Abstract
We consider string theory on AdS3 × S3 × 𝕋4 in the tensionless limit, with one unit of NS-NS flux. This theory is conjectured to describe the symmetric product orbifold CFT. We consider the string on different Euclidean backgrounds such as thermal AdS3, the BTZ black hole, conical defects and wormhole geometries. In simple examples we compute the full string partition function. We find it to be independent of the precise bulk geometry, but only dependent on the geometry of the conformal boundary. For example, the string partition function on thermal AdS3 and the conical defect with a torus boundary is shown to agree, thus giving evidence for the equivalence of the tensionless string on these different background geometries. We also find that thermal AdS3 and the BTZ black hole are dual descriptions and the vacuum of the BTZ black hole is mapped to a single long string winding many times asymptotically around thermal AdS3. Thus the system yields a concrete example of the string-black hole transition. Consequently, reproducing the boundary partition function does not require a sum over bulk geometries, but rather agrees with the string partition function on any bulk geometry with the appropriate boundary. We argue that the same mechanism can lead to a resolution of the factorization problem when geometries with disconnected boundaries are considered, since the connected and disconnected geometries give the same contribution and we do not have to include them separately.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
N. Berkovits and C. Vafa, Towards a Worldsheet Derivation of the Maldacena Conjecture, AIP Conf. Proc. 1031 (2008) 21 [arXiv:0711.1799] [INSPIRE].
N. Berkovits, Sketching a Proof of the Maldacena Conjecture at Small Radius, JHEP 06 (2019) 111 [arXiv:1903.08264] [INSPIRE].
D. J. Gross and P. F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys. B 303 (1988) 407 [INSPIRE].
M. R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018) 085 [arXiv:1803.04423] [INSPIRE].
G. Giribet, C. Hull, M. Kleban, M. Porrati and E. Rabinovici, Superstrings on AdS3 at ∥ = 1, JHEP 08 (2018) 204 [arXiv:1803.04420] [INSPIRE].
L. Eberhardt, M. R. Gaberdiel and R. Gopakumar, The Worldsheet Dual of the Symmetric Product CFT, JHEP 04 (2019) 103 [arXiv:1812.01007] [INSPIRE].
L. Eberhardt, M. R. Gaberdiel and R. Gopakumar, Deriving the AdS3/CFT2 correspondence, JHEP 02 (2020) 136 [arXiv:1911.00378] [INSPIRE].
L. Eberhardt, AdS3/CFT2 at higher genus, JHEP 05 (2020) 150 [arXiv:2002.11729] [INSPIRE].
Y. Hikida and T. Liu, Correlation functions of symmetric orbifold from AdS3 string theory, JHEP 09 (2020) 157 [arXiv:2005.12511] [INSPIRE].
G. Giribet, String theory on AdS3 × M7 in the tensionless limit, Int. J. Mod. Phys. D 29 (2020) 2030005 [arXiv:2003.02868] [INSPIRE].
A. Karlhede and U. Lindström, The Classical Bosonic String in the Zero Tension Limit, Class. Quant. Grav. 3 (1986) L73 [INSPIRE].
J. Isberg, U. Lindström, B. Sundborg and G. Theodoridis, Classical and quantized tensionless strings, Nucl. Phys. B 411 (1994) 122 [hep-th/9307108] [INSPIRE].
H. Gustafsson, U. Lindström, P. Saltsidis, B. Sundborg and R. van Unge, Hamiltonian BRST quantization of the conformal string, Nucl. Phys. B 440 (1995) 495 [hep-th/9410143] [INSPIRE].
U. Lindström and M. Zabzine, Tensionless strings, WZW models at critical level and massless higher spin fields, Phys. Lett. B 584 (2004) 178 [hep-th/0305098] [INSPIRE].
I. Bakas and C. Sourdis, On the tensionless limit of gauged WZW models, JHEP 06 (2004) 049 [hep-th/0403165] [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
X. Yin, Partition Functions of Three-Dimensional Pure Gravity, Commun. Num. Theor. Phys. 2 (2008) 285 [arXiv:0710.2129] [INSPIRE].
S. Giombi, A. Maloney and X. Yin, One-loop Partition Functions of 3D Gravity, JHEP 08 (2008) 007 [arXiv:0804.1773] [INSPIRE].
C. A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
N. Benjamin, H. Ooguri, S.-H. Shao and Y. Wang, Light-cone modular bootstrap and pure gravity, Phys. Rev. D 100 (2019) 066029 [arXiv:1906.04184] [INSPIRE].
