An exact AdS/CFT duality

Matthias Gaberdiel ETH Zürich

DIAS 8 July 2020

Based mainly on work with and Lorenz Eberhardt AAACBXicbVDLSsNAFJ3UV62vqEtdDBbBVUmkRd1IxY3LSu0Dmhgmk0k7dPJgZiKUkI0bf8WNC0Xc+g/u/BunaRbaeuDC4Zx7ufceN2ZUSMP41kpLyyura+X1ysbm1vaOvrvXFVHCMengiEW87yJBGA1JR1LJSD/mBAUuIz13fD31ew+ECxqFd3ISEztAw5D6FCOpJEc/TC0ewCuvnTkNaEkaEAFzqZ3dNxy9atSMHHCRmAWpggItR/+yvAgnAQklZkiIgWnE0k4RlxQzklWsRJAY4TEakoGiIVLr7DT/IoPHSvGgH3FVoYS5+nsiRYEQk8BVnQGSIzHvTcX/vEEi/XM7pWGcSBLi2SI/YVBGcBoJ9CgnWLKJIghzqm6FeIQ4wlIFV1EhmPMvL5Luac2s1y5u69XmZRFHGRyAI3ACTHAGmuAGtEAHYPAInsEreNOetBftXfuYtZa0YmYf/IH2+QMDApeh AdS / CFT duality

Much recent progress in theory has been related to AdS/CFT duality [Maldacena ’97, ...]

superstrings on SU(N) super Yang-Mills = AdS S5 theory in 4 dimensions 5 ⇥ 4d non-abelian similar to that appearing in the standard model of particle physics. Relation of parameters

The relation between the parameters of the two theories is

4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥

AdS radius in AdS radius in Planck units string units ‘t Hooft parameter Strong weak duality

For example, in the large N limit of gauge theory at large ‘t Hooft coupling

4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥ large small large (point particle) approximation is good for AdS description. This is interesting since it gives insights into strongly coupled gauge theories using supergravity methods. Strategy for a proof

There is very good evidence that at least the original duality is correct. However, we have not yet succeeded in proving it (and we therefore also do not know to which extent it generalises).

In order to make progress in this direction, we will consider

‣ Tensionless regime

‣ Lower dimensional version (AdS3) Weakly coupled gauge theory

The tensionless regime arises in another corner of parameter space where the gauge theory is weakly coupled

4 4 R 2 R 2 = N gstring = g = g N = l YM l YM Pl ⇥ s ⇥

large small

[Sundborg ’01] [Witten ’01] l `tensionless strings’ s ⇥ [Sezgin,Sundell ’01] Tensionless limit

In tensionless limit all string excitations become massless:

mass mass

spin spin

leading Regge trajectory Higher spin theory

Resulting theory has an infinite number of massless higher spin fields, which generate a very large gauge symmetry.

effective description in terms of Vasiliev Higher Spin Theory.

maximally symmetric/unbroken phase of Higher spin CFT duality

Proposal for higher spin subsector: [Klebanov-Polyakov ’02] [Sezgin-Sundell ’02]

higher spin theory 3d O(N) vector model on AdS4 in large N limit

Many precision tests of this proposal have since been performed, in particular by [Giombi, Yin ’09-’10] [Maldacena, Zhiboedov ’11-’12], … Lower dimensions

The other key ingredient is to consider the lower-dimensional case, in particular, string theory on AdS3.

The advantage of going to the 3d case is that

‣ Solvable world-sheet theory for strings on AdS3 exists [sl(2,R) WZW model] [Maldacena, (Son), Ooguri ’00 - ’01]

‣ Much better control over 2d CFTs AdS3

It has long been suspected that the CFT dual of string theory on 3 4

AdS(null) 3 S T ⇥ ⇥ is on the same of CFTs that also contains the symmetric theory

Sym ( 4) ( 4)N /S (null) N T T N ⌘ [Maldacena ’97], …. see e.g. [David et.al. ’02] AdS3

However, it was not known what precise string background is being described by the symmetric orbifold theory itself. see however [Larsen, Martinec ’99]

Conversely, it was not known what the precise CFT dual of the explicitly solvable world-sheet theory for strings in terms of an sl(2,R) WZW model is. [Maldacena, (Son), Ooguri ’00 & ‘01] AdS3

In fact, the only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model… AdS3

Long andIn fact, short the strings only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model…

Short string solution Long string solution The basic reason for this is that the WZW model describes the t background with pure NS-NS flux, which is known to have long string solutions.

