Abstract
We provide a precise relation between an ensemble of Narain conformal field theories (CFTs) with central charge c = n, and a sum of (U(1) × U(1))n Chern-Simons theories on different handlebody topologies. We begin by reviewing the general relation of additive codes to Narain CFTs. Then we describe a holographic duality between any given Narain theory and a pure Chern-Simons theory on a handlebody manifold. We proceed to consider an ensemble of Narain theories, defined in terms of an ensemble of codes of length n over ℤk × ℤk for prime k. We show that averaging over this ensemble is holographically dual to a level-k (U(1) × U(1))n Chern-Simons theory, summed over a finite number of inequivalent classes of handlebody topologies. In the limit of large k the ensemble approaches the ensemble of all Narain theories, and its bulk dual becomes equivalent to “U(1)-gravity” — the sum of the pertubative part of the Chern-Simons wavefunction over all possible handlebodies — providing a bulk microscopic definition for this theory. Finally, we reformulate the sum over handlebodies in terms of Hecke operators, paving the way for generalizations.
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References
J.M. Maldacena and L. Maoz, Wormholes in AdS, JHEP 02 (2004) 053 [hep-th/0401024] [INSPIRE].
P. Saad, S.H. Shenker and D. Stanford, JT gravity as a matrix integral, arXiv:1903.11115 [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. Pérez and R. Troncoso, Gravitational dual of averaged free CFT’s over the Narain lattice, arXiv:2006.08216 [https://doi.org/10.1007/JHEP11(2020)015].
A. Dymarsky and A. Shapere, Comments on the holographic description of Narain theories, JHEP 10 (2021) 197 [arXiv:2012.15830] [INSPIRE].
S. Datta et al., Adding flavor to the Narain ensemble, JHEP 05 (2022) 090 [arXiv:2102.12509] [INSPIRE].
N. Benjamin, C.A. Keller, H. Ooguri and I.G. Zadeh, Narain to Narnia, Commun. Math. Phys. 390 (2022) 425 [arXiv:2103.15826] [INSPIRE].
N. Benjamin et al., Harmonic analysis of 2d CFT partition functions, JHEP 09 (2021) 174 [arXiv:2107.10744] [INSPIRE].
V. Meruliya and S. Mukhi, AdS3 gravity and RCFT ensembles with multiple invariants, JHEP 08 (2021) 098 [arXiv:2104.10178] [INSPIRE].
J. Dong, T. Hartman and Y. Jiang, Averaging over moduli in deformed WZW models, JHEP 09 (2021) 185 [arXiv:2105.12594] [INSPIRE].
M. Ashwinkumar et al., Chern-Simons invariants from ensemble averages, JHEP 08 (2021) 044 [arXiv:2104.14710] [INSPIRE].
S. Collier and A. Maloney, Wormholes and spectral statistics in the Narain ensemble, JHEP 03 (2022) 004 [arXiv:2106.12760] [INSPIRE].
S. Chakraborty and A. Hashimoto, Weighted average over the Narain moduli space as a \( T\overline{T} \) deformation of the CFT target space, Phys. Rev. D 105 (2022) 086018 [arXiv:2109.10382] [INSPIRE].
J. Raeymaekers, A note on ensemble holography for rational tori, JHEP 12 (2021) 177 [arXiv:2110.08833] [INSPIRE].
F. Benini, C. Copetti and L. Di Pietro, Factorization and global symmetries in holography, SciPost Phys. 14 (2023) 019 [arXiv:2203.09537] [INSPIRE].
J. Kames-King, A. Kanargias, B. Knighton and M. Usatyuk, The Lion, the Witch, and the Wormhole: Ensemble averaging the symmetric product orbifold, arXiv:2306.07321 [INSPIRE].
M. Ashwinkumar, J.M. Leedom and M. Yamazaki, Duality Origami: Emergent Ensemble Symmetries in Holography and Swampland, arXiv:2305.10224 [INSPIRE].
