June 29th, 2020

Deriving the CorrespondenceRajesh Gopakumar, ICTS-TIFR, Bengaluru; (Virtual) Strings 2020, CapetownBased on: arXiv:1911.00378 (w/ L. Eberhardt & M. Gaberdiel);

AdS3/CFT2

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Derivation vs. Verification

✤ What does it mean to derive the AdS/CFT correspondence?

✤ Top down construction of dual pairs via Maldacena’s near horizon limit.

✤ Match (BPS/integrable) spectrum, compute correlators, Wilson lines, EE…

✤ Verification of equality of both sides appears miraculous/mysterious.

✤ Also does not help in delineating the scope of gauge-string duality - how and to what extent, can we systematically come up with new dual pairs?

Can We Tautologise AdS/CFT?

✤ Thus derivation of AdS/CFT is not mathematical fastidiousness.

✤ Rather a physics necessity to lay bare the inner workings of the duality.

✤ Perhaps ultimately a geometrisation of quantum information (tensor networks).

✤ Can aim at a less ambitious but concrete goal: equivalence of worldsheet CFT description of certain backgrounds with (large N) boundary .

✤ Dictionary that relates two (separately) mathematically well defined entities - can we make the equality manifest?

AdSd+1

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CFTd

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From Worldsheet CFT to Spacetime CFT

✤ A large part of the dictionary is the equality of (euclidean) correlators.

✤ Assumes a matching of spectrum: on shell vertex operators of physical states in the worldsheet CFT single trace operators in spacetime CFT

Can we transform the LHS correlator into the RHS correlator (or vice versa)?

Vwh (x; z) $ O

(w)h (x)

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$

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spacetime position

worldsheet position

addl. labels

spacetime operator dim.

Z

Mg,n

⌦Vw1h1

(x1; z1)Vw2h2

(x2; z2) . . .Vwnhn

(xn; zn)↵⌃g,n

=⌦O

(w1)h1

(x1)O(w2)h2

(x2) . . .O(wn)hn

(xn)↵Sd

���g

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A Testing Ground

✤ Need an example where this program can be carried through.

✤ PROPOSAL [Eberhardt-Gaberdiel-RG (’18)]: String theory on with ( ) unit of NS-NS flux (in a perturbative expansion) = Large limit of - Symmetric orbifold 2d CFT.

✤ Worldsheet theory with NS-NS flux can be quantised - in RNS formalism essentially a WZW model: ( ).

✤ Spectrum organised in terms of sectors with `spectral flow’ - asymptotic winding of string on . Also continuum of long strings with mom. .

AdS3 ⇥ S3 ⇥ T 4

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k = 1

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(T 4)N/SN

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k � 2

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sl(2,R)k+2 � su(2)k�2 � T4 � fermions� ghosts

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(w)

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AdS3

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p

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N

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[Maldacena-Ooguri (’00)]

A Tractable Case

✤ For ( ) - `tensionless limit’ - can employ the hybrid formalism of Berkovits-Vafa-Witten based on a WZW model.

✤ Spectrum truncates to short reps of .

✤ Only states at the bottom of the long string continuum survives: .

✤ Match of the entire perturbative spectrum with dual CFT (and other checks). [Eberhardt-Gaberdiel-R.G. (’18); see Lorenz@Strings 2019]

psu(1, 1|2)1

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k = 1

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psu(1, 1|2)1

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j =1

2+ i(p = 0)

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sl(2) qtm no.

Correlating Correlators

✤ Thus we have ; spectral flow sector twisted sector .

✤ GOAL: to derive the relation between correlators rather than verify.

✤ Restrict to ground states in each sector: - states at the bottom of the would-be continuum (i.e. no torus oscillators excited).

✤ Also to genus zero (for generalisation to higher genus see Eberhardt (’20)).

Vwh (x; z) $ O

(w)h (x)

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(w)

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(w)

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$

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h =w2 � 1

4w

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Z

M0,n

⌦Vw1h1

(x1; z1)Vw2h2

(x2; z2) . . .Vwnhn

(xn; zn)↵⌃0,n

=⌦O

(w1)h1

(x1)O(w2)h2

(x2) . . .O(wn)hn

(xn)↵S2

���g=0

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✤ Uncover the structural reason why correlators in this worldsheet CFT match with those of the spacetime CFT.

✤ Localisation on moduli space to holomorphic covering maps: with specified branching ( ) at the insertions .

✤ CLAIM: Worldsheet correlator - discrete set of points.

✤ Leads to a realization of covering space computation of symmetric product orbifold correlators (and its identification with worldsheet).

