dS3/CFT2 correspondence(3D de Sitter space/2D CFT correspondence)
Yasuaki Hikida (YITP, Kyoto U., CoI:C01)
Ref. YH, T. Nishioka (PI:D01), T. Takayanagi (HI&PI:C01), Y. Taki,arXiv:2110.03197; arXiv:2203.02852
March 8th, 2022@First Annual Meeting at YITP
Grant-in-Aid for Transformative Research Areas (A)The Natural Laws of Extreme Universe

de Sitter gravity and dS/CFT correspondence• de Sitter gravity• Quantum gravity in de Sitter (dS) space isimportant since the space approximately
appears at inflation era and current universe• dS/CFT correspondence• dS/CFT should play important roles onunderstanding quantum gravity on dS as
AdS/CFT does
• Unfortunately, dS/CFT has not been
understood yet compared with AdS/CFTFrom wikipedia

dS/CFT and its example• dS/CFT is poorly understood as very few concrete examples are available• de Sitter space is not allowed as supersymmetric solution to supergravity[Maldacena-Nunez’00], and it is very difficult to realize it in superstring theory• Dual CFT is known to be exotic like with negative/imaginary central charge• A concrete example is given by higher-spin holography
Higher-spin gravity on dS4
at the classical limit

3d Sp(N) vector modelat the large N limit[Anninos-Hartman-Strominger’11]• Analytic continuation of duality between higher-spin gravity on AdS4 and 3d O(N)vector model [Klebanov-Polyakov’02]

Our proposal• We propose a new duality involving 3d Einstein gravity
Einstein gravity on dS3
at the classical limit(+ matters)

2d SU(2) WZW modelwith large central charge(+ extra sectors)• Evidence• Partition functions are computed both from gravity theory and dual CFT withlarge central charge (k ! 2) and find perfect match at the limit• Our proposal can be regarded as an analytic continuation of duality with(higher-spin) gravity on AdS3 [Gaberdiel-Gopakumar’10]<latexit sha1_base64="w16Aw6tUPoC3XetCwnGWw8YpAhI=">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</latexit>

Plan of this talk• Introduction• dS/CFT correspondence• Our proposal and gravity partition functions• Analytic continuation of Gaberdiel-Gopakumar duality• Relation to quantum information theory• Conclusion

dS/CFT correspondenceAdS/CFT correspondence• AdS/CFT correspondence [Maldacena’97](d+1)-dim. gravity theory
on anti-de Sitter space(e.g. string theory on AdS5xS5)①②d-dim. conformal field theory(e.g. 4d N=4 U(N) gauge theory)• Two approaches① Strongly coupled gauge theory from classical gravity② Quantum gravity from well-defined CFT

AdS/CFT duality map• Map between CFT operators and AdS bulk fields
CFT operators

AdS bulk fields• Poincare coordinates for AdS space (boundary at)• GKP-Witten relation’98• CFT correlation functions ó gravity scattering amplitudes

Wave functional of de Sitter universe• A way to describe gravity theory on dS space isutilized wave functional of universe<latexit sha1_base64="sjmJbE1/NPzGxMY1V6o2h86We08=">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</latexit>dS [h,0] =Z

DgD exp iS[g, ]• Hartle-Hawking wave functional is used as aparticularly useful choice• Universe starts from the hemisphere• continues to Lorentzian dS space

dS/CFT correspondence• Wave functional of universe is proposed to be the same as
generating function of correlation functions in dual CFT• Late time correlators can be computed as expectation values[Maldacena’02]
Correlators are computedby dual Euclidean CFT

Gravity partition function on sphere• Gravity partition function is computed from square of wave functional• Sphere partition function from Hartle-Hawking wave functional• Lorentzian dS part leads to only a pure phase which cancels out2Our proposal andgravity partition functions

Our proposal• We propose a new duality involving 3d Einstein gravity
Einstein gravity on dS3
at the classical limit(+ matters)

2d SU(2) WZW modelwith large central charge(+ extra sectors)• Gravity partition functions• Partition functions on
computed from Einstein gravity are reproduced
from 2d SU(2) WZW model large central charge (k ! 2)
• We utilize Witten’s method to compute partition functions on
from2d SU(2) WZW model [Witten’89]<latexit sha1_base64="w16Aw6tUPoC3XetCwnGWw8YpAhI=">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</latexit>

Central charge and the level of WZW model• Virasoro symmetry appears near thefuture infinity with central charge [Strominger’01]
Radius of de Sitter space
Newton constant (classical limit)• Dual CFT is quite strange as it has pure imaginary central charge• 2d SU(2) WZW model with level kwith• To reproduce the requirement from dual gravity, we consider a strange limit

Sketch of Witten’s method[Witten’89]• We use Witten’s method to compute partition functions on
modular S-matrix of 2d SU(2) WZW model• Wilson loop in spin-j rep

can be inserted• Modular S-matrix relates two amplitudes
of solid tori related by S-transformation• Sphere can be obtained bygluing two tori related by
S-transformation via surgeryvia

Partition function on 3-sphere[YH-Nishioka-Takayanagi-Taki’21;’22]• CFT computation• Modular S-matrix of 2d SU(2) WZW model• Vacuum partition function at the leading order in2≈≈• Gravity computation• Definition of classical partition function• Classical action on:Reproduces CFT computation!!

Wilson loop and bulk excitation• Partition function onwith Wilson loop in rep.