M. Ashwinkumar, A. Kidambi, J.M. Leedom and M. Yamazaki, Generalized Narain Theories Decoded: Discussions on Eisenstein series, Characteristics, Orbifolds, Discriminants and Ensembles in any Dimension, arXiv:2311.00699 [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
A. Dymarsky and A. Shapere, Quantum stabilizer codes, lattices, and CFTs, JHEP 03 (2020) 160 [arXiv:2009.01244] [INSPIRE].
S. Yahagi, Narain CFTs and error-correcting codes on finite fields, JHEP 08 (2022) 058 [arXiv:2203.10848] [INSPIRE].
N. Angelinos, D. Chakraborty and A. Dymarsky, Optimal Narain CFTs from codes, JHEP 11 (2022) 118 [arXiv:2206.14825] [INSPIRE].
I. Frenkel, J. Lepowsky and A. Meurman, Vertex operator algebras and the monster, Academic Press (1989) [ISBN: 9780080874548].
L. Dolan, P. Goddard and P. Montague, Conformal field theories, representations and lattice constructions, Commun. Math. Phys. 179 (1996) 61 [hep-th/9410029] [INSPIRE].
M. Miyamoto, Binary codes and vertex operator (super) algebras, J. Algebra 181 (1996) 207.
C. Dong, R.L. Griess Jr. and G. Hoehn, Framed vertex operator algebras, codes and the moonshine module, Commun. Math. Phys. 193 (1998) 407 [q-alg/9707008] [INSPIRE].
C.H. Lam and H. Yamada, Z2 × Z2 Codes and Vertex Operator Algebras, J. Algebra 224 (2000) 268.
D. Gaiotto and T. Johnson-Freyd, Holomorphic SCFTs with small index, Can. J. Math. 74 (2022) 573 [arXiv:1811.00589] [INSPIRE].
Y. Moriwaki, Code conformal field theory and framed algebra, arXiv:2104.10094 [INSPIRE].
A. Almheiri, X. Dong and D. Harlow, Bulk Locality and Quantum Error Correction in AdS/CFT, JHEP 04 (2015) 163 [arXiv:1411.7041] [INSPIRE].
A. Dymarsky and A. Shapere, Solutions of modular bootstrap constraints from quantum codes, Phys. Rev. Lett. 126 (2021) 161602 [arXiv:2009.01236] [INSPIRE].
A. Dymarsky and A. Sharon, Non-rational Narain CFTs from codes over F4, JHEP 11 (2021) 016 [arXiv:2107.02816] [INSPIRE].
M. Buican, A. Dymarsky and R. Radhakrishnan, Quantum codes, CFTs, and defects, JHEP 03 (2023) 017 [arXiv:2112.12162] [INSPIRE].
J. Henriksson, A. Kakkar and B. McPeak, Classical codes and chiral CFTs at higher genus, JHEP 05 (2022) 159 [arXiv:2112.05168] [INSPIRE].
J. Henriksson and B. McPeak, Averaging over codes and an SU(2) modular bootstrap, JHEP 11 (2023) 035 [arXiv:2208.14457] [INSPIRE].
J. Henriksson, A. Kakkar and B. McPeak, Narain CFTs and quantum codes at higher genus, JHEP 04 (2023) 011 [arXiv:2205.00025] [INSPIRE].
A. Dymarsky and R.R. Kalloor, Fake Z, JHEP 06 (2023) 043 [arXiv:2211.15699] [INSPIRE].
K. Kawabata, T. Nishioka and T. Okuda, Narain CFTs from qudit stabilizer codes, SciPost Phys. Core 6 (2023) 035 [arXiv:2212.07089] [INSPIRE].
Y. Furuta, On the Rationality and the Code Structure of a Narain CFT, and the Simple Current Orbifold, arXiv:2307.04190 [INSPIRE].
Y.F. Alam et al., Narain CFTs from nonbinary stabilizer codes, JHEP 12 (2023) 127 [arXiv:2307.10581] [INSPIRE].