Localisation

x = �(z)

<latexit sha1_base64="+iCZIZNXLwVmkytNjkr5lPM6Vv0=">AAAB8nicbVBNSwMxEJ31s9avqkcvwSLUS9mVgnoQih70WMF+wLaUbJptQ5PskmTFuvRnePGgiFd/jTf/jWm7B219MPB4b4aZeUHMmTau++0sLa+srq3nNvKbW9s7u4W9/YaOEkVonUQ8Uq0Aa8qZpHXDDKetWFEsAk6bwfB64jcfqNIskvdmFNOOwH3JQkawsZL/eNm+wULg0tNJt1B0y+4UaJF4GSlChlq38NXuRSQRVBrCsda+58amk2JlGOF0nG8nmsaYDHGf+pZKLKjupNOTx+jYKj0URsqWNGiq/p5IsdB6JALbKbAZ6HlvIv7n+YkJzzspk3FiqCSzRWHCkYnQ5H/UY4oSw0eWYKKYvRWRAVaYGJtS3obgzb+8SBqnZa9SvrirFKtXWRw5OIQjKIEHZ1CFW6hBHQhE8Ayv8OYY58V5dz5mrUtONnMAf+B8/gBnDpCz</latexit>

wi

<latexit sha1_base64="kVxL5/yVhJvvymHlh2eek9gXJKM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4r2g9oQ9lsJ+3SzSbsbpQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaLSPJYPZpygH9GB5CFn1Fjp/qnHe+WKW3VnIMvEy0kFctR75a9uP2ZphNIwQbXueG5i/Iwqw5nASambakwoG9EBdiyVNELtZ7NTJ+TEKn0SxsqWNGSm/p7IaKT1OApsZ0TNUC96U/E/r5Oa8NLPuExSg5LNF4WpICYm079JnytkRowtoUxxeythQ6ooMzadkg3BW3x5mTTPqt559eruvFK7zuMowhEcwyl4cAE1uIU6NIDBAJ7hFd4c4bw4787HvLXg5DOH8AfO5w9i6o3j</latexit>

/n�3Y

i=1

�(xi � �(zi))

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[Lunin-Mathur(’00); Pakman-Rastelli-Razamat(’09)]

x ⇡ xi + a�i (z � zi)wi ; (i = 1 . . . n)

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The Worldsheet as a Covering Space

A Two-Pronged Strategy

✤ Argue (i) that worldsheet correlators localise and (ii) give the right contributions that reproduce the correlators of the spacetime CFT.

✤ (i) Ward identities for correlators of spectrally flowed vertex operators have the special delta function solution provided: - satisfied for .

✤ (ii) An exact classical solution of the sigma model corresponding to this covering space map. Gives precisely the contribution to the path integral as in Lunin-Mathur computation of symmetric orbifold correlators.

nX

i=1

ji =k

2(n� 2) + 1

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ji =1

2; k = 1; 8n

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sl(2,R)k+2

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Semiclassical Picture

Ward Identities

✤ Ward Identities for are very nontrivial on the spectrally flowed vertex operators.

✤ Under spectral flow:

✤ Therefore, on the flowed primary vertex operator, the OPE with the currents is

�w(J±)(z) = z⌥wJ±(z); �w(J3)(z) = J3(z) +(k + 2)w

2z.

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J+(z)V wh (0; 0) ⇠

w+1X

p=2

(J+p�1V

wh )(0; 0)

zp+

@xV wh (0; 0)

z,

J3(z)V wh (0; 0) ⇠

hV wh (0; 0)

z,

J�(z)V wh (0; 0) ⇠ O(zw�1) ,

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+ve modes with non-zero action

-ve modes which annihilate

sl(2,R)k+2

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Ward Identities (Contd.)

✤ So now have new unknowns from the . But also have new equations from the regularity of OPE upto .

✤ Apply these OPEs to the correlator .

✤ For have the usual

✤ For general { } a complicated set of recursion relations for after eliminating unknowns.

✤ General solution difficult to write down. [See also Hikida-Liu (’20)]

(J+p�1V

wh )

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J�

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O(zw�1)

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hJa(z)nY

i=1

V wihi

(xi; zi)i

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(w)

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(w)

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hnY

i=1

V wihi

(xi; zi)i

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{wi = 0}

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hJa(z)nY

i=1

V 0hi(xi; zi)i = �

X

n=1

Da

z � zih

nY

i=1

V 0hi(xi; zi)i .

<latexit sha1_base64="EppN628/80i2Fl1pAg8g+QXYryA=">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</latexit>

wi

<latexit sha1_base64="kVxL5/yVhJvvymHlh2eek9gXJKM=">AAAB6nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4r2g9oQ9lsJ+3SzSbsbpQS+hO8eFDEq7/Im//GbZuDtj4YeLw3w8y8IBFcG9f9dgorq2vrG8XN0tb2zu5eef+gqeNUMWywWMSqHVCNgktsGG4EthOFNAoEtoLRzdRvPaLSPJYPZpygH9GB5CFn1Fjp/qnHe+WKW3VnIMvEy0kFctR75a9uP2ZphNIwQbXueG5i/Iwqw5nASambakwoG9EBdiyVNELtZ7NTJ+TEKn0SxsqWNGSm/p7IaKT1OApsZ0TNUC96U/E/r5Oa8NLPuExSg5LNF4WpICYm079JnytkRowtoUxxeythQ6ooMzadkg3BW3x5mTTPqt559eruvFK7zuMowhEcwyl4cAE1uIU6NIDBAJ7hFd4c4bw4787HvLXg5DOH8AfO5w9i6o3j</latexit>

global (spacetime)conformal generators

✤ Rather remarkably, there exists a special solution which is essentially determined when there exists a covering map:

✤ This solution exists only when . True for .

