrelevantpartforourgoal,namelyS(M ,T),andrefertotheliterature
(k)
for the other terms[4,10]. To obtain S(M ,T), we compute mixed open/closedstring disk diagrams
(k)
involvinginstanton moduliand bulk fields. The simplest diagramsyield the “classical” instantonaction
−2πikτ supersymmetrizedbyinsertionsofθmoduli,sothatτ getsreplacedbythesuperfieldT,resulting
in−2πikT. OthermixeddiagramscontributingtoS(M ,M)involvethebosonicmodulusχ,whichis
(k)
akintothepositionoftheD(–1)’sinthetransversespacetotheD7’s,butwithanti-symmetricChan-Paton
indices due to the orientifoldprojection. Such diagramsturn out to be exactly computable, even if they
involveanarbitrary(even)numberofχinsertions. Moreover,theyaresupersymmetrizedbyθ-insertions
thatpromoteallτ occurrencestoT.Altogether,themixedmoduli/gravityactionis[4]
∞ (2πα′)2ℓ
S(M ,T)=−2πi tr(χ2ℓ)p¯2ℓT , (10)
(k) (2ℓ)!
X
ℓ=0
wherep¯isthemomentumconjugatetoz.
Giventhecompleteinstantonaction(9),onecanobtainthenon-perturbativeeffectiveactionontheD7
branesbyperforminganintegralovertheD(–1)modulispace,namely
Sn.p. = Z dM(k)e−Sinst(M(k),M,T) =Z d8xd8θFn.p.(M,T), (11)
X
k
where in the last step we have explicitly exhibited the integral over the superspace coordinates x and θ
todefinethenon-perturbativeprepotential. Thelatterthereforearisesfromanintegraloverallremaining
4 M.Billo`,M.Frau,L.Giacone,andA.Lerda:Holographiccouplings
instantonmoduli,alsocalledcenteredmoduli.Suchanintegralcanbeexplicitlycomputedusinglocaliza-
tion techniques[5,6]. Thisamountsto selectoneof thepreservedsuperchargesasa BRST chargeQso
thattheinstantonaction(9)isQ-exact,andtoorganizetheinstantonmoduliinBRSTdoubletssothatthe
integraloverthemreducestotheevaluationofdeterminantsaroundthefixedpointsofQ. Inordertohave
isolatedfixedpoints,theinstantonactionmustbedeformedbysuitableparameters(toberemovedatthe
end)which in ourset-uparise froma particularRR graviphotonbackground[10,11]. Here, we will not
delveintothedetailsbutsimplyrecalltheessentialingredientsoftheprocedure.
Onefirstintroducesthek-instantonpartitionfunctionZ accordingto
k
Zk =Z dM(k)e−Sinst(M(k),M,T;E), (12)
whereE isthedeformationparameter.Then,settingq =e2πiτ0 andZ0 =1,onewritesthegran-canonical
partitionfunctionZ = ∞ qkZ ,fromwhichoneobtainsthenon-perturbativeprepotential
k=0 k
P
∞
F = lim ElogZ = qkF . (13)
n.p. k
E→0 X
k=1
Forexample,F1 =limE→0EZ1,F2 =limE→0E Z2−21Z12 andsoon.Incompleteanalogywitheq.(8),
onethen writestheinstanton-inducedsourcefor(cid:0)the dilaton(cid:1)-axionasJ = −(2πα′)4 δ¯F , so thatits
n.p. 2π n.p
q-expansion involves the variations δ¯F , which in turn are related to the variations δ¯Z . For example,
k k
δ¯F1 = limE→0Eδ¯Z1, δ¯F2 = limE→0E δ¯Z2 −Z1δ¯Z1 and so on. Given the explicit form (10) of the
moduliaction,itreadilyfollowsthat (cid:0) (cid:1)
∞
δ¯Z =4πi (2πα′)2ℓp¯2ℓ+4Z(2ℓ), (14)
k k
X
ℓ=0
whereweintroducedthe“correlators”oftheχ-moduliintheinstantonmatrixtheory
1
Zk(2ℓ) = (2ℓ)! ZdM(k) tr(χ2ℓ)e−Sinst(M(k),M,T;E)(cid:12)(cid:12)T=τ0,M=mcl . (15)
(cid:12)
Atthefirsttwoinstantonnumbersonefinds
∞
δ¯F =4πi (2πα′)2ℓp¯2ℓ+4 lim EZ(2ℓ),
1 X E→0 1
ℓ=0 (16)
∞
δ¯F =4πi (2πα′)2ℓp¯2ℓ+4 lim E Z(2ℓ)−Z Z(2ℓ) ,
2 Xℓ=0 E→0 (cid:0) 2 1 1 (cid:1)