I These long strings live close to the boundary[Seiberg, and Witten give ’99], rise to[Maldacena, a Ooguri ’00] continuum of excitations that are not present in the symmetric orbifold theory. So the spectrum cannot possibly match.

13 AdS3

Long andIn fact, short the strings only consensus was that the actual symmetric orbifold theory cannot be dual to the WZW model…

Short string solution Long string solution These long strings give rise to a continuum of t excitations that are not present in the actual symmetric orbifold theory.

I These long strings live close to the boundary[Seiberg, and Witten give ’99], rise to[Maldacena, a Ooguri ’00] continuum of excitations that are not present in the symmetric orbifold theory. So the spectrum cannot possibly match.

13 Joining forces…

Use insights from the HS/CFT duality to make progress with identifying the worldsheet description of symmetric orbifold. 3d version of Klebanov-Polyakov duality is:

AdS3: 2d CFT: higher spin theory N,k minimal models with a complex Win large N ‘t Hooft limit scalar of mass M with coupling

[MRG,Gopakumar ’10] Susy version

Just as in original Klebanov-Polyakov duality, this is purely bosonic. However, in order to establish relation to string theory, one can consider susy generalisation.

For susy case, the 2d CFT is the so-called Wolf space coset su(N + 2) so(4N + 4) k 1 u(1) , su(N) u(1) 

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 k+2  which is dual to an N=4 susy HS theory on AdS3. [MRG, Gopakumar ’13] hs theory in string theory

In the large k limit, the Wolf space coset becomes a subsector of [MRG, Gopakumar ’14]

3 4 AdS3 S T hs theory based on ⇥ ⇥ string theory shs2[]

0

Wolf space cosets AAACGnicbVDLSgMxFM34rPU16tJNsAiuykwV7LLgxmUF+4BOKZlMpg3NJENyRylDv8ONv+LGhSLuxI1/Y/oAtfVA4HDOPeTeE6aCG/C8L2dldW19Y7OwVdze2d3bdw8Om0ZlmrIGVULpdkgME1yyBnAQrJ1qRpJQsFY4vJr4rTumDVfyFkYp6yakL3nMKQEr9Vw/MEDoUDORB8LGIoIDzfsDIFqre+yNraxk/0ca99ySV/amwMvEn5MSmqPecz+CSNEsYRKoIMZ0fC+Fbk40cCrYuBhkhqV2B9JnHUslSZjp5tPTxvjUKhGOlbZPAp6qvxM5SYwZJaGdTAgMzKI3Ef/zOhnE1W7OZZoBk3T2UZwJDApPesIR14yCGFlCqOZ2V0wHRBMKts2iLcFfPHmZNCtl/7xcubko1arzOgroGJ2gM+SjS1RD16iOGoiiB/SEXtCr8+g8O2/O+2x0xZlnjtAfOJ/fdfOiWw== ! symmetric orbifold ! 4 4 (N) SymN T T ⌦ /SN ⇢ ⌘ ⇣ ⌘ Strategy

This viewpoint therefore suggests that the symmetric orbifold Sym ( 4) ( 4)N /S (null) N T T N ⌘ is dual to string theory at the tensionless point. The tensionless limit arises when the spacetime geometry is of string size, i.e. in the deeply stringy regime. In the context of the WZW description, this should be the situation where the level of the sl(2,R) affine theory takes the smallest possible value. WZW model

This led us to study the spacetime spectrum of the k=1 sl(2,R) WZW model systematically.

We have shown that this worldsheet description is indeed exactly dual to the symmetric orbifold.