✤ Delta function localised to a finite set of points on where exists.

✤ depends only on cross ratios (can constrain further [Dei-Eberhardt-Gaberdiel (’19)]). And depends on . [Note: only holomorphic dependence shown.]

A Special Solution

hV w1h1

(0; 0)V w2h2

(1; 1)V w3h3

(1;1)nY

i=4

V wihi

(xi; zi)i =X

�

nY

i=1

(a�i )�hi

nY

i=4

�(xi � �(zi))W�(z4, . . . , zn) .

<latexit sha1_base64="tIF9PXZ4Htah3FBfwQSYPF3jrIk=">AAAC1nicbZLLjtMwFIadcBvKZQos2VhUSInUqZJOJUDVSCNYwHIQ9CI1reW4TmuN7US2A2SisAAhtjwbOx6Cd8BJyoiZ4UhOfn3nnP84duKMM22C4JfjXrt+4+atvdudO3fv3d/vPng41WmuCJ2QlKdqHmNNOZN0YpjhdJ4pikXM6Sw+fVXnZx+o0iyV702R0aXAG8kSRrCxCHV/RxzLDadwisotCqtV+dE+vWAc+A0ZNmRYeeE4bMlhQw4r6EVMJqYYty8/ylS6RiU7GlUr2dqxppRV3ifExmeI+TBSzbTOUaRzgcroNRYCV/C8N6x7PYzY6m/OX5UHtRO86B+tKTe4Nj5oC73a3w7owxk6J6N+tE6N7sMzJP0IDlC3FwyCJuBVEe5ED+ziBHV/WgOSCyoN4VjrRRhkZlliZRjhtOpEuaYZJqd4QxdWSiyoXpbNtVTwqSVrmKTKLmlgQ//tKLHQuhCxrRTYbPXlXA3/l1vkJnm+LJnMckMlaQclOYcmhfUdwzVTlBheWIGJYnavkGyxwsTYP6FjDyG8/MlXxXQ4CEeDF29HveOXu+PYA4/BE+CBEDwDx+ANOAETQJx3TuF8cb66c/ez+8393pa6zq7nEbgQ7o8/WZzgqw==</latexit>

x = �(z); x ⇡ xi + a�i (z � zi)wi

<latexit sha1_base64="7xjPFzUfInXigDB6u0VXVBG6eQs=">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</latexit>

nX

i=1

ji =k

2(n� 2) + 1

<latexit sha1_base64="lGwAWRBo23QyFSQj2PEHHb9goD4=">AAACDXicbVDLSsNAFJ3UV62vqEs3g1WoiCUpBXVRKLpxWcE+oKlhMp3UsZNJmJkIJeQH3Pgrblwo4ta9O//GaZuFth64cDjnXu69x4sYlcqyvo3cwuLS8kp+tbC2vrG5ZW7vtGQYC0yaOGSh6HhIEkY5aSqqGOlEgqDAY6TtDS/HfvuBCElDfqNGEekFaMCpTzFSWnLNA0fGgZvQmp3ecnjv0prjC4STYZpU0hI/qRwd29A1i1bZmgDOEzsjRZCh4ZpfTj/EcUC4wgxJ2bWtSPUSJBTFjKQFJ5YkQniIBqSrKUcBkb1k8k0KD7XSh34odHEFJ+rviQQFUo4CT3cGSN3JWW8s/ud1Y+Wf9RLKo1gRjqeL/JhBFcJxNLBPBcGKjTRBWFB9K8R3SKehdIAFHYI9+/I8aVXKdrV8fl0t1i+yOPJgD+yDErDBKaiDK9AATYDBI3gGr+DNeDJejHfjY9qaM7KZXfAHxucPYnaaeA==</latexit>

ji =1

2; k = 1; 8n

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M0,n

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�(z)

<latexit sha1_base64="Ojsjoy7d7mbilWKF2juOsQWqWNA=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRahXsquFNRb0YMeK9gPaZeSTbNtaJJdkqxQl/4KLx4U8erP8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOooUYQ2SMQj1Q6wppxJ2jDMcNqOFcUi4LQVjK6nfuuRKs0ieW/GMfUFHkgWMoKNlR66N1gIXH467RVLbsWdAS0TLyMlyFDvFb+6/YgkgkpDONa647mx8VOsDCOcTgrdRNMYkxEe0I6lEguq/XR28ASdWKWPwkjZkgbN1N8TKRZaj0VgOwU2Q73oTcX/vE5iwgs/ZTJODJVkvihMODIRmn6P+kxRYvjYEkwUs7ciMsQKE2MzKtgQvMWXl0nzrOJVK5d31VLtKosjD0dwDGXw4BxqcAt1aAABAc/wCm+Ocl6cd+dj3ppzsplD+APn8wcD7I/q</latexit>