and similar expressionscan be easily obtained for any k. The same combinationsof partition functions
Z andχ-correlatorsZ(2ℓ)appearinthecomputationoftheD-instantoncontributionstoaratherdifferent
k k
classofobservables,namelytheprotectedcorrelatorshtrmJiformingthechiralringoftheSO(8)gauge
theory defined in the 8d world-volume of the D7 branes. The non-perturbative part of the chiral ring
elementshaveaq-expansion,htrmJi = ∞ qkhtrmJi , whichcanbeexplicitlycomputedusing
n.p. k=1 k
localizationtechniquesasdiscussedinRef.[1P2]. Atthefirsttwoinstantonnumbersonefinds
(−1)ℓ (−1)ℓ
lim EZ(2ℓ) = htrm(2ℓ+4)i , lim E Z(2ℓ)−Z Z(2ℓ) = htrm(2ℓ+4)i (17)
E→0 1 (2ℓ+4)! 1 E→0 (cid:0) 2 1 1 (cid:1) (2ℓ+4)! 2
and so on. Using these results, we therefore find a very strict relation between the δ¯ variation of the
prepotentialandthenon-perturbativeSO(8)chiralring[13–15],namely
∞
(−1)ℓ
δ¯F =4πi (2πα′)2ℓp¯2ℓ+4 htrm2ℓ+4i , (18)
k (2ℓ+4)! k
X
ℓ=0
Holographiccouplings 5
which,takingintoaccountthefactthathtrm2i =0forallk,impliesthat
k
(2πα′)4 ∞ ∞ (2πα′)2ℓp¯2ℓ
J =− qkδ¯F =−2i (−1)ℓ htrm2ℓi . (19)
n.p. 2π k (2ℓ)! n.p.
X X
k=1 ℓ=1
Addingthisexpression(rewritteninthe z coordinatespace)tothe classicaltermJ ofeq.(8) yieldsthe
cl
completesourceJ. Solvingthefieldequation(cid:3)τ =Jδ2(z),wegetthentheexactdilaton-axionprofile
∞ (2πα′)2ℓ htrm2ℓi (2πα′)m
2πiτ(z)=2πiτ − =2πiτ + logdet 1− . (20)
0 X 2ℓ z2ℓ 0 D (cid:16) z (cid:17)E
ℓ=1
At the perturbative level, eq.(4) expressed that fact that the quantities trm2ℓ of the D7 theory act as a
cl
source for the dilaton-axion. This source, however, is non-perturbativelycorrected and the exact result
is obtainedbyreplacingthe classicalvacuumexpectationvalueswiththe fullquantumcorrelatorsin the
D7-branetheory,namelytrm2ℓ ≡ trhm2ℓi → htrm2ℓi. Furthermore,introducingtheoperator
cl
1 (2πα′)m
O (z)=τ + logdet 1− (21)
τ 0 2πi (cid:16) z (cid:17)
wecanrewriteeq.(20)asτ(z)= O (z) ,whichhasthetypicalformofaholographicrelation.
τ
(cid:10) (cid:11)
3 Comparisonwithgaugetheory results
Aswesaidabove,thechiralringelementshtrm2ℓiareexplicitlycomputablevialocalizationandforthe
first few valuesofℓ theirinstantonexpansioncan be foundin Ref. [12]. Using these resultsin (20) and
parametrizingthe transversedirectionswith z = 2πα′a in such a way thatall α′ factorsin the τ profile
are reabsorbed, we get an expression τ(a) that, by direct comparison, can be seen to be exactly equal
to the large-a expansion of the low-energy effective coupling of the SU(2) SYM theory with N = 4
f
massiveflavors,asderivedfromtheSeiberg-Wittencurve[13]. Wecanthereforerephrasethisresultinthe
followingrelation
τ (z)⇔τ (a) (22)
sugra gauge
wherea = z representsthe Coulombbranchparameter. Itis interestingtoremarkthatonthesuper-
2πα′
gravity side the non-perturbativecontributions to the dilaton-axion profile τ (z) are due to “exotic”
sugra
instantonconfigurationsinthe8dworld-volumetheoryontheD7branes,whileonthegaugetheoryside
thenon-perturbativeeffectsinτ (a)arisefromstandardgaugeinstantonsinthe4dSYMtheory.These
gauge
twotypesofcontributionsagreebecausetheyactuallyhavethesamemicroscopicorigin:inbothcasesthey
areduetoD(–1)branes,whichrepresent“exotic”instantonsfortheD7/O7systemandordinaryinstantons
fortheprobeD3branesupportingthe4dSYMtheory.