• The Wilson line onó Operator in 2d WZW model [Witten’89]• Conformal dimension of CFT operator• dS/CFT map to bulk excitation energy

Partition function on Euclidean dS3 black hole
[YH-Nishioka-Takayanagi-Taki’21;’22]• CFT computation• The modular S-matrix leads at the leading order in2• Gravity computation• Bulk excitation creates Euclidean dS3 black hole
• Classical action on the geometry
Reproduces CFT computation!!

Analytic continuation ofGaberdiel-Gopakumar duality

Gaberdiel-Gopakumar duality for AdS3[Castro-Gopakumar-Gutperle-Raeymaekers’11;Gaberdiel-Gopakumar’12](see [Gaberdiel-Gopakumar’10] for original proposal)• A version of Gaberdiel-Gopakumar duality
Higher-spin gravity on AdS3
at the classical limit
Spins of gauge fields

2d coset model atlarge central charge
The coset describes analytic continuation ofWN-minimal model, which was shown to
reduce to Toda theory [Creutzig-YH’21]• The simplest case with N=2
Einstein gravity on AdS3 withmatter at the classical limit

2d coset model atlarge central charge

Central charge and the level of coset model• A version of Gaberdiel-Gopakumar duality
2d coset model atlarge central charge

Einstein gravity on AdS3 withmatter at the classical limit• Comparison of central charge• Near the boundary of AdS3 thereappears Virasoro symmetry withcentral charge [Brown-Henneaux’86]•The central charge of the coset is•

To have large central charge, we have to setAnalytic continuation from AdS3 to dS3[YH-Nishioka-Takayanagi-Taki’21;’22] (see [Ouyang’11] for previous attempt)• Formally we can move from AdS3 to dS3 by replacing LAdS ! i LdS<latexit sha1_base64="Ax3lOdgbcklLFqUgdCr20880Gl4=">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</latexit>• Note that this is possible only for (higher-spin) holography• Gaberdiel-Gopakumar duality becomes
Einstein gravity on dS3 withmatter at the classical limit

2d coset model withimaginary central charge• Comparison of central charge

At the leading order inonly SU(2)k part dominates andthe duality reduces to our proposal of dS3/CFT2 correspondenceRelation to quantum informationtheory

Holographic entanglement entropy[YH-Nishioka-Takayanagi-Taki’22](see also [Narayan’15;Sato’15;Miyaji-Takayanagi’15] for previous works)• Entanglement entropy of dual CFT at• Metric of dS3:
• Subsystem A:• Holographic computationA• Geodesic distanceReal part• Holographic entanglement entropy [Ryu-Takayanagi’06]• Reproduce the standard CFT result [Calabrese-Cardy’04]
Analytic continuation from AdS to dS

de Sitter entropy[YH-Nishioka-Takayanagi-Taki’21;’22](see also [McGough-Verlinde’13] for AdS3)• Quantum gravity should explain the origin ofblack hole entropy• Similarly dS gravity should explain dS entropyho• The metric of dS3 black hole

• dS entropy associated with horizonis• Topological entanglement entropy for our system is⇡
Stop = log |S0j |2 ' c(g)<latexit sha1_base64="G7/tYHGnAoxyQRD/sUBYjDHn/Cs=">AAACuHichVFNaxQxGH479aNu1a71UvASXCr14PJOXbQUhEIRPUnbcdtCpzvOpNkx7Xw1k12s6fgD/AMePCmIiL/As5f+AQ89ebV4rODFg+/ODogW9Q1Jnjx5nzdPkiCLZK4RD0es0VOnz5wdO1cbP3/h4kT90uRqnvYUF22eRqlaD/xcRDIRbS11JNYzJfw4iMRasLM42F/rC5XLNHmo9zKxGfthIruS+5oor+44nqvFE210mhXsDnOjNGT7jocd82y72O/Mspqby1jsMrerfG7cTBbmZsF4x8yE1wvm5rtKG5vdYHPsnveA3fW2C6/ewCaWwU4CuwINqGIprb8FF7YgBQ49iEFAAppwBD7k1DbABoSMuE0wxClCstwXUECNtD3KEpThE7tDY0irjYpNaD2omZdqTqdE1BUpGUzjJ3yHx3iA7/EIf/y1lilrDLzs0RwMtSLzJp5POd//q4pp1vD4l+qfnjV0Ya70Ksl7VjKDW/Chvv/0xbEzvzJtruFr/Er+X+EhfqQbJP1v/M2yWHkJNfoA+8/nPglWZ5v2rWZrudVYaFVfMQZX4CrM0HvfhgW4D0vQpnM/wGf4AkfWvPXICi05TLVGKs1l+C0s9RNUwKje</latexit>

p1 8GN Ej
3[Kitaev-Preskill’05;Levin-Wen’05]

reproduces the gravity resultnozrieurtFuPa
sthorizonConclusion

Summary & future problems• Summary• dS/CFT correspondence should play important roles on understanding
quantum gravity on dS space• New dS/CFT is proposed between 3d classical Einstein gravity and 2d SU(2)WZW model with k ! 2 limit• Evidence is provided by computing gravity partition functions and relating to(higher-spin) AdS3 holography• Lorentzian contributions and quantum corrections in our dS/CFT were also
analyzed [YH-Nishioka-Takayanagi-Taki’22]<latexit sha1_base64="eQZXgQ1LJc0pSSZGXwS+XUBRCkc=">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</latexit>• Future problems• Pursue on relation to quantum information theory furthermore• Bulk correlation functions at late time and relation to in-in formulation are
now investigated [Chen-YH-Nishioka in progress] (see also [Sleight-Taronna’20;’21])