W�

<latexit sha1_base64="9aba3oZ/An+ATUNUna02ZtDUkRo=">AAAB8XicbVBNS8NAEJ3Ur1q/qh69BIvgqSQiqLeiBz1WsB/YhjLZbtqlu5uwuxFK6L/w4kERr/4bb/4bt20O2vpg4PHeDDPzwoQzbTzv2ymsrK6tbxQ3S1vbO7t75f2Dpo5TRWiDxDxW7RA15UzShmGG03aiKIqQ01Y4upn6rSeqNIvlgxknNBA4kCxiBI2VHlu9rHuLQuCkV654VW8Gd5n4OalAjnqv/NXtxyQVVBrCUeuO7yUmyFAZRjidlLqppgmSEQ5ox1KJguogm108cU+s0nejWNmSxp2pvycyFFqPRWg7BZqhXvSm4n9eJzXRZZAxmaSGSjJfFKXcNbE7fd/tM0WJ4WNLkChmb3XJEBUSY0Mq2RD8xZeXSfOs6p9Xr+7PK7XrPI4iHMExnIIPF1CDO6hDAwhIeIZXeHO08+K8Ox/z1oKTzxzCHzifP4UXkNc=</latexit>

{wi, xi, zi}

<latexit sha1_base64="lj4evxHJmMEq2k4y3sJxiYTaamc=">AAAB+nicbVDLSsNAFL2pr1pfqS7dDBbBhZRECuqu6MZlBfuANoTJdNIOnTyYmVhr7Ke4caGIW7/EnX/jpM1CWw/cy+Gce5k7x4s5k8qyvo3Cyura+kZxs7S1vbO7Z5b3WzJKBKFNEvFIdDwsKWchbSqmOO3EguLA47Ttja4zv31PhWRReKcmMXUCPAiZzwhWWnLNci8du+wUPWTt0WW9qWtWrKo1A1omdk4qkKPhml+9fkSSgIaKcCxl17Zi5aRYKEY4nZZ6iaQxJiM8oF1NQxxQ6aSz06foWCt95EdCV6jQTP29keJAykng6ckAq6Fc9DLxP6+bKP/CSVkYJ4qGZP6Qn3CkIpTlgPpMUKL4RBNMBNO3IjLEAhOl0yrpEOzFLy+T1lnVrlUvb2uV+lUeRxEO4QhOwIZzqMMNNKAJBMbwDK/wZjwZL8a78TEfLRj5zgH8gfH5A1H4k2o=</latexit>

a�i / @wi�(zi)

<latexit sha1_base64="DFTbm2YhmiqitozcOzKGsR3tK6g=">AAACH3icbVDLSgMxFM34rPVVdekmWATdlBkpPnZFF7qsYB/Q1uFOmtFgkhmSjFKH/okbf8WNC0XEXf/GTDsLtV4IOZxzLvfeE8ScaeO6I2dmdm5+YbGwVFxeWV1bL21sNnWUKEIbJOKRagegKWeSNgwznLZjRUEEnLaCu7NMb91TpVkkr8wgpj0BN5KFjICxlF86xOCz67R7DkLAEHdjFcUmsj8ow4BbxTA5wA8+G048e48+2y/6pbJbcceFp4GXgzLKq+6Xvrr9iCSCSkM4aN3x3Nj00mwK4XRY7CaaxkDu4IZ2LJQgqO6l4/uGeNcyfRxGyj5p8Jj92ZGC0HogAusUYG71Xy0j/9M6iQmPeymTcWKoJJNBYcKxTSALC/eZosTwgQVAFLO7YnILCoixkWYheH9PngbNg4pXrZxcVsu10zyOAtpGO2gPeegI1dAFqqMGIugJvaA39O48O6/Oh/M5sc44ec8W+lXO6Bvy7qLr</latexit>

Classical Action

✤ Now need to figure out the exact contribution to the worldsheet correlator at these discrete points on the moduli space which admit a covering map.

✤ We look for a classical solution to the sigma model (first order form).

✤ Might seem a foolhardy thing to do when one is at . Highly curved sigma model.

✤ However, action is also quadratic when i.e. worldsheet at boundary.

SAdS3 =k

4⇡

Zd2z (4@�@̄�+ �̄@�̄ + �@̄� � e�2���̄ � k�1R�) .

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Radial direction Boundary direction

k = 1

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� ! 1

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[deBoer-Ooguri-Robbins-Tannenhauser (’98)]

AdS3

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[Giveon-Kutasov-Seiberg(’98)]

✤ Ground state in -spectrally flowed sector given by

✤ Obtained by taking a scaling limit ( ) to the bottom of continuum - worldsheet essentially glued to the boundary.

✤ Can now find general solution corresponding to the correlator (with right boundary conditions at insertions) - in terms of a covering map:

Classical Solution

�(z) = zw; �(z, z̄) = � ln ✏� (w � 1)

2ln z � (w � 1)

2ln z̄

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✏ / p ! 0

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�(z) = �(z); �(z, z̄) = �1

2ln(@�)� 1

2ln(@̄�̄)� ln ✏

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w

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at the boundary

Semiclassical Picture

✤ Since these solutions are at the boundary, we can evaluate their contribution to the path integral semiclassically and connect with Ward identity solution.