An exact AdS/CFT duality

[MRG, Gopakumar ’18] [Eberhardt, MRG, Gopakumar ’18] [Eberhardt, MRG, Gopakumar ’19] Plan of talk

1. Introduction and Motivation

2. The k=1 WZW model

3. The formulation

4. Correlation functions

5. Conclusions and Outlook Plan of talk

1. Introduction and Motivation

2. The k=1 WZW model

3. The supergroup formulation

4. Correlation functions

5. Conclusions and Outlook NS-R WZW model

The perturbatively solvable world-sheet theory for AdS3 is formulated in terms of a WZW model based on the sl(2,R). For the case with the relevant algebra is [Maldacena, (Son), Ooguri ’00 & ‘01]

(1) sl(2, ) = sl(2, )k+2 3 free fermions (null) R R k ⇠ 3 bosonic: J ,J± (null) n n decoupled NS-R WZW model

The representations of the bosonic sl(2,R) affine algebra are characterised by the sl(2,R) reps of the highest weights. There are 2 classes of sl(2,R) reps that appear: [Maldacena, Ooguri ’00]

Discrete lowest weight reps: [short strings] + : C = j(j 1) ,J j, j =0 Dj 0 | i

Continuous reps: [long strings] Cj : C = j(j 1) = 1 + p2 , j, m with m ↵ +

(null) Z ↵ 4 | i 2 (j = 1 + ip) (null) 2 Spectral flow

Because of the Maldacena-Ooguri (unitarity) bound,

[Petropoulos ’90] 1 k +1 [Hwang ’91] MO-bound : (null) 2 2 [Maldacena, Ooguri ’00] the (discrete) spectrum is bounded from above. Additional states are spectrally flowed images of these two classes of representations. [Maldacena, Ooguri ’00] see also [Henningson et.al. ’91] outer automorphism of affine algebra Physical states

This description is covariant, i.e. we need to impose the physical state condition, e.g. in NS sector tot Gr =0 (r>0) (Ltot 1 ) =0. (null) 0 2 In particular, the second condition (mass-shell condition) implies that C 1 + h0 + N = (NS-sector)

(null) k 2 World-sheet conformal Casimir of sl(2,R) dim. of internal CFT Dual CFT

The dual (`spacetime’) CFT lives on the boundary of AdS3, and we have the identifications

CFT 3 CFT CFT + L0 = J0 ,L1 = J0 ,L1 = J0 , with a similar relation for the right-movers.

We are interested in the `tensionless’ regime of this theory. Since the level k is proportional to the size of the AdS3 space in string units, this should correspond to the smallest (non-trivial) value of k: k=1. Dual CFT

Thus we are led to study the physical spectrum of the (spacetime) theory for k=1 systematically.

As we shall see, the interesting part of the spectrum comes from the spectrally flowed continuous reps. Continuous reps

For the spectrally flowed continuous reps, the mass-shell condition (in the NS sector) at k=1 is

1 2 1 1 2 C wm w + N = where C = 4 + p (null) 4 2

Here m is the J 3 eigenvalue before spectral flow, (null) 0

and we have set h(null) 0 =0 (for simplicity).

For the continuous representations we can simply solve this equation for m. For the case of p=0 we then get Continuous reps

1 2 1 1 C wm w + N = with C = 4 (null) 4 2 1 w2 +1 m = N

(null) w 4 h i Continuous reps

1 2 1 1 C wm w + N = with C = 4 (null) 4 2 1 w2 +1 m = N

(null) w 4 h i Then observing that the actual J 3 eigenvalue is (null) 0

w N w2 1 h = m + = + . 2 w 4w Full spectrum

w N w2 1 h = m + = + . 2 w 4w

ground state energy in w-twisted modes w-twisted sector Symmetric orbifold formula for cycle length w! [MRG, Gopakumar ’18] [Giribet, Hull, Kleban, Porrati, Rabinovici ’18] see also [Giveon, Kutasov, Rabinovici, Sever ’05]. Note that for w=1 and N=0, this includes in particular chiral states (h=0) that correspond to massless higher spin fields! [MRG, Gopakumar, Hull ’17] [Ferreira, MRG, Jottar ’17] Full symmetric orbifold

Thus we recover the full single-particle spectrum of the symmetric orbifold (in the large N limit).

[MRG, Gopakumar ’18] [Giribet, Hull, Kleban, Porrati, Rabinovici ’18]

3 4 For AdS 3 S T at k=1, criticality implies that the bosonic⇥ su(2)⇥ factor appears at level -1, and thus the analysis in the NS-R sector is a bit formal.

