Abstract
We use entanglement entropy to define a central charge associated to a twodimensional defect or boundary in a conformal field theory (CFT). We present holographic calculations of this central charge for several maximally supersymmetric CFTs dual to eleven-dimensional supergravity in Anti-de Sitter space, namely the M5-brane theory with a Wilson surface defect and three-dimensional CFTs related to the M2-brane theory with a boundary. Our results for the central charge depend on a partition of N M2-branes ending on M M5-branes. For the Wilson surface, the partition specifies a representation of the gauge algebra, and we write our result for the central charge in a compact form in terms of the algebra’s Weyl vector and the representation’s highest weight vector. We explore how the central charge scales with N and M for some examples of partitions. In general the central charge does not scale as M3 or N3/2, the number of degrees of freedom of the M5- or M2-brane theory at large M or N , respectively.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
K. Becker, M. Becker and J.H. Schwarz, String theory and M-theory: A modern introduction, Cambridge University Press (2006) [INSPIRE].
P. West, Introduction to strings and branes, Cambridge University Press (2012) [INSPIRE].
A. Strominger, Open p-branes, Phys. Lett. B 383 (1996) 44 [hep-th/9512059] [INSPIRE].
E. Witten, Bound states of strings and p-branes, Nucl. Phys. B 460 (1996) 335 [hep-th/9510135] [INSPIRE].
J. Bagger and N. Lambert, Modeling Multiple M2’s, Phys. Rev. D 75 (2007) 045020 [hep-th/0611108] [INSPIRE].
A. Gustavsson, Algebraic structures on parallel M2-branes, Nucl. Phys. B 811 (2009) 66 [arXiv:0709.1260] [INSPIRE].
J. Bagger and N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008 [arXiv:0711.0955] [INSPIRE].
J. Bagger and N. Lambert, Comments on multiple M2-branes, JHEP 02 (2008) 105 [arXiv:0712.3738] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [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].
I.R. Klebanov and A.A. Tseytlin, Entropy of near extremal black p-branes, Nucl. Phys. B 475 (1996) 164 [hep-th/9604089] [INSPIRE].
N. Drukker, M. Mariño and P. Putrov, From weak to strong coupling in ABJM theory, Commun. Math. Phys. 306 (2011) 511 [arXiv:1007.3837] [INSPIRE].
D.M. Kaplan and J. Michelson, Zero modes for the D = 11 membrane and five-brane, Phys. Rev. D 53 (1996) 3474 [hep-th/9510053] [INSPIRE].
P.S. Howe, N.D. Lambert and P.C. West, The Selfdual string soliton, Nucl. Phys. B 515 (1998) 203 [hep-th/9709014] [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
P. Pasti, D.P. Sorokin and M. Tonin, Covariant action for a D = 11 five-brane with the chiral field, Phys. Lett. B 398 (1997) 41 [hep-th/9701037] [INSPIRE].
I.A. Bandos, K. Lechner, A. Nurmagambetov, P. Pasti, D.P. Sorokin and M. Tonin, Covariant action for the superfive-brane of M-theory, Phys. Rev. Lett. 78 (1997) 4332 [hep-th/9701149] [INSPIRE].
W. Nahm, Supersymmetries and their Representations, Nucl. Phys. B 135 (1978) 149 [INSPIRE].
J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [INSPIRE].
O.J. Ganor, Six-dimensional tensionless strings in the large N limit, Nucl. Phys. B 489 (1997) 95 [hep-th/9605201] [INSPIRE].
M. Bullimore and H.-C. Kim, The Superconformal Index of the (2, 0) Theory with Defects, JHEP 05 (2015) 048 [arXiv:1412.3872] [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in M-theory. I: Local Solutions, JHEP 08 (2008) 028 [arXiv:0806.0605] [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS 7 × S 4, JHEP 12 (2008) 044 [arXiv:0810.4647] [INSPIRE].
J. Estes, R. Feldman and D. Krym, Exact half-BPS flux solutions in M theory with D(2, 1; c ′; 0)2 symmetry: Local solutions, Phys. Rev. D 87 (2013) 046008 [arXiv:1209.1845] [INSPIRE].
C. Bachas, E. D’Hoker, J. Estes and D. Krym, M-theory Solutions Invariant under D(2, 1; γ) ⊕ D(2, 1; γ), Fortsch. Phys. 62 (2014) 207 [arXiv:1312.5477] [INSPIRE].
