Abstract
Superstring theory on \( {\mathrm{AdS}}_3 \times {S}^3\times {\mathbb{T}}^4 \) with the smallest amount of NS-NS flux (“k = 1”) is shown to be dual to the spacetime CFT given by the large N limit of the free symmetric product orbifold SymN\( \left({\mathbb{T}}^4\right) \). To define the worldsheet theory at k = 1, we employ the hybrid formalism in which the AdS3 × S3 part is described by the \( \mathfrak{p}\mathfrak{s}\mathfrak{u}{\left(1,1\Big|2\right)}_1 \) WZW model (which is well defined). Unlike the case for k ≥ 2, it turns out that the string spectrum at k = 1 does not exhibit the long string continuum, and perfectly matches with the large N limit of the symmetric product. We also demonstrate that the fusion rules of the symmetric orbifold are reproduced from the worldsheet perspective. Our proposal therefore affords a tractable worldsheet description of a tensionless limit in string theory, for which the dual CFT is also explicitly known.
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J.M. Maldacena and H. Ooguri, Strings in AdS3 and SL(2, ℝ) WZW model 1.: The Spectrum, J. Math. Phys. 42 (2001) 2929 [hep-th/0001053] [INSPIRE].
J.M. Maldacena, H. Ooguri and J. Son, Strings in AdS3 and the SL(2, ℝ) WZW model. Part 2. Euclidean black hole, J. Math. Phys. 42 (2001) 2961 [hep-th/0005183] [INSPIRE].
J.M. Maldacena and H. Ooguri, Strings in AdS3 and the SL(2, ℝ) WZW model. Part 3. Correlation functions, Phys. Rev. D 65 (2002) 106006 [hep-th/0111180] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
J.R. David, G. Mandal and S.R. Wadia, Microscopic formulation of black holes in string theory, Phys. Rept. 369 (2002) 549 [hep-th/0203048] [INSPIRE].
L. Eberhardt, M.R. Gaberdiel and W. Li, A holographic dual for string theory on AdS3 × S3 × S3 × S1, JHEP 08 (2017) 111 [arXiv:1707.02705] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
F. Larsen and E.J. Martinec, U(1) charges and moduli in the D1-D5 system, JHEP 06 (1999) 019 [hep-th/9905064] [INSPIRE].
M.R. Gaberdiel, R. Gopakumar and C. Hull, Stringy AdS3 from the worldsheet, JHEP 07 (2017) 090 [arXiv:1704.08665] [INSPIRE].
K. Ferreira, M.R. Gaberdiel and J.I. Jottar, Higher spins on AdS3 from the worldsheet, JHEP 07 (2017) 131 [arXiv:1704.08667] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
N. Berkovits, C. Vafa and E. Witten, Conformal field theory of AdS background with Ramond-Ramond flux, JHEP 03 (1999) 018 [hep-th/9902098] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3 at k = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
R. Argurio, A. Giveon and A. Shomer, Superstrings on AdS3 and symmetric products, JHEP 12 (2000) 003 [hep-th/0009242] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Stringy Symmetries and the Higher Spin Square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
D. Israel, C. Kounnas and M.P. Petropoulos, Superstrings on NS5 backgrounds, deformed AdS3 and holography, JHEP 10 (2003) 028 [hep-th/0306053] [INSPIRE].
S. Raju, Counting giant gravitons in AdS3, Phys. Rev. D 77 (2008) 046012 [arXiv:0709.1171] [INSPIRE].
M. Cho, S. Collier and X. Yin, Strings in Ramond-Ramond Backgrounds from the Neveu-Schwarz-Ramond Formalism, arXiv:1811.00032 [INSPIRE].
M. Bershadsky, S. Zhukov and A. Vaintrob, PSL(n|n) σ-model as a conformal field theory, Nucl. Phys. B 559 (1999) 205 [hep-th/9902180] [INSPIRE].
S.K. Ashok, R. Benichou and J. Troost, Conformal Current Algebra in Two Dimensions, JHEP 06 (2009) 017 [arXiv:0903.4277] [INSPIRE].
R. Benichou and J. Troost, The Conformal Current Algebra on Supergroups with Applications to the Spectrum and Integrability, JHEP 04 (2010) 121 [arXiv:1002.3712] [INSPIRE].
L. Eberhardt and K. Ferreira, The plane-wave spectrum from the worldsheet, JHEP 10 (2018) 109 [arXiv:1805.12155] [INSPIRE].
L. Eberhardt and K. Ferreira, Long strings and chiral primaries in the hybrid formalism, JHEP 02 (2019) 098 [arXiv:1810.08621] [INSPIRE].
J. Troost, Massless particles on supergroups and AdS3 × S3 supergravity, JHEP 07 (2011) 042 [arXiv:1102.0153] [INSPIRE].
M.R. Gaberdiel and S. Gerigk, The massless string spectrum on AdS3 × S3 from the supergroup, JHEP 10 (2011) 045 [arXiv:1107.2660] [INSPIRE].