To improve this, consider the same theory in so-called hybrid formalism. Plan of talk

1. Introduction and Motivation

2. The k=1 WZW model

3. The supergroup formulation

4. Correlation functions

5. Conclusions and Outlook Hybrid formalism

[Berkovits, Vafa, Witten ’99] In the hybrid formalism the world-sheet theory is described (for pure NS-NS flux) by the WZW model based on psu(1, 1 2) (null) | k together with the (topologically twisted) sigma model for T4. For generic k, this description agrees with the NS-R description a la MO. [Troost ’11], [MRG, Gerigk ’11] [Gerigk ’12] Hybrid formalism

For the following it will be important to understand the representation theory of psu(1, 1 2) (null) | 1 The bosonic subalgebra of this superaffine algebra is sl(2) su(2) (null) 1 1

Thus only n=1 and n=2 are allowed for the highest weight states. Short representations

A generic representation of the zero mode algebra psu(1, 1 2) (null) has the form | j (C↵, n)

j+ 1 j+ 1 j 1 j 1 2 2 2 2 (C↵+ 1 , n + 1)(C↵+ 1 , n 1)(C↵+ 1 , n + 1)(C↵+ 1 , n 1) 2 2 2 2 (Cj+1, n)(Cj , n + 2)2(Cj , n)(Cj , n 2)(Cj+1, n) ↵ ↵ · ↵ ↵ ↵ j+ 1 j+ 1 j 1 j 1 2 2 2 2 (C↵+ 1 , n + 1)(C↵+ 1 , n 1)(C↵+ 1 , n + 1)(C↵+ 1 , n 1) (null) 2 2 2 2

(Cj , n) (null) continuous rep ↵ rep of su(2) of of sl(2,R) dim = n+1. Short representations

A generic representation of the zero mode algebra psu(1, 1 2) (null) has the form | j (C↵, n)

j+ 1 j+ 1 j 1 j 1 2 2 2 2 (C↵+ 1 , n + 1)(C↵+ 1 , n 1)(C↵+ 1 , n + 1)(C↵+ 1 , n 1) 2 2 2 2 (Cj+1, n)(Cj , n + 2)2(Cj , n)(Cj , n 2)(Cj+1, n) ↵ ↵ · ↵ ↵ ↵ j+ 1 j+ 1 j 1 j 1 2 2 2 2 (C↵+ 1 , n + 1)(C↵+ 1 , n 1)(C↵+ 1 , n + 1)(C↵+ 1 , n 1) (null) 2 2 2 2

(Cj , n) (null) continuous rep ↵ Thus for k=1 need of sl(2,R) a short rep! Short representations

In fact, the only representations that are allowed are

j (C↵, 2) j+ 1 j 1 2 2 (C↵+ 1 , 1)(C↵+ 1 , 1) (null) 2 2 and the shortening condition actually implies that this is only possible provided that

1 j = NO CONTINUUM!

(null) 2 [Eberhardt, MRG, Gopakumar ’18] Short representations

The corresponding affine representation (null) has in fact many null-vectors, and thus, after includingF the ghosts, the contribution from [Gotz, Quella, Schomerus ’06] [Ridout ’10] psu(1, 1 2) (null) | 1 just reduces to the zero-modes

2 2 m mw+ w 3 2 ZAdS3 S = x q ⇥ 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 m Z+ X2 [Morally similar to su(2) at level 1, which only describes the degrees of freedom of a single boson!] Short representations

2 2 m mw+ w 3 2 ZAdS3 S = x q ⇥ 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 m Z+ X2 Finally, for every state from T4, exactly one term in this sum is picked out by the mass-shell condition.

One also finds that these are the only representations, i.e. no discrete representations appear. Physical spectrum

The rest of the analysis works essentially as in the NS-R formulation. Because now the continuum has disappeared (and there are no discrete representations ) we get exactly the (single-particle) spectrum of [Eberhardt, MRG, Gopakumar ’18] Sym ( 4) (null) N T where, as before, the spectral flow parameter w is to be identified with the length of the single cycle twisted sector in the symmetric orbifold (in the large N limit). Plan of talk

1. Introduction and Motivation

2. The k=1 WZW model

3. The supergroup formulation

4. Correlation functions

5. Conclusions and Outlook Correlators in sym orbifold

Correlation functions of (twist) fields in the symmetric orbifold can be described in terms of covering maps.

