On the minimal coloring number of the minimal diagram of ...
Minimal surfaces in q-deformed AdS₅ S⁵soken.editorial/sokendenshi/...1 Minimal surfaces in...
Transcript of Minimal surfaces in q-deformed AdS₅ S⁵soken.editorial/sokendenshi/...1 Minimal surfaces in...
11
Minimal surfaces in
q-deformed AdS₅×S⁵
13/11/2015@YITPWorkshopFieldTheoryandStringTheory
TakashiKameyama (Dept.ofPhys.,KyotoU.)
Basedon:[arXiv:1408.2189]and[arXiv:1410.5544]+α
collaboratedwithKentarohYoshida(Dept.ofPhys.,KyotoU.)
I.Introduction
2
• TypeIIBsuperstringonAdS₅×S⁵isrealizedasacosetsigmamodel[Metsaev-Tseytlin,’98]
[Bena-Polchinski-Roiban,’03]
Aremarkablefeature:anintegrablestructurebehindAdS/CFT
typeIIBsuperstringonAdS₅×S⁵ SU(N)SYM(largeNlimit)
• TheintegrabilityplaysanimportantroleintesangtheconjecturedrelaaonsintheAdS/CFT
e.g. anomalousdimensions,Wilsonloops….
AdS/CFTcorrespondence
• TheexistenceofLaxpairtheclassicalintegrablity
3
Inthistalk,wewillfocusontheclassicalintegrabilityonthestring-theoryside
4
IntegrabledeformaaonsoftheAdS/CFT
Nextstep
• Preservingtheintegrabilitywhiledeformingthebackground(symmetry)inanon-trivialway
• Herewefocusonaq-deformaaonofAdS₅×S⁵superstring
• Itwouldbesignificanttorevealadeeperintegrablestructurebehindgauge/gravitydualiaesbeyondtheconformalinvariance
[Delduc-Magro-Vicedo,’13]
5
II.q-deformation of
AdS5×S5 superstring
66
R:asoluaonofmodifiedclassicalYang-Baxterequaaon(mCYBE)
[Klimcik,’02,’08]
Deformedprinciplechiralmodels
mCYBE(non-split):
η:deformaaonparameter
• TheexistenceofaLaxpairclassicalintegrability
[Delduc-Magro-Vicedo,’13]
Integrabledeformaaon
Integrabledeformaaons:Yang-Baxtersigmamodels
• Generalizedtosymmetriccosetmodels
• typeIIBsuperstringonAdS₅×S⁵ [Delduc-Magro-Vicedo,’13]
NOTE:Anotherkindofintegrabledeformaaonsbasedon(non-modified)CYBE
[Kawaguchi-Matsumoto-Yoshida,’14][Matsumoto-Yoshida,’15]
Manyr-matriceshavebeenidenafiedwithsoluaonsoftypeIIBSUGRA
7
q-deformedsuperstringacaon [Delduc-Magro-Vicedo,’13,’14]
Deformaaonparameter:η Integrabledeformaaon
(ifXisaposiaveroot)
(ifXisanegaaveroot)
R-operator:
• TheexistenceofLaxpairsclassicalintegrablity
• SU(2,2|4)symmetryq-deformedSU(2,2|4)
• kappa-invariance
(Drinfeld-Jimbotype)
Groupelement:
0 (ifXisaCartan)
• Theq-deformedmetric(inthestringframe)andtheB-fieldwerederived
• SomeargumentstowardsthecompleteSUGRAsoluaon
8
Deformaaonparameter:
• Asingularitysurface(curvaturesingularity)existsat
Aq-deformedAdS₅×S⁵background
[Arutyunov-Borsato-Frolov,’13]
[Lunin-Roiban-Tseytlin,’14][Arutyunov-Borsato-Frolov,’15][Hoare-Tseytlin,’15]
• Apossiblegauge-theorydualhasnotbeenuncoveredyet
RRcouplingsfailtosaasfyeomofIIBSUGRA,despitethepresenceofκ-symmetry
9
Revealingthenatureofthesingularitysurface
Aninteresangissue
• GKP-likerotaangstringsoluaonshavebeenconsideredasprobes.
