Minimal surfaces in q-deformed AdS₅ S⁵soken.editorial/sokendenshi/...1 Minimal surfaces in...

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1 Minimal surfaces in q-deformed AdS×S13/11/2015 @ YITP Workshop Field Theory and String Theory Takashi Kameyama (Dept. of Phys., Kyoto U.) Based on: [arXiv:1408.2189] and [arXiv:1410.5544] + α collaborated with Kentaroh Yoshida (Dept. of Phys., Kyoto U.)

Transcript of Minimal surfaces in q-deformed AdS₅ S⁵soken.editorial/sokendenshi/...1 Minimal surfaces in...

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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.)

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I.Introduction

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•  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

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Inthistalk,wewillfocusontheclassicalintegrabilityonthestring-theoryside

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IntegrabledeformaaonsoftheAdS/CFT

Nextstep

•  Preservingtheintegrabilitywhiledeformingthebackground(symmetry)inanon-trivialway

•  Herewefocusonaq-deformaaonofAdS₅×S⁵superstring

•  Itwouldbesignificanttorevealadeeperintegrablestructurebehindgauge/gravitydualiaesbeyondtheconformalinvariance

[Delduc-Magro-Vicedo,’13]

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II.q-deformation of

AdS5×S5 superstring

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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

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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)

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•  Theq-deformedmetric(inthestringframe)andtheB-fieldwerederived

•  SomeargumentstowardsthecompleteSUGRAsoluaon

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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

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Revealingthenatureofthesingularitysurface

Aninteresangissue

•  GKP-likerotaangstringsoluaonshavebeenconsideredasprobes.

“GKP-likestringsneverstretchbeyondthesingularitysurface”

[Frolov,IGST14][T.K.,Yoshida,’14]

•  TheVirasoroconstraintsimplyω

singularitysurface

ρ(σ)•  WeconsideredtwokindsoflimitstoexpresstheenergyEasafuncaonofthespinSexplicitly

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Inthelargeωcase:

•  ThestringisconfinedtoanarrowregionneartheoriginofdeformedAdS

ω

singularitysurface

ρ₀

with

Theundeformedresultisreproducedprecisely[Gubser-Klebanov-Polyakov,’02]

Ashortstringlimit

Theundeformedlimit

•  Spinbehavesas

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Inthelimit:with,

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thelengthofthestringbecomesmaximum

TheresultisquitedifferentfromtheGKPrelaaon

[Gubser-Klebanov-Polyakov,’02]

Alongstringlimit

ω

ρ₀

singularitysurface

ρ(σ)

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III.A holographic setup

for q-deformed geometry

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Observaaon

•  Itwouldbeworthtryingtolookforacoordinatesystemwhichdescribesspaceameonlyinsidethesingularitysurface

Thed.o.f.areconfinedintotheregionenclosedbythesingularitysurface?

•  ThecausalstructurearoundthesingularitysurfaceisverysimilartotheboundaryoftheglobalAdSspace

•  ClassicalstringsoluaonssuchasGKP-likestringscannotstretchbeyondthesingularitysurface

e.g.Formasslessparacles,ittakesinfiniteaffineametoreachthesingularity

Thesingularitysurfacemightbetreatedastheholographicscreen

Ourconjecture

analogue:thetortoisecoordinatesforblackholes

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Anothercoordinatesystemforq-deformedAdS5•  Performingthecoordinatetransformaaon:

ismappedto

•  Thesingularitysurfaceisnowlocatedatinfinityoftheradialdirecaon

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IV.Minimal surfaces

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Minimalsurfacesfortheq-deformedbackground

•  Forthedeformedcase,weconsiderminimalsurfaceswhichendonthe‘‘boundary’’(singularitysurface)

•  WithintheusualAdS/CFTcase,WilsonloopsarecalculatedbyanareaofanopenstringextendingtotheboundaryofAdS(minimalsurface)

•  Forthispurpose,itishelpfultousePoincarécoordinatesforq-deformedAdS5

•  Thesesoluaonsreducetousualsoluaonsintheundeformedlimit

Toseekforthemysteriousgauge-theorydual,minimalsurfacesmightbeagoodclue

Thesingularitysurfaceisnowlocatedatz=0(boundary)

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whoseboundary(=0)shapeisacircle(radius=)

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1)q-deformedAdS₂:aminimalsurfacewithacircularboundary

Ansatz:

withtheconformalgauge

Inducedmetric:

Soluaon:

•  Weconstructedaminimalsurfacewhichendsattheboundaryoftheq-deformedAdSwiththePoincarécoordinates

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NOTE:Anaddiaonalcontribuaon(totalderivaave)comingfromtheboundaryvanisheswhen

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•  Theminimalsurfaceareacanbecomputedwithoutanyregularizaaonincontrastwiththeundeformedcase

q-deformaaonmayberegardedasaUVregularizaaon

•  EvaluaangtheclassicalEuclideanacaon(areaoftheminimalsurface)

Theresultwouldcomefromthefinitenessofthespace-likeproperdistancetothesingularitysurface

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2)q-deformedAdS3×S1:acuspedminimalsurface

•  Thebcistwolinesseparatedbyontheboundaryoftheq-deformedAdSandonthespherepart

•  Thetwoconservedquanaaesare

•  Theresulangequaaonsareellipacandtheclassicalsoluaonisexpressedasellipacintegralsoffirstandthirdkind

•  Thestringsoluaonfitsinsideq-deformedAdS3×S1:

•  Asworld-sheetcoordinateswecantakerandandtheansatzfortheothercoordinatesis

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C=0.02

•  Inthelimit:φ→π,thetwocurvesapproachanaparallellines

•  InthecaseC<<1,theclassicalacaonleadstoarepulsivepotenaalUndeformedcase:[Drukker-Forini,’11]

•  Astrongrepulsiveforcebetweenquarkandanaquarkiftheyarecloseenough

analogytogravitydualsfornon-commutaavegaugetheories

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V. Summary

&Discussion

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Summary

Wehavediscussedthenatureofthesingularitysurfaceoftheq-deformedAdS₅×S⁵superstringandclassicalstringsoluaons

•  GKP-likestringscannotstretchbeyondthesingularitysurface

•  Wehaveintroducedacoordinatesystemwhichdescribesthespaceameonlyinsidethesingularitysurface

•  Areaofminimalsurfacesdoesnothavealineardivergence,incontrastwiththeundeformedcase

•  Thesingularitysurfacemayberegardedastheholographicscreen

•  Aquark-anaquarkpotenaalfromtheq-deformedAdS₅×S⁵hasananalogytogravitydualsfornon-commutaavegaugetheories

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Outlook

�Apossiblegauge-theorydual?

�One-loopbetafuncaon?

•  Tofindmoresupportfortheconjectureofthesingularitysurfaceacangasaholographicscreen

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