✤ Define . Then the classical action on-shell is

✤ Combining with Ward identity, the worldsheet correlator has the form

Semiclassically Exact

�cl = �2�+ constant = ln(|@�|2)

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(Liouville action for the conformal factor.)

SAdS3 = SL[�cl] =1

8⇡

Zd2z

pg⇣2 @�cl @̄�cl +R�cl

⌘

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DV w1h1

(0; 0)V w2h2

(1; 1)V w3h3

(1;1)nY

i=4

V wihi

(xi; zi)E=

X

�

fW�e�SL[�cl]

nY

i=1

|@wi�(zi)|�2(hi�h0i )

nY

i=4

�(2)(zi � ��1(xi))

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⇣h0 =

w2 � 1

4w

⌘

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Symmetric Orbifold Correlators

✤ The worldsheet correlators are now in a form which is manifestly that of the spacetime orbifold CFT correlators.

✤ Recall Lunin-Mathur (’00) approach to computing symmetric orbifold correlators (e.g. of -twisted sector ground states).

✤ Lift the correlator to the covering space (each . -twisted sector insertion lifts to a -sheeted branch cover locally)

w

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⌦O

(w1)h1

(x1)O(w2)h2

(x2) . . .O(wn)hn

(xn)↵S2

���g=0

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w

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w

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✤ Contribution from each such covering map.

✤ Twisted sector ground state lifts to vacuum.

✤ Vacuum path integral on covering space.

✤ Coord. dependence from conformal factor .�(z)

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Making Equality Manifest

✤ Thus symmetric orbifold CFT correlators (for ) take the form

✤ We see that the computation of the integrated worldsheet CFT correlators . reduce to the same Lunin-Mathur path integral.

✤ With exactly the same Liouville action for the same conformal factor . ( should be understood in an appropriately regularised sense.)

✤ Worldsheet CFT on the covering space of the spacetime CFT.

[See also Dei-Eberhardt (’19)]

Z

M0,n

⌦Vw1h1

(x1; z1)Vw2h2

(x2; z2) . . .Vwnhn

(xn; zn)↵⌃0,n

<latexit sha1_base64="Uk4DFZxURPSezx9vNBEX1eOny3g=">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</latexit>

[Pakman, Rastelli, Razamat (’09)]

e�cl = |@�|2

<latexit sha1_base64="bqvtox8LGHubbCHPE/7GeHhMF9c=">AAACDXicbVC7SgNBFJ31GeMramkzGAWrsBsCaiEELbSMYB6Q3YS7k0kyZGZ3mZkVwiY/YOOv2FgoYmtv5984SbbQxAMXDufcO3Pv8SPOlLbtb2tpeWV1bT2zkd3c2t7Zze3t11QYS0KrJOShbPigKGcBrWqmOW1EkoLwOa37g+uJX3+gUrEwuNfDiHoCegHrMgLaSO3cMW0lbtRn7cSVAhM+Hl+O3AikZsDdGxACRq1iO5e3C/YUeJE4KcmjFJV27svthCQWNNCEg1JNx460l0yeJZyOs26saARkAD3aNDQAQZWXTK8Z4xOjdHA3lKYCjafq74kEhFJD4ZtOAbqv5r2J+J/XjHX33EtYEMWaBmT2UTfmWId4Eg3uMEmJ5kNDgEhmdsWkDxKINgFmTQjO/MmLpFYsOKXCxV0pX75K48igQ3SETpGDzlAZ3aIKqiKCHtEzekVv1pP1Yr1bH7PWJSudOUB/YH3+AC0GnEU=</latexit>

SL[�cl]

<latexit sha1_base64="O4UTePrXMnWUklbm+BTSxTOZ+Bk=">AAAB/HicbVDLSsNAFJ3UV62vaJduBovgqiQiqLuiGxcuKtoHJCFMppN26MwkzEyEEOqvuHGhiFs/xJ1/47TNQlsPXDiccy/33hOljCrtON9WZWV1bX2julnb2t7Z3bP3D7oqySQmHZywRPYjpAijgnQ01Yz0U0kQjxjpRePrqd97JFLRRDzoPCUBR0NBY4qRNlJo1+/DW89PRzQsfMkhZpMAhnbDaTozwGXilqQBSrRD+8sfJDjjRGjMkFKe66Q6KJDUFDMyqfmZIinCYzQknqECcaKCYnb8BB4bZQDjRJoSGs7U3xMF4krlPDKdHOmRWvSm4n+el+n4IiioSDNNBJ4vijMGdQKnScABlQRrlhuCsKTmVohHSCKsTV41E4K7+PIy6Z423bPm5d1Zo3VVxlEFh+AInAAXnIMWuAFt0AEY5OAZvII368l6sd6tj3lrxSpn6uAPrM8fD/eUaQ==</latexit>