A. Sevrin, W. Troost and A. Van Proeyen, Superconformal Algebras in Two-Dimensions with N = 4, Phys. Lett. B 208 (1988) 447 [INSPIRE].
L. Frappat, P. Sorba and A. Sciarrino, Dictionary on Lie superalgebras, hep-th/9607161 [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle, D. Krym and P. Sorba, Half-BPS supergravity solutions and superalgebras, JHEP 12 (2008) 047 [arXiv:0810.1484] [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle and D. Krym, Janus solutions in M-theory, JHEP 06 (2009) 018 [arXiv:0904.3313] [INSPIRE].
N. Bobev, K. Pilch and N.P. Warner, Supersymmetric Janus Solutions in Four Dimensions, JHEP 06 (2014) 058 [arXiv:1311.4883] [INSPIRE].
K. Jensen and A. O’Bannon, Holography, Entanglement Entropy and Conformal Field Theories with Boundaries or Defects, Phys. Rev. D 88 (2013) 106006 [arXiv:1309.4523] [INSPIRE].
J. Estes, K. Jensen, A. O’Bannon, E. Tsatis and T. Wrase, On Holographic Defect Entropy, JHEP 05 (2014) 084 [arXiv:1403.6475] [INSPIRE].
S.A. Gentle, M. Gutperle and C. Marasinou, Entanglement entropy of Wilson surfaces from bubbling geometries in M-theory, JHEP 08 (2015) 019 [arXiv:1506.00052] [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].
D.V. Fursaev, Quantum Entanglement on Boundaries, JHEP 07 (2013) 119 [arXiv:1305.2334] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, Anomalies, entropy and boundaries, Phys. Rev. D 93 (2016) 084021 [arXiv:1601.06418] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on Defect and Boundary Renormalization Grou Flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [INSPIRE].
S.P. Kumar and D. Silvani, Holographic flows and thermodynamics of Polyakov loop impurities, JHEP 03 (2017) 107 [arXiv:1611.06033] [INSPIRE].
S.P. Kumar and D. Silvani, Entanglement of heavy quark impurities and generalized gravitational entropy, JHEP 01 (2018) 052 [arXiv:1711.01554] [INSPIRE].
N. Kobayashi, T. Nishioka, Y. Sato and K. Watanabe, Towards a C-theorem in defect CFT, JHEP 01 (2019) 039 [arXiv:1810.06995] [INSPIRE].
R. Rodgers, Holographic entanglement entropy from probe M-theory branes, JHEP 03 (2019) 092 [arXiv:1811.12375] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, \( \mathcal{W} \) symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
C. Cordova, T.T. Dumitrescu and X. Yin, Higher Derivative Terms, Toroidal Compactification and Weyl Anomalies in Six-Dimensional (2, 0) Theories, arXiv:1505.03850 [INSPIRE].
D. Freed, J.A. Harvey, R. Minasian and G.W. Moore, Gravitational anomaly cancellation for M-theory five-branes, Adv. Theor. Math. Phys. 2 (1998) 601 [hep-th/9803205] [INSPIRE].
J.A. Harvey, R. Minasian and G.W. Moore, NonAbelian tensor multiplet anomalies, JHEP 09 (1998) 004 [hep-th/9808060] [INSPIRE].
M. Henningson and K. Skenderis, The Holographic Weyl anomaly, JHEP 07 (1998) 023 [hep-th/9806087] [INSPIRE].
K. Jensen, A. O’Bannon, B. Robinson and R. Rodgers, From the Weyl Anomaly to Entropy of Two-Dimensional Boundaries and Defects, arXiv:1812.08745 [INSPIRE].
C.R. Graham and E. Witten, Conformal anomaly of submanifold observables in AdS/CFT correspondence, Nucl. Phys. B 546 (1999) 52 [hep-th/9901021] [INSPIRE].
M. Henningson and K. Skenderis, Weyl anomaly for Wilson surfaces, JHEP 06 (1999) 012 [hep-th/9905163] [INSPIRE].
A. Schwimmer and S. Theisen, Entanglement Entropy, Trace Anomalies and Holography, Nucl. Phys. B 801 (2008) 1 [arXiv:0802.1017] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central Charges for BCFTs and Holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
S.N. Solodukhin, Boundary terms of conformal anomaly, Phys. Lett. B 752 (2016) 131 [arXiv:1510.04566] [INSPIRE].
C. Herzog, K.-W. Huang and K. Jensen, Displacement Operators and Constraints on Boundary Central Charges, Phys. Rev. Lett. 120 (2018) 021601 [arXiv:1709.07431] [INSPIRE].