S. Gerigk, String States on AdS3 × S3 from the Supergroup, JHEP 10 (2012) 084 [arXiv:1208.0345] [INSPIRE].
G. Götz, T. Quella and V. Schomerus, The WZNW model on PSU(1, 1|2), JHEP 03 (2007) 003 [hep-th/0610070] [INSPIRE].
M. Sugiura, Unitary Representations and Harmonic Analysis, an Introduction, North-Holland (1990).
A. Kitaev, Notes on \( \overset{\sim }{\mathrm{SL}}\left(2,\mathbb{R}\right) \) representations, arXiv:1711.08169 [INSPIRE].
I. Bars, Free fields and new cosets of current algebras, Phys. Lett. B 255 (1991) 353 [INSPIRE].
P. Bowcock, B.L. Feigin, A.M. Semikhatov and A. Taormina, Affine \( \mathfrak{s}\mathfrak{l}\left(2\Big|1\right) \) and affine \( \mathfrak{d}\left(2,1;\ \alpha \right) \) as vertex operator extensions of dual affine \( \mathfrak{s}\mathfrak{l}(2) \) algebras, Commun. Math. Phys. 214 (2000) 495 [hep-th/9907171] [INSPIRE].
M. Henningson, S. Hwang, P. Roberts and B. Sundborg, Modular invariance of SU(1, 1) strings, Phys. Lett. B 267 (1991) 350 [INSPIRE].
M.R. Gaberdiel, Fusion rules and logarithmic representations of a WZW model at fractional level, Nucl. Phys. B 618 (2001) 407 [hep-th/0105046] [INSPIRE].
G.M. Sotkov and M.S. Stanishkov, \( \mathcal{N}=1 \) Superconformal Operator Product Expansions and Superfield Fusion Rules, Phys. Lett. B 177 (1986) 361 [INSPIRE].
P. Goddard, D.I. Olive and G. Waterson, Superalgebras, Symplectic Bosons and the Sugawara Construction, Commun. Math. Phys. 112 (1987) 591 [INSPIRE].
D. Ridout, Fusion in Fractional Level \( \mathfrak{s}\mathfrak{l}(2) \) -Theories with \( k=-\frac{1}{2} \), Nucl. Phys. B 848 (2011) 216 [arXiv:1012.2905] [INSPIRE].
M.R. Gaberdiel and H.G. Kausch, A Local logarithmic conformal field theory, Nucl. Phys. B 538 (1999) 631 [hep-th/9807091] [INSPIRE].
M. Miyamoto, Modular invariance of vertex operator algebras satisfying C2 cofiniteness, Duke Math. J. 122 (2004) 51 [math/0209101].
M. Flohr and M.R. Gaberdiel, Logarithmic torus amplitudes, J. Phys. A 39 (2006) 1955 [hep-th/0509075] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [INSPIRE].
N. Seiberg, Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. B 303 (1988) 286 [INSPIRE].
A. Jevicki, M. Mihailescu and S. Ramgoolam, Gravity from CFT on S N (X): Symmetries and interactions, Nucl. Phys. B 577 (2000) 47 [hep-th/9907144] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
A. Pakman, L. Rastelli and S.S. Razamat, Extremal Correlators and Hurwitz Numbers in Symmetric Product Orbifolds, Phys. Rev. D 80 (2009) 086009 [arXiv:0905.3451] [INSPIRE].
A. Pressley and G. Segal, Loop Groups, Clarendon Press (1986).
M.R. Gaberdiel, WZW models of general simple groups, Nucl. Phys. B 460 (1996) 181 [hep-th/9508105] [INSPIRE].
L. Eberhardt and M.R. Gaberdiel, Strings on AdS3 × S3 × S3 × S1, arXiv:1904.01585 [INSPIRE].
D.J. Gross, High-Energy Symmetries of String Theory, Phys. Rev. Lett. 60 (1988) 1229 [INSPIRE].
B. Sundborg, Stringy gravity, interacting tensionless strings and massless higher spins, Nucl. Phys. Proc. Suppl. 102 (2001) 113 [hep-th/0103247] [INSPIRE].
E. Witten, Spacetime Reconstruction, talk at The John Schwarz 60-th birthday symposium, November 2001 [http://theory.caltech.edu/jhs60/witten/1.html].
A. Mikhailov, Notes on higher spin symmetries, hep-th/0201019 [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, String Theory as a Higher Spin Theory, JHEP 09 (2016) 085 [arXiv:1512.07237] [INSPIRE].