L. F. Alday and J.-B. Bae, Rademacher Expansions and the Spectrum of 2d CFT, JHEP 11 (2020) 134 [arXiv:2001.00022] [INSPIRE].
N. Benjamin, S. Collier and A. Maloney, Pure Gravity and Conical Defects, JHEP 09 (2020) 034 [arXiv:2004.14428] [INSPIRE].
H. Maxfield and G. J. Turiaci, The path integral of 3D gravity near extremality; or, JT gravity with defects as a matrix integral, JHEP 01 (2021) 118 [arXiv:2006.11317] [INSPIRE].
A. Castro, M. R. Gaberdiel, T. Hartman, A. Maloney and R. Volpato, The Gravity Dual of the Ising Model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].
C.-M. Jian, A. W. W. Ludwig, Z.-X. Luo, H.-Y. Sun and Z. Wang, Establishing strongly-coupled 3D AdS quantum gravity with Ising dual using all-genus partition functions, JHEP 10 (2020) 129 [arXiv:1907.06656] [INSPIRE].
S. W. Hawking and D. N. Page, Thermodynamics of Black Holes in anti-de Sitter Space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
C. A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [INSPIRE].
P. Saad, S. H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [INSPIRE].
D. Stanford and E. Witten, JT Gravity and the Ensembles of Random Matrix Theory, arXiv:1907.03363 [INSPIRE].
J. Cotler and K. Jensen, AdS3 gravity and random CFT, arXiv:2006.08648 [INSPIRE].
A. Belin and J. de Boer, Random Statistics of OPE Coefficients and Euclidean Wormholes, arXiv:2006.05499 [INSPIRE].
N. Afkhami-Jeddi, H. Cohn, T. Hartman and A. Tajdini, Free partition functions and an averaged holographic duality, JHEP 01 (2021) 130 [arXiv:2006.04839] [INSPIRE].
A. Maloney and E. Witten, Averaging over Narain moduli space, JHEP 10 (2020) 187 [arXiv:2006.04855] [INSPIRE].
A. Altland and J. Sonner, Late time physics of holographic quantum chaos, arXiv:2008.02271 [INSPIRE].
L. Susskind, Some speculations about black hole entropy in string theory, hep-th/9309145 [INSPIRE].
G. T. Horowitz and J. Polchinski, A correspondence principle for black holes and strings, Phys. Rev. D 55 (1997) 6189 [hep-th/9612146] [INSPIRE].
A. Giveon, D. Kutasov, E. Rabinovici and A. Sever, Phases of quantum gravity in AdS3 and linear dilaton backgrounds, Nucl. Phys. B 719 (2005) 3 [hep-th/0503121] [INSPIRE].
A. Pakman, L. Rastelli and S. S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
B. Maskit, Kleinian groups, vol. 287, Springer Science & Business Media (2012).
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].
A. Giveon, D. Kutasov and N. Seiberg, Comments on string theory on AdS3, Adv. Theor. Math. Phys. 2 (1998) 733 [hep-th/9806194] [INSPIRE].
J. de Boer, H. Ooguri, H. Robins and J. Tannenhauser, String theory on AdS3, JHEP 12 (1998) 026 [hep-th/9812046] [INSPIRE].
E. Witten and S.-T. Yau, Connectedness of the boundary in the AdS/CFT correspondence, Adv. Theor. Math. Phys. 3 (1999) 1635 [hep-th/9910245] [INSPIRE].
K. Krasnov, Holography and Riemann surfaces, Adv. Theor. Math. Phys. 4 (2000) 929 [hep-th/0005106] [INSPIRE].
L. A. Takhtajan and L.-P. Teo, Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography, Commun. Math. Phys. 239 (2003) 183 [math/0204318] [INSPIRE].
X. Yin, On Non-handlebody Instantons in 3D Gravity, JHEP 09 (2008) 120 [arXiv:0711.2803] [INSPIRE].
L. Bers, Simultaneous uniformization, Bull. Am. Math. Soc. 66 (1960) 94.
G. Götz, T. Quella and V. Schomerus, The WZNW model on PSU(1, 1|2), JHEP 03 (2007) 003 [hep-th/0610070] [INSPIRE].
E. J. Martinec and W. McElgin, String theory on AdS orbifolds, JHEP 04 (2002) 029 [hep-th/0106171] [INSPIRE].
E. J. Martinec and W. McElgin, Exciting AdS orbifolds, JHEP 10 (2002) 050 [hep-th/0206175] [INSPIRE].
J. Kim and M. Porrati, On the central charge of spacetime current algebras and correlators in string theory on AdS3, JHEP 05 (2015) 076 [arXiv:1503.07186] [INSPIRE].