Therelation(22)canobviouslybeusedintwoways. Ontheonehand,itcanbeusedtoreadthegauge
couplingconstantofthe4dSU(2)theoryintermsofthequantumcorrelatorsinthe8dtheorywhichgauges
itsSO(8)flavorsymmetry. Thisistheapproachwehavediscussedsofar. Ontheotherhand,therelation
(22)canbeusedtoreadthe8dchiralringelementsintermsofthe4dgaugecoupling.Actuallythiscanbe
doneinanexactway,i.e. toallordersinq. Infact,usingrecursionrelationsofMatonetype,itispossible
toextractfromtheSeiberg-Wittencurvetheexactexpressionofanygivencoefficientoftheexpansionof
τ (a)ininversepowersofa. Forexample,fromtheexactcoefficientof 1 wecandeducethat
gauge a4
htrm4i=E (q)R2−6θ4(q)T +6θ4(q)T (23)
2 4 1 2 2
whereE istheEisensteinseriesof(almost)weight2,θ andθ areJacobiθ-functionsandR,T andT
2 2 4 1 2
arethequadraticandquarticSO(8)massinvariants(seeRef.[15]fordetails). Thisexpressionresumsthe
instantonexpansion ∞ qkhtrm4i inwhichonlythefirstfewtermswereknownbydirectevaluation,
k=1 k
andcanbegeneralizePdtoallotherchiralringelements.
6 M.Billo`,M.Frau,L.Giacone,andA.Lerda:Holographiccouplings
4 The pure SU(2)theory
The gauge/gravity relation (22) can be established also when some or all flavors are decoupled to re-
cover the asymptotically free theories with N = 3,2,1,0. In particular, from the gauge theory side
f
one can reach the pure SU(2) case by sending q → 0 and m → ∞, while keeping the combination
i
qm m m m ≡ Λ4 finite. Λ4 isthedimensionfulcountingparameterintheinstantonexpansionwhich
1 2 3 4
replacesthedimensionlessq of thesuperconformalN = 4 theory; in otherwordsΛ canbe interpreted
f
asthedynamicallygeneratedscaleofthepureSU(2)theory.Fromthesupergravitysidethedecouplingof
theflavorscorrespondstosendingallD7branesfarawayfromtheorigin,orequivalentlytoevaluatethe
dilaton-axionτ(z)atazmuchsmallerthantheD7branepositionsinsuchawaythatonlytheorientifold
O7planeactsasasourceforτ. InthiscasewecanthereforerepeatthesamestepsdescribedinSection2
andevaluatethedilaton-axionfieldemittedbyjusttheO7plane.
Atthenon-perturbativelevelwhenkD-instantonsareputontheorientifoldplane,themaindifference
withrespecttothecasewiththeD7’sisthatthespectrumofinstantonmodulicontainsonlyneutralmoduli
corresponding to open strings of type D(–1)/D(–1). The rest of the derivation remains as before. In
particulareq.s(12)and(15)arewell-definedevenintheabsenceoftheD7’s,andthusformulaslike(16)
continuetohold. Forexample,onefinds
12 105
lim EZ(2ℓ) =− δ , lim E Z(2ℓ)−Z Z(2ℓ) =− δ (24)
E→0 1 4! ℓ,0 E→0 (cid:0) 2 1 1 (cid:1) 4·8! ℓ,2
which are the analogue of eq.(17) for the pure SU(2) theory. Of course the interpretationas chiralring
elementsin a flavortheoryis no longerpossible since thereare no flavors. Using these results andtheir
generalizationsathigherinstantonnumbersk,wecanstillfindthevariationsδ¯F oftheprepotentialand
k
obtainfromthesethenon-perturbativesourcecurrentJ ,whosefirstinstantontermsare
n.p.
12 105
J =2i Λ4(2πα′p¯)4 +Λ8(2πα′p¯)8 +··· . (25)
n.p.
h 4! 4·8! i
Solvingthefieldequation(cid:3)τ (z)=J δ2(z)andexpressingtheresultintermsofa= z ,weget
n.p. n.p. 2πα′
Λ4 105 Λ8
2πiτ (a)=3 + +··· (26)
n.p. a4 32 a8
whichexactlycoincideswiththefirsttwoinstantoncontributionsasderivedfromtheSeiberg-Wittencurve
[6,16,17]. After adding the perturbativepiece 2πiτ −4log 4a2 , we can therefore concludethat the
0 Λ2
full dilaton-axionprofile sourcedby an O7 plane plus its D-in(cid:0)stant(cid:1)onscompletely agreeswith the exact
couplingofthepureSU(2)N =2SYMtheoryind=4.
ThisworkwassupportedinpartbytheMIUR-PRINcontract2009-KHZKRX.
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