Abstract
In this paper we study the properties of two-dimensional CFTs defined by cyclic and symmetric orbifolds of free Dirac fermions, especially by focusing on the partition function and entanglement entropy. Via the bosonization, we construct the twist operators which glue two complex planes to calculate the partition function of ℤ2 orbifold CFT on a torus. We also find an expression of ℤN cyclic orbifold in terms of Hecke operators, which provides an explicit relation between the partition functions of cyclic orbifolds and those of symmetric ones. We compute the entanglement entropy and Renyi entropy in cyclic orbifolds on a circle both for finite temperature states and for time-dependent states under quantum quenches. We find that the replica method calculation is highly non-trivial and new because of the contributions from replicas with different boundary conditions. We find the full expression for the ℤ2 orbifold and show that the periodicity gets doubled. Finally, we discuss extensions of our results on entanglement entropy to symmetric orbifold CFTs and make a heuristic argument towards holographic CFTs.
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References
A. Strominger and C. Vafa, Microscopic origin of the Bekenstein-Hawking entropy, Phys. Lett. B 379 (1996) 99 [hep-th/9601029] [INSPIRE].
C. Vafa, Instantons on D-branes, Nucl. Phys. B 463 (1996) 435 [hep-th/9512078] [INSPIRE].
C. Vafa, Gas of D-branes and Hagedorn density of BPS states, Nucl. Phys. B 463 (1996) 415 [hep-th/9511088] [INSPIRE].
M.R. Douglas, Branes within branes, NATO Sci. Ser. C 520 (1999) 267 [hep-th/9512077] [INSPIRE].
R. Dijkgraaf, Instanton strings and hyperKähler geometry, Nucl. Phys. B 543 (1999) 545 [hep-th/9810210] [INSPIRE].
P.S. Aspinwall, Enhanced gauge symmetries and K3 surfaces, Phys. Lett. B 357 (1995) 329 [hep-th/9507012] [INSPIRE].
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [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].
S.G. Avery, B.D. Chowdhury and S.D. Mathur, Deforming the D1D5 CFT away from the orbifold point, JHEP 06 (2010) 031 [arXiv:1002.3132] [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].
J. de Boer, Large N elliptic genus and AdS/CFT correspondence, JHEP 05 (1999) 017 [hep-th/9812240] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from Conformal Field Theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
C.A. Keller, Phase transitions in symmetric orbifold CFTs and universality, JHEP 03 (2011) 114 [arXiv:1101.4937] [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].
A. Pakman, L. Rastelli and S.S. Razamat, Diagrams for Symmetric Product Orbifolds, JHEP 10 (2009) 034 [arXiv:0905.3448] [INSPIRE].
A. Belin, C.A. Keller and A. Maloney, Permutation Orbifolds in the large N Limit, arXiv:1509.01256 [INSPIRE].
E. Perlmutter, Bounding the Space of Holographic CFTs with Chaos, JHEP 10 (2016) 069 [arXiv:1602.08272] [INSPIRE].
P. Caputa, Y. Kusuki, T. Takayanagi and K. Watanabe, Evolution of Entanglement Entropy in Orbifold CFTs, J. Phys. A 50 (2017) 244001 [arXiv:1701.03110] [INSPIRE].
S. Giusto and R. Russo, Entanglement Entropy and D1-D5 geometries, Phys. Rev. D 90 (2014) 066004 [arXiv:1405.6185] [INSPIRE].
V. Balasubramanian, B.D. Chowdhury, B. Czech and J. de Boer, Entwinement and the emergence of spacetime, JHEP 01 (2015) 048 [arXiv:1406.5859] [INSPIRE].
V. Balasubramanian, B. Craps, B. Czech and G. Sárosi, Echoes of chaos from string theory black holes, JHEP 03 (2017) 154 [arXiv:1612.04334] [INSPIRE].
V. Balasubramanian, B. Craps, T. De Jonckheere and G. Sárosi, Entanglement versus entwinement in symmetric product orbifolds, JHEP 01 (2019) 190 [arXiv:1806.02871] [INSPIRE].
L. Apolo, A. Belin, S. Bintanja, A. Castro and C.A. Keller, Deforming symmetric product orbifolds: a tale of moduli and higher spin currents, JHEP 08 (2022) 159 [arXiv:2204.07590] [INSPIRE].
A. Belin, N. Benjamin, A. Castro, S.M. Harrison and C.A. Keller, \( \mathcal{N} \) = 2 Minimal Models: A Holographic Needle in a Symmetric Orbifold Haystack, SciPost Phys. 8 (2020) 084 [arXiv:2002.07819] [INSPIRE].
A. Belin, S. Biswas and J. Sully, The spectrum of boundary states in symmetric orbifolds, JHEP 01 (2022) 123 [arXiv:2110.05491] [INSPIRE].
M.R. Gaberdiel and R. Gopakumar, Higher Spins & Strings, JHEP 11 (2014) 044 [arXiv:1406.6103] [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].