C.P. Herzog and K.-W. Huang, Boundary Conformal Field Theory and a Boundary Central Charge, JHEP 10 (2017) 189 [arXiv:1707.06224] [INSPIRE].
R.C. Myers, Dielectric branes, JHEP 12 (1999) 022 [hep-th/9910053] [INSPIRE].
O. Lunin, 1/2-BPS states in M-theory and defects in the dual CFTs, JHEP 10 (2007) 014 [arXiv:0704.3442] [INSPIRE].
D. Marolf, Chern-Simons terms and the three notions of charge, in Quantization, gauge theory and strings. Proceedings, International Conference dedicated to the memory of Professor Efim Fradkin, Moscow, Russia, June 5-10, 2000, Vol. 1+2, pp. 312-320 (2000) [hep-th/0006117] [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].
T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42 (2009) 504008 [arXiv:0905.0932] [INSPIRE].
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
D.S. Berman, M-theory branes and their interactions, Phys. Rept. 456 (2008) 89 [arXiv:0710.1707] [INSPIRE].
D.S. Berman and J.A. Harvey, The Self-dual string and anomalies in the M5-brane, JHEP 11 (2004) 015 [hep-th/0408198] [INSPIRE].
V. Niarchos and K. Siampos, Entropy of the self-dual string soliton, JHEP 07 (2012) 134 [arXiv:1206.2935] [INSPIRE].
O. Ganor and L. Motl, Equations of the (2, 0) theory and knitted five-branes, JHEP 05 (1998) 009 [hep-th/9803108] [INSPIRE].
K.A. Intriligator, Anomaly matching and a Hopf-Wess-Zumino term in 6d, N = (2, 0) field theories, Nucl. Phys. B 581 (2000) 257 [hep-th/0001205] [INSPIRE].
F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
V. Niarchos and K. Siampos, M2-M5 blackfold funnels, JHEP 06 (2012) 175 [arXiv:1205.1535] [INSPIRE].
R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, Essentials of Blackfold Dynamics, JHEP 03 (2010) 063 [arXiv:0910.1601] [INSPIRE].
R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, Blackfolds in Supergravity and String Theory, JHEP 08 (2011) 154 [arXiv:1106.4428] [INSPIRE].
R. Emparan, Blackfolds, in Black holes in higher dimensions, G.T. Horowitz ed., pp. 180-212, (2012) [arXiv:1106.2021] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
D.S. Berman, M.J. Perry, E. Sezgin and D.C. Thompson, Boundary Conditions for Interacting Membranes, JHEP 04 (2010) 025 [arXiv:0912.3504] [INSPIRE].
V. Niarchos, A Lagrangian for self-dual strings, JHEP 12 (2015) 060 [arXiv:1509.07676] [INSPIRE].
A. Basu and J.A. Harvey, The M2-M5 brane system and a generalized Nahm’s equation, Nucl. Phys. B 713 (2005) 136 [hep-th/0412310] [INSPIRE].
B. Haghighat, A. Iqbal, C. Kozçaz, G. Lockhart and C. Vafa, M-Strings, Commun. Math. Phys. 334 (2015) 779 [arXiv:1305.6322] [INSPIRE].
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
N. Hama and K. Hosomichi, Seiberg-Witten Theories on Ellipsoids, JHEP 09 (2012) 033 [arXiv:1206.6359] [INSPIRE].
L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [INSPIRE].
C. Kozcaz, S. Pasquetti and N. Wyllard, A & B model approaches to surface operators and Toda theories, JHEP 08 (2010) 042 [arXiv:1004.2025] [INSPIRE].
M. Bullimore, M. Fluder, L. Hollands and P. Richmond, The superconformal index and an elliptic algebra of surface defects, JHEP 10 (2014) 062 [arXiv:1401.3379] [INSPIRE].
J. Gomis and B. Le Floch, M2-brane surface operators and gauge theory dualities in Toda, JHEP 04 (2016) 183 [arXiv:1407.1852] [INSPIRE].
J. Gomis, B. Le Floch, Y. Pan and W. Peelaers, Intersecting Surface Defects and Two-Dimensional CFT, Phys. Rev. D 96 (2017) 045003 [arXiv:1610.03501] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1812.00923
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Estes, J., Krym, D., O’Bannon, A. et al. Wilson surface central charge from holographic entanglement entropy. J. High Energ. Phys. 2019, 32 (2019). https://doi.org/10.1007/JHEP05(2019)032
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2019)032