“GKP-likestringsneverstretchbeyondthesingularitysurface”
[Frolov,IGST14][T.K.,Yoshida,’14]
• TheVirasoroconstraintsimplyω
singularitysurface
ρ(σ)• WeconsideredtwokindsoflimitstoexpresstheenergyEasafuncaonofthespinSexplicitly
10
Inthelargeωcase:
• ThestringisconfinedtoanarrowregionneartheoriginofdeformedAdS
ω
singularitysurface
ρ₀
with
Theundeformedresultisreproducedprecisely[Gubser-Klebanov-Polyakov,’02]
Ashortstringlimit
Theundeformedlimit
• Spinbehavesas
Inthelimit:with,
11
thelengthofthestringbecomesmaximum
TheresultisquitedifferentfromtheGKPrelaaon
[Gubser-Klebanov-Polyakov,’02]
Alongstringlimit
ω
ρ₀
singularitysurface
ρ(σ)
12
III.A holographic setup
for q-deformed geometry
13
Observaaon
• Itwouldbeworthtryingtolookforacoordinatesystemwhichdescribesspaceameonlyinsidethesingularitysurface
Thed.o.f.areconfinedintotheregionenclosedbythesingularitysurface?
• ThecausalstructurearoundthesingularitysurfaceisverysimilartotheboundaryoftheglobalAdSspace
• ClassicalstringsoluaonssuchasGKP-likestringscannotstretchbeyondthesingularitysurface
e.g.Formasslessparacles,ittakesinfiniteaffineametoreachthesingularity
Thesingularitysurfacemightbetreatedastheholographicscreen
Ourconjecture
analogue:thetortoisecoordinatesforblackholes
14
Anothercoordinatesystemforq-deformedAdS5• Performingthecoordinatetransformaaon:
ismappedto
• Thesingularitysurfaceisnowlocatedatinfinityoftheradialdirecaon
15
IV.Minimal surfaces
16
Minimalsurfacesfortheq-deformedbackground
• Forthedeformedcase,weconsiderminimalsurfaceswhichendonthe‘‘boundary’’(singularitysurface)
• WithintheusualAdS/CFTcase,WilsonloopsarecalculatedbyanareaofanopenstringextendingtotheboundaryofAdS(minimalsurface)
• Forthispurpose,itishelpfultousePoincarécoordinatesforq-deformedAdS5
• Thesesoluaonsreducetousualsoluaonsintheundeformedlimit
Toseekforthemysteriousgauge-theorydual,minimalsurfacesmightbeagoodclue
Thesingularitysurfaceisnowlocatedatz=0(boundary)
whoseboundary(=0)shapeisacircle(radius=)
17
1)q-deformedAdS₂:aminimalsurfacewithacircularboundary
Ansatz:
withtheconformalgauge
Inducedmetric:
Soluaon:
• Weconstructedaminimalsurfacewhichendsattheboundaryoftheq-deformedAdSwiththePoincarécoordinates
NOTE:Anaddiaonalcontribuaon(totalderivaave)comingfromtheboundaryvanisheswhen
18
• Theminimalsurfaceareacanbecomputedwithoutanyregularizaaonincontrastwiththeundeformedcase
q-deformaaonmayberegardedasaUVregularizaaon
• EvaluaangtheclassicalEuclideanacaon(areaoftheminimalsurface)
Theresultwouldcomefromthefinitenessofthespace-likeproperdistancetothesingularitysurface
19
2)q-deformedAdS3×S1:acuspedminimalsurface
• Thebcistwolinesseparatedbyontheboundaryoftheq-deformedAdSandonthespherepart
• Thetwoconservedquanaaesare
• Theresulangequaaonsareellipacandtheclassicalsoluaonisexpressedasellipacintegralsoffirstandthirdkind
• Thestringsoluaonfitsinsideq-deformedAdS3×S1:
• Asworld-sheetcoordinateswecantakerandandtheansatzfortheothercoordinatesis
20
C=0.02
• Inthelimit:φ→π,thetwocurvesapproachanaparallellines
• InthecaseC<<1,theclassicalacaonleadstoarepulsivepotenaalUndeformedcase:[Drukker-Forini,’11]
• Astrongrepulsiveforcebetweenquarkandanaquarkiftheyarecloseenough
analogytogravitydualsfornon-commutaavegaugetheories
21
V. Summary
&Discussion
22
Summary
Wehavediscussedthenatureofthesingularitysurfaceoftheq-deformedAdS₅×S⁵superstringandclassicalstringsoluaons
• GKP-likestringscannotstretchbeyondthesingularitysurface
• Wehaveintroducedacoordinatesystemwhichdescribesthespaceameonlyinsidethesingularitysurface
• Areaofminimalsurfacesdoesnothavealineardivergence,incontrastwiththeundeformedcase
• Thesingularitysurfacemayberegardedastheholographicscreen
• Aquark-anaquarkpotenaalfromtheq-deformedAdS₅×S⁵hasananalogytogravitydualsfornon-commutaavegaugetheories
23
Outlook
�Apossiblegauge-theorydual?
�One-loopbetafuncaon?
• Tofindmoresupportfortheconjectureofthesingularitysurfaceacangasaholographicscreen
24