M.R. Gaberdiel and R. Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018) 085 [arXiv:1803.04423] [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, Summing over Geometries in String Theory, JHEP 05 (2021) 233 [arXiv:2102.12355] [INSPIRE].
C. Holzhey, F. Larsen and F. Wilczek, Geometric and renormalized entropy in conformal field theory, Nucl. Phys. B 424 (1994) 443 [hep-th/9403108] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
S. Furukawa, V. Pasquier and J. Shiraishi, Mutual Information and Compactification Radius in a c=1 Critical Phase in One Dimension, Phys. Rev. Lett. 102 (2009) 170602 [arXiv:0809.5113] [INSPIRE].
M. Headrick, Entanglement Renyi entropies in holographic theories, Phys. Rev. D 82 (2010) 126010 [arXiv:1006.0047] [INSPIRE].
T. Azeyanagi, T. Nishioka and T. Takayanagi, Near Extremal Black Hole Entropy as Entanglement Entropy via AdS2/CFT1, Phys. Rev. D 77 (2008) 064005 [arXiv:0710.2956] [INSPIRE].
N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi Surfaces and Entanglement Entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].
S. Mukhi, S. Murthy and J.-Q. Wu, Entanglement, Replicas, and Thetas, JHEP 01 (2018) 005 [arXiv:1706.09426] [INSPIRE].
T. Takayanagi and T. Ugajin, Measuring Black Hole Formations by Entanglement Entropy via Coarse-Graining, JHEP 11 (2010) 054 [arXiv:1008.3439] [INSPIRE].
P. Calabrese and J.L. Cardy, Evolution of entanglement entropy in one-dimensional systems, J. Stat. Mech. 0504 (2005) P04010 [cond-mat/0503393] [INSPIRE].
J. Abajo-Arrastia, J. Aparicio and E. Lopez, Holographic Evolution of Entanglement Entropy, JHEP 11 (2010) 149 [arXiv:1006.4090] [INSPIRE].
T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014 [arXiv:1303.1080] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
V.E. Hubeny, M. Rangamani and T. Takayanagi, A Covariant holographic entanglement entropy proposal, JHEP 07 (2007) 062 [arXiv:0705.0016] [INSPIRE].
T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602 [arXiv:1105.5165] [INSPIRE].
M. Fujita, T. Takayanagi and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043 [arXiv:1108.5152] [INSPIRE].
L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Strings on Orbifolds, Nucl. Phys. B 261 (1985) 678 [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].
F.M. Haehl and M. Rangamani, Permutation orbifolds and holography, JHEP 03 (2015) 163 [arXiv:1412.2759] [INSPIRE].
A. Klemm and M.G. Schmidt, Orbifolds by Cyclic Permutations of Tensor Product Conformal Field Theories, Phys. Lett. B 245 (1990) 53 [INSPIRE].
H. Casini, C.D. Fosco and M. Huerta, Entanglement and alpha entropies for a massive Dirac field in two dimensions, J. Stat. Mech. 0507 (2005) P07007 [cond-mat/0505563] [INSPIRE].
O. Lunin and S.D. Mathur, Correlation functions for MN/SN orbifolds, Commun. Math. Phys. 219 (2001) 399 [hep-th/0006196] [INSPIRE].
B. Chen and J.-q. Wu, Universal relation between thermal entropy and entanglement entropy in conformal field theories, Phys. Rev. D 91 (2015) 086012 [arXiv:1412.0761] [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Graduate Texts in Contemporary Physics, Springer-Verlag, New York, U.S.A. (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
D.-E. Diaconescu, M.R. Douglas and J. Gomis, Fractional branes and wrapped branes, JHEP 02 (1998) 013 [hep-th/9712230] [INSPIRE].
M. Billó, B. Craps and F. Roose, On D-branes in type 0 string theory, Phys. Lett. B 457 (1999) 61 [hep-th/9902196] [INSPIRE].
D.-E. Diaconescu and J. Gomis, Fractional branes and boundary states in orbifold theories, JHEP 10 (2000) 001 [hep-th/9906242] [INSPIRE].
M. Billó, B. Craps and F. Roose, Orbifold boundary states from Cardy’s condition, JHEP 01 (2001) 038 [hep-th/0011060] [INSPIRE].
T. Takayanagi and T. Uesugi, D-branes in Melvin background, JHEP 11 (2001) 036 [hep-th/0110200] [INSPIRE].
J.M. Maldacena and L. Susskind, D-branes and fat black holes, Nucl. Phys. B 475 (1996) 679 [hep-th/9604042] [INSPIRE].
J.M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021 [hep-th/0106112] [INSPIRE].
G. Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, Graduate Studies in Mathematics, American Mathematical Society (2015).
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Takayanagi, T., Tsuda, T. Free fermion cyclic/symmetric orbifold CFTs and entanglement entropy. J. High Energ. Phys. 2022, 4 (2022). https://doi.org/10.1007/JHEP12(2022)004
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DOI: https://doi.org/10.1007/JHEP12(2022)004