D. Kutasov and N. Seiberg, More comments on string theory on AdS3, JHEP 04 (1999) 008 [hep-th/9903219] [INSPIRE].
H. Erbin, J. Maldacena and D. Skliros, Two-Point String Amplitudes, JHEP 07 (2019) 139 [arXiv:1906.06051] [INSPIRE].
J. M. Maldacena and A. Strominger, AdS3 black holes and a stringy exclusion principle, JHEP 12 (1998) 005 [hep-th/9804085] [INSPIRE].
J. D. Brown and M. Henneaux, Central Charges in the Canonical Realization of Asymptotic Symmetries: An Example from Three-Dimensional Gravity, Commun. Math. Phys. 104 (1986) 207 [INSPIRE].
L. Eberhardt and M. R. Gaberdiel, String theory on AdS3 and the symmetric orbifold of Liouville theory, Nucl. Phys. B 948 (2019) 114774 [arXiv:1903.00421] [INSPIRE].
M. Rangamani and S. F. Ross, Winding tachyons in BTZ, Phys. Rev. D 77 (2008) 026010 [arXiv:0706.0663] [INSPIRE].
J. Polchinski, Evaluation of the One Loop String Path Integral, Commun. Math. Phys. 104 (1986) 37 [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].
R. Dijkgraaf, G. W. Moore, E. P. Verlinde and H. L. Verlinde, Elliptic genera of symmetric products and second quantized strings, Commun. Math. Phys. 185 (1997) 197 [hep-th/9608096] [INSPIRE].
J. M. Maldacena, G. W. Moore and A. Strominger, Counting BPS black holes in toroidal Type II string theory, hep-th/9903163 [INSPIRE].
T. Hartman, C. A. Keller and B. Stoica, Universal Spectrum of 2d Conformal Field Theory in the Large c Limit, JHEP 09 (2014) 118 [arXiv:1405.5137] [INSPIRE].
F. M. Haehl and M. Rangamani, Permutation orbifolds and holography, JHEP 03 (2015) 163 [arXiv:1412.2759] [INSPIRE].
C. F. E. Holzhey and F. Wilczek, Black holes as elementary particles, Nucl. Phys. B 380 (1992) 447 [hep-th/9202014] [INSPIRE].
A. Sen, Extremal black holes and elementary string states, Mod. Phys. Lett. A 10 (1995) 2081 [hep-th/9504147] [INSPIRE].
E. Witten, On string theory and black holes, Phys. Rev. D 44 (1991) 314 [INSPIRE].
D. Kutasov and D. A. Sahakyan, Comments on the thermodynamics of little string theory, JHEP 02 (2001) 021 [hep-th/0012258] [INSPIRE].
Y. Nakayama, S.-J. Rey and Y. Sugawara, D-brane propagation in two-dimensional black hole geometries, JHEP 09 (2005) 020 [hep-th/0507040] [INSPIRE].
J. J. Atick and E. Witten, The Hagedorn Transition and the Number of Degrees of Freedom of String Theory, Nucl. Phys. B 310 (1988) 291 [INSPIRE].
M. Berkooz, Z. Komargodski and D. Reichmann, Thermal AdS3, BTZ and competing winding modes condensation, JHEP 12 (2007) 020 [arXiv:0706.0610] [INSPIRE].
F.-L. Lin, T. Matsuo and D. Tomino, Hagedorn Strings and Correspondence Principle in AdS3, JHEP 09 (2007) 042 [arXiv:0705.4514] [INSPIRE].
P. De Lange, A. Maloney and E. Verlinde, Monstrous Product CFTs in the Grand Canonical Ensemble, arXiv:1807.06200 [INSPIRE].
C. Vafa, Modular Invariance and Discrete Torsion on Orbifolds, Nucl. Phys. B 273 (1986) 592 [INSPIRE].
C. Vafa and E. Witten, On orbifolds with discrete torsion, J. Geom. Phys. 15 (1995) 189 [hep-th/9409188] [INSPIRE].
O. Lunin, S. D. Mathur and A. Saxena, What is the gravity dual of a chiral primary?, Nucl. Phys. B 655 (2003) 185 [hep-th/0211292] [INSPIRE].
R. C. Kirby, The topology of 4-manifolds, vol. 1374, Springer (2006).
R. Dijkgraaf, J. M. Maldacena, G. W. Moore and E. P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [INSPIRE].
J. Manschot, AdS3 Partition Functions Reconstructed, JHEP 10 (2007) 103 [arXiv:0707.1159] [INSPIRE].
J. Manschot and G. W. Moore, A Modern Farey Tail, Commun. Num. Theor. Phys. 4 (2010) 103 [arXiv:0712.0573] [INSPIRE].
P. Bantay, Orbifoldization, covering surfaces and uniformization theory, Lett. Math. Phys. 57 (2001) 1 [hep-th/9808023] [INSPIRE].