[Lunin, Mathur ’00] [Pakman, Rastelli, Razamat ’09]

zAAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHZ9RI4kXjxCIo8ENmR26IWR2dnNzKwJEr7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eTW1jc2t/LbhZ3dvf2D4uFRU8epYthgsYhVO6AaBZfYMNwIbCcKaRQIbAWj25nfekSleSzvzThBP6IDyUPOqLFS/alXLLlldw6ySryMlCBDrVf86vZjlkYoDRNU647nJsafUGU4EzgtdFONCWUjOsCOpZJGqP3J/NApObNKn4SxsiUNmau/JyY00nocBbYzomaol72Z+J/XSU1Y8SdcJqlByRaLwlQQE5PZ16TPFTIjxpZQpri9lbAhVZQZm03BhuAtv7xKmhdl77J8Xb8qVStZHHk4gVM4Bw9uoAp3UIMGMEB4hld4cx6cF+fd+Vi05pxs5hj+wPn8AeiDjPs= Locally, the effect of a w-cycle twist field is to introduce a w-fold covering. x = (z) z (z)=x + a(z z )w +

AAACGnicbVDLSgMxFM34rPVVdekmWISKWGZ8oBuh4EKXFWwrdMbhTprWYDIzJBm1HfodbvwVNy4UcSdu/BvTdhZaPXDh5Jx7yb0niDlT2ra/rInJqemZ2dxcfn5hcWm5sLJaV1EiCa2RiEfyMgBFOQtpTTPN6WUsKYiA00ZwczLwG7dUKhaFF7obU09AJ2RtRkAbyS84PVdArHSE3VMQAkq9rWN879t4GwMu9XZ6vr11ld71zdslrUgr7BeKdtkeAv8lTkaKKEPVL3y4rYgkgoaacFCq6dix9lKQmhFO+3k3UTQGcgMd2jQ0BEGVlw5P6+NNo7RwO5KmQo2H6s+JFIRSXRGYTgH6Wo17A/E/r5no9pGXsjBONA3J6KN2wrFJYpATbjFJieZdQ4BIZnbF5BokEG3SzJsQnPGT/5L6btnZKx+c7xcrR1kcObSONlAJOegQVdAZqqIaIugBPaEX9Go9Ws/Wm/U+ap2wspk19AvW5zf29Z5I AAAB9HicbVDLSgNBEOz1GeMr6tHLYBDiJez6wFyEgAc9RjAPSJYwO5lNhszMrjOzwRjyHV48KOLVj/Hm3zhJ9qCJBQ1FVTfdXUHMmTau++0sLa+srq1nNrKbW9s7u7m9/ZqOEkVolUQ8Uo0Aa8qZpFXDDKeNWFEsAk7rQf964tcHVGkWyXszjKkvcFeykBFsrOQ/oivUusFC4MLTSTuXd4vuFGiReCnJQ4pKO/fV6kQkEVQawrHWTc+NjT/CyjDC6TjbSjSNMenjLm1aKrGg2h9Njx6jY6t0UBgpW9Kgqfp7YoSF1kMR2E6BTU/PexPxP6+ZmLDkj5iME0MlmS0KE45MhCYJoA5TlBg+tAQTxeytiPSwwsTYnLI2BG/+5UVSOy16Z8WLu/N8uZTGkYFDOIICeHAJZbiFClSBwAM8wyu8OQPnxXl3PmatS046cwB/4Hz+ABEekPg= 7! 0 0 ··· Correlators in sym orbifold

To describe the full correlator combine these local coverings into a global covering surface.

Then use the covering map to lift correlator to covering surface.

covering map: conformal map from covering surface to base surface. Correlators in sym orbifold

The situation is particularly simple if the w-cycle twist field generates the ground state of the twisted sector: then the correlator on the covering surface is just the vacuum correlator.