K. Kawabata, T. Nishioka and T. Okuda, Supersymmetric conformal field theories from quantum stabilizer codes, Phys. Rev. D 108 (2023) L081901 [arXiv:2307.14602] [INSPIRE].
K. Kawabata and S. Yahagi, Fermionic CFTs from classical codes over finite fields, JHEP 05 (2023) 096 [arXiv:2303.11613] [INSPIRE].
K. Kawabata and S. Yahagi, Elliptic genera from classical error-correcting codes, JHEP 01 (2024) 130 [arXiv:2308.12592] [INSPIRE].
K. Kawabata, T. Nishioka and T. Okuda, Narain CFTs from quantum codes and their ℤ2 gauging, JHEP 05 (2024) 133 [arXiv:2308.01579] [INSPIRE].
M. Buican and R. Radhakrishnan, Invertibility of Condensation Defects and Symmetries of 2 + 1d QFTs, arXiv:2309.15181 [INSPIRE].
M. Buican and R. Radhakrishnan, Qudit Stabilizer Codes, CFTs, and Topological Surfaces, arXiv:2311.13680 [INSPIRE].
L. Clozel, H. Oh and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001) 327.
J.H. Conway and N.J.A. Sloane, Sphere packings, lattices and groups, Springer Science & Business Media (2013) [https://doi.org/10.1007/978-1-4757-2249-9].
P. Kraus and F. Larsen, Partition functions and elliptic genera from supergravity, JHEP 01 (2007) 002 [hep-th/0607138] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Bos and V.P. Nair, U(1) Chern-Simons theory and c = 1 conformal blocks, Phys. Lett. B 223 (1989) 61 [INSPIRE].
S. Elitzur, G.W. Moore, A. Schwimmer and N. Seiberg, Remarks on the Canonical Quantization of the Chern-Simons-Witten Theory, Nucl. Phys. B 326 (1989) 108 [INSPIRE].
S. Gukov, E. Martinec, G.W. Moore and A. Strominger, Chern-Simons gauge theory and the AdS3/CFT2 correspondence, in the proceedings of the From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan, Oxford, U.K., January 8–10 (2004) [https://doi.org/10.1142/9789812775344_0036] [hep-th/0403225] [INSPIRE].
L.D. Faddeev and R. Jackiw, Hamiltonian Reduction of Unconstrained and Constrained Systems, Phys. Rev. Lett. 60 (1988) 1692 [INSPIRE].
R. Jackiw, (Constrained) quantization without tears, in the proceedings of the 2nd Workshop on Constraint Theory and Quantization Methods, Montepulciano, Italy, June 28 – July 01 (1993) [hep-th/9306075] [INSPIRE].
M. Porrati and C. Yu, Partition functions of Chern-Simons theory on handlebodies by radial quantization, JHEP 07 (2021) 194 [arXiv:2104.12799] [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
L. Eberhardt, Partition functions of the tensionless string, JHEP 03 (2021) 176 [arXiv:2008.07533] [INSPIRE].
L. Eberhardt, Summing over Geometries in String Theory, JHEP 05 (2021) 233 [arXiv:2102.12355] [INSPIRE].
A. Dymarsky, J. Henriksson and B. McPeak, in progress.
B. Schoeneberg, Elliptic modular functions: an introduction, Springer Science & Business Media, (2012) [https://doi.org/10.1007/978-3-642-65663-7].
R. Dijkgraaf, J.M. Maldacena, G.W. Moore and E.P. Verlinde, A Black hole Farey tail, hep-th/0005003 [INSPIRE].
A. Maloney and E. Witten, Quantum Gravity Partition Functions in Three Dimensions, JHEP 02 (2010) 029 [arXiv:0712.0155] [INSPIRE].
C.A. Keller and A. Maloney, Poincaré Series, 3D Gravity and CFT Spectroscopy, JHEP 02 (2015) 080 [arXiv:1407.6008] [INSPIRE].