c = 6

<latexit sha1_base64="h+O6DIkIk9a8XQQd/axLn5Wsia8=">AAAB6nicbVDLSgNBEOyNrxhfUY9eBoPgKexK8HEQgl48RjQPSJYwO+kkQ2Znl5lZISz5BC8eFPHqF3nzb5wke9DEgoaiqpvuriAWXBvX/XZyK6tr6xv5zcLW9s7uXnH/oKGjRDGss0hEqhVQjYJLrBtuBLZihTQMBDaD0e3Ubz6h0jySj2Ycox/SgeR9zqix0gO7Pu8WS27ZnYEsEy8jJchQ6xa/Or2IJSFKwwTVuu25sfFTqgxnAieFTqIxpmxEB9i2VNIQtZ/OTp2QE6v0SD9StqQhM/X3REpDrcdhYDtDaoZ60ZuK/3ntxPQv/ZTLODEo2XxRPxHERGT6N+lxhcyIsSWUKW5vJWxIFWXGplOwIXiLLy+TxlnZq5Sv7iul6k0WRx6O4BhOwYMLqMId1KAODAbwDK/w5gjnxXl3PuatOSebOYQ/cD5/AMNtjXo=</latexit>

⌦O

(w1)h1

(x1)O(w2)h2

(x2) . . .O(wn)hn

(xn)↵S2

���g=0

=X

�

W�e�SL[�cl]

nY

i=1

|@wi�i|�2(hi�h0

i )

<latexit sha1_base64="D5Dgue0m5VYQar/dlwSpUIKRJTI=">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</latexit>

⇣h0 =

w2 � 1

4w

⌘

<latexit sha1_base64="4fmVGvABsbRx3KsFSygXt4HNe1k=">AAACCnicbVDLSsNAFJ3UV62vqEs3o0WoC0tSCupCKHXjsoJ9QJOWyXTSDp08mJlYSsjajb/ixoUibv0Cd/6NkzYLbT1w4XDOvdx7jxMyKqRhfGu5ldW19Y38ZmFre2d3T98/aIkg4pg0ccAC3nGQIIz6pCmpZKQTcoI8h5G2M75J/fYD4YIG/r2chsT20NCnLsVIKqmvH1t1OizBUc+A15bLEY4nvcq5mcTVSQJT76yvF42yMQNcJmZGiiBDo69/WYMARx7xJWZIiK5phNKOEZcUM5IUrEiQEOExGpKuoj7yiLDj2SsJPFXKALoBV+VLOFN/T8TIE2LqOarTQ3IkFr1U/M/rRtK9tGPqh5EkPp4vciMGZQDTXOCAcoIlmyqCMKfqVohHSAUiVXoFFYK5+PIyaVXKZrV8dVct1upZHHlwBE5ACZjgAtTALWiAJsDgETyDV/CmPWkv2rv2MW/NadnMIfgD7fMHNzmYuQ==</latexit>

Scattered Remarks

✤ Delta function localisation of correlators reminiscent of a topological string. Also carries over to higher genus correlators [Eberhardt (’20)].

✤ The single particle contribution to the torus partition function also showed localisation to holomorphic maps: [Eberhardt-Gaberdiel-R.G.(’18)].

✤ Both of these have an analogue for . One loop answer as well as 4-pt. correlators have singularities exactly when holomorphic coverings exist.

✤ The integration over the continuum (radial momentum) smears out the delta functions into a singularity.

Z(g=1) / �(2)(t� w⌧ �m)

<latexit sha1_base64="mWeP3RNyqJEeZo+dYtR4VsIQsVU=">AAACEnicbVC7SgNBFJ31GeNr1dJmMAibImE3BNRCCNpYRjAPzMZldjJJhsw+mLmrhCXfYOOv2FgoYmtl5984eRSaeGDgcM693DnHjwVXYNvfxtLyyuraemYju7m1vbNr7u3XVZRIymo0EpFs+kQxwUNWAw6CNWPJSOAL1vAHl2O/cc+k4lF4A8OYtQPSC3mXUwJa8sz8rZdavXMnP8JuLKMYIux2mAByl1ql/MiCwoMLJCkEec/M2UV7ArxInBnJoRmqnvnldiKaBCwEKohSLceOoZ0SCZwKNsq6iWIxoQPSYy1NQxIw1U4nkUb4WCsd3I2kfiHgifp7IyWBUsPA15MBgb6a98bif14rge5pO+VhnAAL6fRQNxFYBx/3gztcMgpiqAmhkuu/YtonklDQLWZ1Cc585EVSLxWdcvHsupyrXMzqyKBDdIQs5KATVEFXqIpqiKJH9Ixe0ZvxZLwY78bHdHTJmO0coD8wPn8AyVicUA==</latexit>