R. Gopakumar, From free fields to AdS, Phys. Rev. D 70 (2004) 025009 [hep-th/0308184] [INSPIRE].
R. Gopakumar, From free fields to AdS. II, Phys. Rev. D 70 (2004) 025010 [hep-th/0402063] [INSPIRE].
R. Gopakumar, From free fields to AdS: III, Phys. Rev. D 72 (2005) 066008 [hep-th/0504229] [INSPIRE].
O. Aharony, J.R. David, R. Gopakumar, Z. Komargodski and S.S. Razamat, Comments on worldsheet theories dual to free large N gauge theories, Phys. Rev. D 75 (2007) 106006 [hep-th/0703141] [INSPIRE].
S.S. Razamat, On a worldsheet dual of the Gaussian matrix model, JHEP 07 (2008) 026 [arXiv:0803.2681] [INSPIRE].
R. Gopakumar, What is the Simplest Gauge-String Duality?, arXiv:1104.2386 [INSPIRE].
R. Gopakumar and R. Pius, Correlators in the Simplest Gauge-String Duality, JHEP 03 (2013) 175 [arXiv:1212.1236] [INSPIRE].
M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B 405 (1993) 279 [hep-th/9302103] [INSPIRE].
N. Berkovits, A New Limit of the AdS5 × S5 σ-model, JHEP 08 (2007) 011 [hep-th/0703282] [INSPIRE].
N. Berkovits and C. Vafa, Towards a Worldsheet Derivation of the Maldacena Conjecture, JHEP 03 (2008) 031 [arXiv:0711.1799] [INSPIRE].
N. Berkovits, Perturbative Super-Yang-Mills from the Topological AdS5 × S5 σ-model, JHEP 09 (2008) 088 [arXiv:0806.1960] [INSPIRE].
N. Berkovits, Simplifying and Extending the AdS5 × S5 Pure Spinor Formalism, JHEP 09 (2009) 051 [arXiv:0812.5074] [INSPIRE].
G. Bonelli, P.A. Grassi and H. Safaai, Exploring Pure Spinor String Theory on AdS4 × ℂℙ3, JHEP 10 (2008) 085 [arXiv:0808.1051] [INSPIRE].
Y. Sugawara, Topological string on \( Ad{S}_3\times \mathcal{N} \), Nucl. Phys. B 576 (2000) 265 [hep-th/9909146] [INSPIRE].
L. Rastelli and M. Wijnholt, Minimal AdS3, Adv. Theor. Math. Phys. 11 (2007) 291 [hep-th/0507037] [INSPIRE].
M. Baggio and A. Sfondrini, Strings on NS-NS Backgrounds as Integrable Deformations, Phys. Rev. D 98 (2018) 021902 [arXiv:1804.01998] [INSPIRE].
O. Ohlsson Sax and B. Stefanski, Closed strings and moduli in AdS3 /CFT2, JHEP 05 (2018) 101 [arXiv:1804.02023] [INSPIRE].
A. Dei and A. Sfondrini, Integrable spin chain for stringy Wess-Zumino-Witten models, JHEP 07 (2018) 109 [arXiv:1806.00422] [INSPIRE].
M.R. Gaberdiel, C. Peng and I.G. Zadeh, Higgsing the stringy higher spin symmetry, JHEP 10 (2015) 101 [arXiv:1506.02045] [INSPIRE].
S. Ribault, Minisuperspace limit of the AdS3 WZNW model, JHEP 04 (2010) 096 [arXiv:0912.4481] [INSPIRE].
F. Lesage, P. Mathieu, J. Rasmussen and H. Saleur, The \( \mathfrak{s}\mathfrak{u}{(2)}_{-1/2} \) WZW model and the βγ system, Nucl. Phys. B 647 (2002) 363 [hep-th/0207201] [INSPIRE].
T. Creutzig and D. Ridout, Modular Data and Verlinde Formulae for Fractional Level WZW Models I, Nucl. Phys. B 865 (2012) 83 [arXiv:1205.6513] [INSPIRE].
M.R. Gaberdiel, T. Hartman and K. Jin, Higher Spin Black Holes from CFT, JHEP 04 (2012) 103 [arXiv:1203.0015] [INSPIRE].
T. Quella and V. Schomerus, Free fermion resolution of supergroup WZNW models, JHEP 09 (2007) 085 [arXiv:0706.0744] [INSPIRE].
M.R. Gaberdiel and I. Runkel, From boundary to bulk in logarithmic CFT, J. Phys. A 41 (2008) 075402 [arXiv:0707.0388] [INSPIRE].
R. Blumenhagen, D. Lüst and S. Theisen, Basic concepts of string theory, Springer (2013).
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Eberhardt, L., Gaberdiel, M.R. & Gopakumar, R. The worldsheet dual of the symmetric product CFT. J. High Energ. Phys. 2019, 103 (2019). https://doi.org/10.1007/JHEP04(2019)103
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DOI: https://doi.org/10.1007/JHEP04(2019)103