In this case, the correlator is completely determined by the conformal factor that comes from the covering map.

In turn this is described by a Liouville term [Lunin, Mathur ’00]

c 2 ¯ 2 SL[]= d zpg 2@@ + R e = @ with AAACCXicbVC7TsMwFHV4lvIqMLJYVEhMVVJAsCBVMMBYJPqQmjS6cd3Wqp1EtoNU2q4s/AoLAwix8gds/A1umwFajnSlo3Pute89QcyZ0rb9bS0sLi2vrGbWsusbm1vbuZ3dqooSSWiFRDyS9QAU5SykFc00p/VYUhABp7WgdzX2a/dUKhaFd7ofU09AJ2RtRkAbyc9h2nTjLsMXeIjdGKRmwP0H7F6DEICHzaKfy9sFewI8T5yU5FGKsp/7clsRSQQNNeGgVMOxY+0Nxk8TTkdZN1E0BtKDDm0YGoKgyhtMLhnhQ6O0cDuSpkKNJ+rviQEIpfoiMJ0CdFfNemPxP6+R6Pa5N2BhnGgakulH7YRjHeFxLLjFJCWa9w0BIpnZFZMuSCDahJc1ITizJ8+TarHgHBdOb0/ypcs0jgzaRwfoCDnoDJXQDSqjCiLoET2jV/RmPVkv1rv1MW1dsNKZPfQH1ucPvBmZGA== z

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 48⇡ | | Z covering map Correlators in sym orbifold

The covering surface has, in general, a non-trivial genus, i.e. it is not necessarily a sphere. The genus captures the 1/N corrections of the symmetric orbifold correlators since 1 g n N 2 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 1 n hO ···O i⇠ [Lunin, Mathur ’00] [Pakman, Rastelli, Razamat ’09] Since string coupling is related to 1/N via 1 g s p AAACBHicbVDLSsNAFJ3UV62vqMtuBovgqiQ+0GXRjSupYB/QhDCZTtqhM5M4MxFKyMKNv+LGhSJu/Qh3/o3TNgttPXDhcM693HtPmDCqtON8W6Wl5ZXVtfJ6ZWNza3vH3t1rqziVmLRwzGLZDZEijArS0lQz0k0kQTxkpBOOriZ+54FIRWNxp8cJ8TkaCBpRjLSRArs6CBT0FOXQiyTCmZtnnrqXOrvJ88CuOXVnCrhI3ILUQIFmYH95/RinnAiNGVKq5zqJ9jMkNcWM5BUvVSRBeIQGpGeoQJwoP5s+kcNDo/RhFEtTQsOp+nsiQ1ypMQ9NJ0d6qOa9ifif10t1dOFnVCSpJgLPFkUpgzqGk0Rgn0qCNRsbgrCk5laIh8iEoU1uFROCO//yImkf192T+tntaa1xWcRRBlVwAI6AC85BA1yDJmgBDB7BM3gFb9aT9WK9Wx+z1pJVzOyDP7A+fwAQ2Jhj ⇠ N this suggests that covering surface = world-sheet. [Pakman, Rastelli, Razamat ’09] Correlators from worldsheet

We can now test this idea from worldsheet perspective, by calculating the correlators for ground states of twisted sector n V wi (x ; z ) hi i i

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 i=1 DY E Here

spectral flow sector V w(x; z) [Eberhardt, MRG, Gopakumar, ’19] AAAB8XicbVBNT8JAEJ3iF+IX6tHLRmKCF9KCiSZeiF48YiIfESrZLlvYsN02u1sVG/6FFw8a49V/481/4wI9KPiSSV7em8nMPC/iTGnb/rYyS8srq2vZ9dzG5tb2Tn53r6HCWBJaJyEPZcvDinImaF0zzWkrkhQHHqdNb3g58Zv3VCoWihs9iqgb4L5gPiNYG+m20R3cPRQfz5+Ou/mCXbKnQIvESUkBUtS6+a9OLyRxQIUmHCvVduxIuwmWmhFOx7lOrGiEyRD3adtQgQOq3GR68RgdGaWH/FCaEhpN1d8TCQ6UGgWe6QywHqh5byL+57Vj7Z+5CRNRrKkgs0V+zJEO0eR91GOSEs1HhmAimbkVkQGWmGgTUs6E4My/vEga5ZJTKZWvTwrVizSOLBzAIRTBgVOowhXUoA4EBDzDK7xZynqx3q2PWWvGSmf24Q+szx/EYpBS h

labels conf worldsheet dim of spacetime spacetime CFT coordinate field coordinate Covering maps