H. L. Verlinde, Bits, matrices and 1/N, JHEP 12 (2003) 052 [hep-th/0206059] [INSPIRE].
C. Teitelboim, Gravitation and Hamiltonian Structure in Two Space-Time Dimensions, Phys. Lett. B 126 (1983) 41 [INSPIRE].
R. Jackiw, Lower dimensional gravity, Nuclear Physics B 252 (1985) 343.
J. Liu and J. Polchinski, Renormalization of the Mobius Volume, Phys. Lett. B 203 (1988) 39 [INSPIRE].
A. A. Tseytlin, Mobius Infinity Subtraction and Effective Action in σ Model Approach to Closed String Theory, Phys. Lett. B 208 (1988) 221 [INSPIRE].
B. Balthazar, V. A. Rodriguez and X. Yin, ZZ Instantons and the Non-Perturbative Dual of c = 1 String Theory, arXiv:1907.07688 [INSPIRE].
B. Balthazar, V. A. Rodriguez and X. Yin, Multi-Instanton Calculus in c = 1 String Theory, arXiv:1912.07170 [INSPIRE].
A. Sen, Fixing an Ambiguity in Two Dimensional String Theory Using String Field Theory, JHEP 03 (2020) 005 [arXiv:1908.02782] [INSPIRE].
A. Sen, D-instanton Perturbation Theory, JHEP 08 (2020) 075 [arXiv:2002.04043] [INSPIRE].
S. Li and J. Troost, Twisted String Theory in Anti-de Sitter Space, JHEP 11 (2020) 047 [arXiv:2005.13817] [INSPIRE].
S. Li and J. Troost, The Topological Symmetric Orbifold, JHEP 10 (2020) 201 [arXiv:2006.09346] [INSPIRE].
S. Gukov, E. Martinec, G. W. Moore and A. Strominger, The search for a holographic dual to AdS3 × S3 × S3 × S1, Adv. Theor. Math. Phys. 9 (2005) 435 [hep-th/0403090] [INSPIRE].
D. Tong, The holographic dual of AdS3 × S3 × S3 × S1, JHEP 04 (2014) 193 [arXiv:1402.5135] [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].
L. Eberhardt and M. R. Gaberdiel, Strings on AdS3 × S3 × S3 × S1, JHEP 06 (2019) 035 [arXiv:1904.01585] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
J. M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [INSPIRE].
B. Sundborg, Stringy gravity, interacting tensionless strings and massless higher spins, Nucl. Phys. B Proc. Suppl. 102 (2001) 113 [hep-th/0103247] [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and N = 4 SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
N. Seiberg and E. Witten, The D1/D5 system and singular CFT, JHEP 04 (1999) 017 [hep-th/9903224] [INSPIRE].
A. Dei, L. Eberhardt and M. R. Gaberdiel, Three-point functions in AdS3/CFT2 holography, JHEP 12 (2019) 012 [arXiv:1907.13144] [INSPIRE].
Z. Carson, S. Hampton, S. D. Mathur and D. Turton, Effect of the deformation operator in the D1D5 CFT, JHEP 01 (2015) 071 [arXiv:1410.4543] [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].
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].
K. Skenderis and B. C. van Rees, Holography and wormholes in 2+1 dimensions, Commun. Math. Phys. 301 (2011) 583 [arXiv:0912.2090] [INSPIRE].
D. Jafferis, Stringy ER=EPR, Talk given at the KITP conference “Order from Chaos”, December 2018.
M. R. Gaberdiel and R. Gopakumar, Stringy Symmetries and the Higher Spin Square, J. Phys. A 48 (2015) 185402 [arXiv:1501.07236] [INSPIRE].
M. R. Gaberdiel and R. Gopakumar, String Theory as a Higher Spin Theory, JHEP 09 (2016) 085 [arXiv:1512.07237] [INSPIRE].
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].
K. Gawędzki, Noncompact WZW conformal field theories, in NATO Advanced Study Institute: New Symmetry Principles in Quantum Field Theory, (1991), pp. 247–274 [hep-th/9110076] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2008.07533
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Eberhardt, L. Partition functions of the tensionless string. J. High Energ. Phys. 2021, 176 (2021). https://doi.org/10.1007/JHEP03(2021)176
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2021)176