k � 2

<latexit sha1_base64="8tTStDZOcS2V3wZsN8ssdLdL/ig=">AAAB73icbVDLSgNBEOyNrxhfUY9eBoPgKeyGgHoLevEYwTwgWcLspDcZMju7mZkVQshPePGgiFd/x5t/4yTZgyYWNBRV3XR3BYng2rjut5Pb2Nza3snvFvb2Dw6PiscnTR2nimGDxSJW7YBqFFxiw3AjsJ0opFEgsBWM7uZ+6wmV5rF8NJME/YgOJA85o8ZK7RHpDnBMKr1iyS27C5B14mWkBBnqveJXtx+zNEJpmKBadzw3Mf6UKsOZwFmhm2pMKBvRAXYslTRC7U8X987IhVX6JIyVLWnIQv09MaWR1pMosJ0RNUO96s3F/7xOasJrf8plkhqUbLkoTAUxMZk/T/pcITNiYgllittbCRtSRZmxERVsCN7qy+ukWSl71fLNQ7VUu83iyMMZnMMleHAFNbiHOjSAgYBneIU3Z+y8OO/Ox7I152Qzp/AHzucP+vmPTA==</latexit>

[Maldacena-Ooguri (’00-’01)]

Tensionless Folklore

✤ Tensionless limit of string theory ought to be a topological string - see a realisation also in reduction in d.o.f. to four bosonic and fermionic oscillators.

✤ Also a given correlator gets contributions from only up to finite genus.

✤ Enhanced Higher Spin symmetries in spacetime CFT [Gaberdiel-R. G (‘14)] arising from free worldsheet theory (sigma model free when localised to boundary).

✤ Gross-Mende like saddle point for worldsheet path integral - from interior of moduli space.

[Gaberdiel-R. G. (in progress)]

Radial direction has same Coulomb gas profile. �(z, z̄) = �

nX

i=1

(wi � 1) ln |z � zi|+ 2MX

a=1

ln |z � ua|+ const.

<latexit sha1_base64="jAYvj45wNN++199R8xbeFlLN+Ec=">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</latexit>

Poles of �(z)

<latexit sha1_base64="Ojsjoy7d7mbilWKF2juOsQWqWNA=">AAAB8HicbVBNSwMxEJ2tX7V+VT16CRahXsquFNRb0YMeK9gPaZeSTbNtaJJdkqxQl/4KLx4U8erP8ea/MW33oK0PBh7vzTAzL4g508Z1v53cyura+kZ+s7C1vbO7V9w/aOooUYQ2SMQj1Q6wppxJ2jDMcNqOFcUi4LQVjK6nfuuRKs0ieW/GMfUFHkgWMoKNlR66N1gIXH467RVLbsWdAS0TLyMlyFDvFb+6/YgkgkpDONa647mx8VOsDCOcTgrdRNMYkxEe0I6lEguq/XR28ASdWKWPwkjZkgbN1N8TKRZaj0VgOwU2Q73oTcX/vE5iwgs/ZTJODJVkvihMODIRmn6P+kxRYvjYEkwUs7ciMsQKE2MzKtgQvMWXl0nzrOJVK5d31VLtKosjD0dwDGXw4BxqcAt1aAABAc/wCm+Ocl6cd+dj3ppzsplD+APn8wcD7I/q</latexit>

[cf. Aharony-Komargodski-Razamat(’06)]

Drawing Lessons from Drawing Diagrams

✤ identified with worldsheet conformal factor. Hence worldsheet curvature

✤ Exactly as in `Strebel Gauge’: curvature of dual string worldsheet at vertices as well as centres of faces of Feynman diagrams of free Yang-Mills theory.

✤ In fact, can draw Feynman-like diagrams for symm. orbifold correlators with vertices and colour loops.

✤ Can associate precisely one diagram for each branched cover.

[Pakman, Rastelli, Razamat (’09)]

�(z, z̄)

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(Localised to insertions of vertex operators and poles.)

[R.G. (’03-’05), (’11)]

/ r2�(z, z̄) ⇠nX

i=1

(wi � 1)�(2)(z � zi)� 2MX

a=1

�(2)(z � ua)

<latexit sha1_base64="BU7ptYHLEuDbAPMtsxwLI7e2zJU=">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</latexit>

Feynman Coverings?

✤ Bifundamental edges give rise to a fatgraph on worldsheet.

✤ Vertices have insertions (and curvature).

✤ Coloured and dashed faces: former contain preimages of .

✤ Colour loops lead to curvature defects when Feynman strips glue together.

✤ Dashed loops only have information on merging different Riemann sheets.

1

<latexit sha1_base64="pYYX8aawcUZQ5cT9tU22CFRjr4s=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4r2A9oQ9lsN+3SzSbsToRQ+iO8eFDEq7/Hm//GTZuDtj4YeLw3w8y8IJHCoOt+O6W19Y3NrfJ2ZWd3b/+genjUNnGqGW+xWMa6G1DDpVC8hQIl7yaa0yiQvBNM7nK/88S1EbF6xCzhfkRHSoSCUbRSpy9UiFllUK25dXcOskq8gtSgQHNQ/eoPY5ZGXCGT1Jie5yboT6lGwSSfVfqp4QllEzriPUsVjbjxp/NzZ+TMKkMSxtqWQjJXf09MaWRMFgW2M6I4NsteLv7n9VIMr/2pUEmKXLHFojCVBGOS/06GQnOGMrOEMi3srYSNqaYMbUJ5CN7yy6ukfVH3Lus3D5e1xm0RRxlO4BTOwYMraMA9NKEFDCbwDK/w5iTOi/PufCxaS04xcwx/4Hz+APyrj1w=</latexit>