The symmetric orbifold perspective suggests that these correlators are expressible in terms of the covering map

(z)=x + a (z z )wi + for z near z 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 i i i ··· i

This requires, in particular, a surprising localisation property since covering maps do not generically exist for given

zAAACI3icbVDLSgMxFM34rPVVdekmWAQXZZjxgSIIRTcuK9gHdMqQyWTa0EwyJhm1lv6LG3/FjQuluHHhv5g+QNt6IHDuOfdyc0+QMKq043xZc/MLi0vLmZXs6tr6xmZua7uiRCoxKWPBhKwFSBFGOSlrqhmpJZKgOGCkGrSvBn71nkhFBb/VnYQ0YtTkNKIYaSP5ufMnn0IPFqB3l6IQPk5U8MGnvxW9cAseC4VWBW6abD+Xd2xnCDhL3DHJgzFKfq7vhQKnMeEaM6RU3XUS3egiqSlmpJf1UkUShNuoSeqGchQT1egOb+zBfaOEMBLSPK7hUP070UWxUp04MJ0x0i017Q3E/7x6qqOzRpfyJNWE49GiKGVQCzgIDIZUEqxZxxCEJTV/hbiFJMLaxJo1IbjTJ8+SyqHtHtknN8f54uU4jgzYBXvgALjgFBTBNSiBMsDgGbyCd/BhvVhvVt/6HLXOWeOZHTAB6/sH/p6gzQ== i ,xi ,wi ,i=1,...,n . Covering maps

For example, for the case of a 4-point function with

wAAACB3icbVDLSsNAFJ3UV62vqEtBBovgIpSkrdiNUHDjsoJ9QBPCZDJth04ezkyUErpz46+4caGIW3/BnX/jNM1CWw9cOHPOvcy9x4sZFdI0v7XCyura+kZxs7S1vbO7p+8fdESUcEzaOGIR73lIEEZD0pZUMtKLOUGBx0jXG1/N/O494YJG4a2cxMQJ0DCkA4qRVJKrHz+49NKCNjSgfZcgH6qXUTVqRj3TXL1sVswMcJlYOSmDHC1X/7L9CCcBCSVmSIi+ZcbSSRGXFDMyLdmJIDHCYzQkfUVDFBDhpNkdU3iqFB8OIq4qlDBTf0+kKBBiEniqM0ByJBa9mfif10/koOGkNIwTSUI8/2iQMCgjOAsF+pQTLNlEEYQ5VbtCPEIcYamiK6kQrMWTl0mnWrFqlfObernZyOMogiNwAs6ABS5AE1yDFmgDDB7BM3gFb9qT9qK9ax/z1oKWzxyCP9A+fwDYTJVs i =1,i=1, 2, 3, 4 , the covering map is just a Moebius transformation. However, this only maps the 4 points to each other provided that the cross-ratios agree

(x x )(x x ) (z z )(z z ) 1 2 3 4 = 1 2 3 4 (x x )(x x ) (z z )(z z ) 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 1 3 2 4 1 3 2 4 Correlation functions