A. Castro et al., The Gravity Dual of the Ising Model, Phys. Rev. D 85 (2012) 024032 [arXiv:1111.1987] [INSPIRE].
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, On moduli spaces of conformal field theories with c ≥ 1, in Perspectives in String Theory, P. Di Vecchia and JL Petersen eds., World Scientific (1988).
N.I. Koblitz, Introduction to elliptic curves and modular forms, Springer Science & Business Media (2012), p. 175 [https://doi.org/10.1007/978-1-4612-0909-6].
N. Benjamin et al., S-duality in \( T\overline{T} \)-deformed CFT, JHEP 05 (2023) 140 [arXiv:2302.09677] [INSPIRE].
D. Goldstein and A. Mayer, On the equidistribution of Hecke points, Forum Math. 15 (2003) 165.
L.J. Dixon, V. Kaplunovsky and J. Louis, Moduli dependence of string loop corrections to gauge coupling constants, Nucl. Phys. B 355 (1991) 649 [INSPIRE].
B. Runge, Codes and Siegel modular forms, Discrete Math. 148 (1996) 175.
B. Runge, On Siegel modular forms, part I, J. Reine Angew. Math. (Crelles Journal) 1993 (1993) 57.
G. Bossard, C. Cosnier-Horeau and B. Pioline, Exact effective interactions and 1/4-BPS dyons in heterotic CHL orbifolds, SciPost Phys. 7 (2019) 028 [arXiv:1806.03330] [INSPIRE].
A. Blommaert, L.V. Iliesiu and J. Kruthoff, Gravity factorized, JHEP 09 (2022) 080 [arXiv:2111.07863] [INSPIRE].
A. Blommaert, L.V. Iliesiu and J. Kruthoff, Alpha states demystified — towards microscopic models of AdS2 holography, JHEP 08 (2022) 071 [arXiv:2203.07384] [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].
E. Witten, Three-Dimensional Gravity Revisited, arXiv:0706.3359 [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].
C. Nayak et al., Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083 [arXiv:0707.1889] [INSPIRE].
F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, JHEP 06 (2015) 149 [arXiv:1503.06237] [INSPIRE].
P. Di Francesco and P. Mathieu and D. Sénéchal, Conformal field theory, Springer Science & Business Media (2012) [https://doi.org/10.1007/978-1-4612-2256-9].
Acknowledgments
We thank Ahmed Barbar, Nathan Benjamin, Debarghya Chakraborty, Mathew Dodelson, Daniel Jafferis, Johan Henriksson, Brian McPeak, Adam Schwimmer and Edward Witten for discussions. The work of OA was supported in part by an Israel Science Foundation (ISF) center for excellence grant (grant number 2289/18), by ISF grant no. 2159/22, by Simons Foundation grant 994296 (Simons Collaboration on Confinement and QCD Strings), by grant no. 2018068 from the United States-Israel Binational Science Foundation (BSF), by the Minerva foundation with funding from the Federal German Ministry for Education and Research, by the German Research Foundation through a German-Israeli Project Cooperation (DIP) grant “Holography and the Swampland”, and by a research grant from Martin Eisenstein. OA is the Samuel Sebba Professorial Chair of Pure and Applied Physics. A.D. is grateful to Weizmann Institute of Science for hospitality and acknowledges sabbatical support of the Schwartz/Reisman Institute for Theoretical Physics, and support by the NSF under grants PHY-2013812 and 2310426. This work was performed in part at Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452. A.S. thanks the Institute for Advanced Study for hospitality and sabbatical support, and the Simons Center for Geometry and Physics for hospitality during the 2023 Simons Summer Workshop.
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Aharony, O., Dymarsky, A. & Shapere, A.D. Holographic description of Narain CFTs and their code-based ensembles. J. High Energ. Phys. 2024, 343 (2024). https://doi.org/10.1007/JHEP05(2024)343
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DOI: https://doi.org/10.1007/JHEP05(2024)343