Analogous (but slightly different from) case of Strebel differentials: poles vertices; zeroes centres of faces. $

<latexit sha1_base64="x0vu3aakTMyI+BTGUOA+2UAHR/8=">AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU0lEUG9FLx4r2A9oQ9lsN+3SzW7YnVRK6D/x4kERr/4Tb/4bN20O2vpg4PHeDDPzwkRwA5737ZTW1jc2t8rblZ3dvf0D9/CoZVSqKWtSJZTuhMQwwSVrAgfBOolmJA4Fa4fju9xvT5g2XMlHmCYsiMlQ8ohTAlbqu25PsAg0H46AaK2eKn236tW8OfAq8QtSRQUafferN1A0jZkEKogxXd9LIMiIBk4Fm1V6qWEJoWMyZF1LJYmZCbL55TN8ZpUBjpS2JQHP1d8TGYmNmcah7YwJjMyyl4v/ed0Uousg4zJJgUm6WBSlAoPCeQx4wDWjIKaWEKq5vRXTEdGEgg0rD8FffnmVtC5q/mXt5uGyWr8t4iijE3SKzpGPrlAd3aMGaiKKJugZvaI3J3NenHfnY9FacoqZY/QHzucPqM2TsQ==</latexit>

$

<latexit sha1_base64="x0vu3aakTMyI+BTGUOA+2UAHR/8=">AAAB+XicbVBNS8NAEN3Ur1q/oh69LBbBU0lEUG9FLx4r2A9oQ9lsN+3SzW7YnVRK6D/x4kERr/4Tb/4bN20O2vpg4PHeDDPzwkRwA5737ZTW1jc2t8rblZ3dvf0D9/CoZVSqKWtSJZTuhMQwwSVrAgfBOolmJA4Fa4fju9xvT5g2XMlHmCYsiMlQ8ohTAlbqu25PsAg0H46AaK2eKn236tW8OfAq8QtSRQUafferN1A0jZkEKogxXd9LIMiIBk4Fm1V6qWEJoWMyZF1LJYmZCbL55TN8ZpUBjpS2JQHP1d8TGYmNmcah7YwJjMyyl4v/ed0Uousg4zJJgUm6WBSlAoPCeQx4wDWjIKaWEKq5vRXTEdGEgg0rD8FffnmVtC5q/mXt5uGyWr8t4iijE3SKzpGPrlAd3aMGaiKKJugZvaI3J3NenHfnY9FacoqZY/QHzucPqM2TsQ==</latexit>

Suggestive of associating branched coverings toFeynman diagrams in higher dim as well e.g. TrZw

<latexit sha1_base64="TGgV5fb5Qebp/D2l3Jy2h+7YpXQ=">AAAB83icbVBNSwMxEJ2tX7V+VT16CRbBU9kVQb0VvXis0C/sriWbpm1okl2SrFKW/g0vHhTx6p/x5r8x2+5BWx8MPN6bYWZeGHOmjet+O4WV1bX1jeJmaWt7Z3evvH/Q0lGiCG2SiEeqE2JNOZO0aZjhtBMrikXIaTsc32R++5EqzSLZMJOYBgIPJRswgo2V/NRXAjXU9P7hqdQrV9yqOwNaJl5OKpCj3it/+f2IJIJKQzjWuuu5sQlSrAwjnE5LfqJpjMkYD2nXUokF1UE6u3mKTqzSR4NI2ZIGzdTfEykWWk9EaDsFNiO96GXif143MYPLIGUyTgyVZL5okHBkIpQFgPpMUWL4xBJMFLO3IjLCChNjY8pC8BZfXiats6p3Xr26O6/UrvM4inAEx3AKHlxADW6hDk0gEMMzvMKbkzgvzrvzMW8tOPnMIfyB8/kDcMWRUA==</latexit>

Wrapping Up

✤ Argued for the structural equivalence, at the level of correlators, of the tensionless string on . with the dual symmetric orbifold CFT.

✤ Localisation of string path integral to a finite # of configurations essential - signifies an underlying novel topological string theory [See Costello-Paquette(’20) for a bulk spacetime approach].

✤ Underlying free field theory description also leads to localisation [Dei-Gaberdiel-R.G.-Knighton].

✤ Worldsheet is a covering space which wraps the boundary. Good playground to study the interplay of worldsheet CFT and spacetime CFT.

✤ Interesting to study the BPS sector further [see Li-Troost (’20)], duals for other symmetric orbifolds [Dei-Gaberdiel-Knighton; see also Belin talk]. Perturb away from small radius limit with RR flux.

✤ Connect with Berkovits’ approach - tensionless limit in higher dim. AdS [Gaberdiel-R.G.(’20)?].

AdS3 ⇥ S3 ⇥ T 4

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AdS3

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Thank You!