We have studied the Ward identities of the correlators n V wi (x ; z ) hi i i

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 i=1 DY E on the sphere, keeping careful track of the OPE behaviour of the spectrally flowed operators. The resulting equations are quite complicated, but very remarkably, the ansatz that reproduces the Lunin-Mathur answer (and that involves delta-functions for the relevant cross-ratios), solves all of these relations 1 ji = provided that k=1 (and all AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiRV0Y1QdOOygn1AG8JkOmnHTh7M3AglxF9x40IRt36IO//GSZuFth64cDjnXu69x4sFV2BZ30ZpZXVtfaO8Wdna3tndM/cPOipKJGVtGolI9jyimOAhawMHwXqxZCTwBOt6k5vc7z4yqXgU3sM0Zk5ARiH3OSWgJdesPrgcX+EB+JLQ1M7SRlZxzZpVt2bAy8QuSA0VaLnm12AY0SRgIVBBlOrbVgxOSiRwKlhWGSSKxYROyIj1NQ1JwJSTzo7P8LFWhtiPpK4Q8Ez9PZGSQKlp4OnOgMBYLXq5+J/XT8C/dFIexgmwkM4X+YnAEOE8CTzkklEQU00IlVzfiumY6BhA55WHYC++vEw6jbp9Wj+/O6s1r4s4yugQHaETZKML1ES3qIXaiKIpekav6M14Ml6Md+Nj3loyipkq+gPj8wdM+JPo 2 ). [Eberhardt, MRG, Gopakumar, ’19] AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0mkYL0VvHisYD+gDWWznbZLdzdhdyPU0L/gxYMiXv1D3vw3Jm0O2vpg4PHeDDPzgkhwY1332ylsbG5t7xR3S3v7B4dH5eOTtgljzbDFQhHqbkANCq6wZbkV2I00UhkI7ATT28zvPKI2PFQPdhahL+lY8RFn1GbS04CXBuWKW3UXIOvEy0kFcjQH5a/+MGSxRGWZoMb0PDeyfkK15UzgvNSPDUaUTekYeylVVKLxk8Wtc3KRKkMyCnVaypKF+nsiodKYmQzSTkntxKx6mfif14vtqO4nXEWxRcWWi0axIDYk2eNkyDUyK2YpoUzz9FbCJlRTZtN4shC81ZfXSfuq6tWqN/e1SqOex1GEMziHS/DgGhpwB01oAYMJPMMrvDnSeXHenY9la8HJZ07hD5zPH5nMjfA= Localisation

Given the restrictive nature of the covering map, the world-sheet integral localises to those finitely many configurations of z i for which covering map exists. (Thus world-sheet integral of string theory becomes simply a finite sum.)

Behaves as a topological theory!

see also [Berkovits, ’08, ’19] Plan of talk

1. Introduction and Motivation

2. The k=1 WZW model

3. The supergroup formulation

4. Correlation functions

5. Conclusions and Outlook Conclusions and Outlook

We have given strong evidence that the symmetric orbifold theory is exactly dual to string theory with one unit of NS-NS flux (k=1):

4 = 3 4

Sym ( ) (null) (null) N T AdS3 S T ⇥ ⇥ 1 unit of NS-NS flux

In particular, the spectrum agrees precisely, and the structure of the symmetric orbifold correlators is reproduced from the worldsheet. Conclusions and Outlook

4 = 3 4

Sym ( ) (null) (null) N T AdS3 S T ⇥ ⇥ 1 unit of NS-NS flux

The worldsheet plays the role of the covering surface of the symmetric orbifold, thus suggesting that the duality can be made manifest in this way.

In particular, this relation implies that the duality actually works to all orders in 1/N! see also [Eberhardt, ’20] Conclusions and Outlook

4 = 3 4

Sym ( ) (null) (null) N T AdS3 S T ⇥ ⇥ 1 unit of NS-NS flux

The world-sheet theory exhibits signs of a topological string theory:

‣ only short representations of psu(1,1|2) appear ‣ correlation functions localise to isolated points

cf [Aharony, David, Gopakumar, Komargodski, Razamat ’07] [Razamat ’08], [Gopakumar ’11], [Gopakumar, Pius ’12] Conclusions and Outlook

4 = 3 4

Sym ( ) (null) (null) N T AdS3 S T ⇥ ⇥ 1 unit of NS-NS flux

Both sides are explicitly solvable and have free field realisations.

This should enable us to tidy up any remaining loose ends in our analysis. Loose ends & future directions

‣ Establish uniqueness of world-sheet correlator solution [Dei, MRG, Knighton, in progress]

‣ Use free field realisation more explicitly [Dei, MRG, Gopakumar, Knighton, in progress]

‣ Understand as topological string theory [MRG, Gopakumar, in progress]

‣ direct description in NS/R formalism

‣ Generalise to AdS5… Thank you!