Abstract
We explore a C-theorem in defect conformal field theories (DCFTs) that unify all the known conjectures and theorems until now. We examine as a candidate C-function the additional contributions from conformal defects to the sphere free energy and the entanglement entropy across a sphere in a number of examples including holographic models. We find the two quantities are equivalent, when suitably regularized, for codimension-one defects (or boundaries), but differ by a universal constant term otherwise. Moreover, we find in a few field theoretic examples that the sphere free energy decreases but the entanglement entropy increases along a certain renormalization group (RG) flow triggered by a defect localized perturbation which is assumed to have a trivial IR fixed point without defects. We hence propose a C-theorem in DCFTs stating that the increment of the regularized sphere free energy due to the defect does not increase under any defect RG flow. We also provide a proof of our proposal in several holographic models of defect RG flows.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.B. Zamolodchikov, Irreversibility of the flux of the renormalization group in a 2D field theory, JETP Lett. 43 (1986) 730 [INSPIRE].
J.L. Cardy, Is there a c theorem in four-dimensions?, Phys. Lett. B 215 (1988) 749 [INSPIRE].
R.C. Myers and A. Sinha, Seeing a c-theorem with holography, Phys. Rev. D 82 (2010) 046006 [arXiv:1006.1263] [INSPIRE].
R.C. Myers and A. Sinha, Holographic c-theorems in arbitrary dimensions, JHEP 01 (2011) 125 [arXiv:1011.5819] [INSPIRE].
Z. Komargodski and A. Schwimmer, On renormalization group flows in four dimensions, JHEP 12 (2011) 099 [arXiv:1107.3987] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-theorem: N = 2 field theories on the three-sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
I.R. Klebanov, S.S. Pufu and B.R. Safdi, F-theorem without supersymmetry, JHEP 10 (2011) 038 [arXiv:1105.4598] [INSPIRE].
S. Giombi and I.R. Klebanov, Interpolating between a and F , JHEP 03 (2015) 117 [arXiv:1409.1937] [INSPIRE].
T. Kawano, Y. Nakaguchi and T. Nishioka, Holographic interpolation between a and F, JHEP 12 (2014) 161 [arXiv:1410.5973] [INSPIRE].
H. Casini, M. Huerta and R.C. Myers, Towards a derivation of holographic entanglement entropy, JHEP 05 (2011) 036 [arXiv:1102.0440] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
H. Liu and M. Mezei, A refinement of entanglement entropy and the number of degrees of freedom, JHEP 04 (2013) 162 [arXiv:1202.2070] [INSPIRE].
H. Casini and M. Huerta, A finite entanglement entropy and the c-theorem, Phys. Lett. B 600 (2004) 142 [hep-th/0405111] [INSPIRE].
H. Casini, E. Testé and G. Torroba, Markov property of the conformal field theory vacuum and the a theorem, Phys. Rev. Lett. 118 (2017) 261602 [arXiv:1704.01870] [INSPIRE].
N. Lashkari, Entanglement at a scale and renormalization monotones, arXiv:1704.05077 [INSPIRE].
I. Affleck and A.W.W. Ludwig, Universal noninteger ‘ground state degeneracy’ in critical quantum systems, Phys. Rev. Lett. 67 (1991) 161 [INSPIRE].
D. Friedan and A. Konechny, On the boundary entropy of one-dimensional quantum systems at low temperature, Phys. Rev. Lett. 93 (2004) 030402 [hep-th/0312197] [INSPIRE].
H. Casini, I.S. Landea and G. Torroba, The g-theorem and quantum information theory, JHEP 10 (2016) 140 [arXiv:1607.00390] [INSPIRE].
M. Nozaki, T. Takayanagi and T. Ugajin, Central charges for BCFTs and holography, JHEP 06 (2012) 066 [arXiv:1205.1573] [INSPIRE].
D. Gaiotto, Boundary F-maximization, arXiv:1403.8052 [INSPIRE].
J. Estes et al., On holographic defect entropy, JHEP 05 (2014) 084 [arXiv:1403.6475] [INSPIRE].
S. Yamaguchi, Holographic RG flow on the defect and g theorem, JHEP 10 (2002) 002 [hep-th/0207171] [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. Girardello, M. Petrini, M. Porrati and A. Zaffaroni, Novel local CFT and exact results on perturbations of N = 4 super-Yang-Mills from AdS dynamics, JHEP 12 (1998) 022 [hep-th/9810126] [INSPIRE].
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
K. Jensen and A. O’Bannon, Constraint on defect and boundary renormalization group flows, Phys. Rev. Lett. 116 (2016) 091601 [arXiv:1509.02160] [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].
D. Gaiotto and H.-C. Kim, Surface defects and instanton partition functions, JHEP 10 (2016) 012 [arXiv:1412.2781] [INSPIRE].
A. Kapustin, Wilson-’t Hooft operators in four-dimensional gauge theories and S-duality, Phys. Rev. D 74 (2006) 025005 [hep-th/0501015] [INSPIRE].
S. Gukov and E. Witten, Gauge theory, ramification, and the geometric Langlands program, hep-th/0612073 [INSPIRE].
D.M. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522 [cond-mat/9505127] [INSPIRE].
M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
J. Armas and J. Tarrio, On actions for (entangling) surfaces and DCFTs, JHEP 04 (2018) 100 [arXiv:1709.06766] [INSPIRE].
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFT d, JHEP 07 (2013) 113 [arXiv:1210.4258] [INSPIRE].
M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP 01 (2018) 013 [arXiv:1710.11165] [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].
D. Rodriguez-Gomez and J.G. Russo, Free energy and boundary anomalies on \( \mathbb{S} \) a × ℍb spaces, JHEP 10 (2017) 084 [arXiv:1708.00305] [INSPIRE].
A. Lewkowycz and J. Maldacena, Exact results for the entanglement entropy and the energy radiated by a quark, JHEP 05 (2014) 025 [arXiv:1312.5682] [INSPIRE].
T. Nishioka and I. Yaakov, Supersymmetric Renyi entropy, JHEP 10 (2013) 155 [arXiv:1306.2958] [INSPIRE].
S.N. Solodukhin, Entanglement entropy of black holes, Living Rev. Rel. 14 (2011) 8 [arXiv:1104.3712] [INSPIRE].
T. Nishioka, Entanglement entropy: holography and renormalization group, Rev. Mod. Phys. 90 (2018) 035007 [arXiv:1801.10352] [INSPIRE].
C.P. Herzog, K.-W. Huang and K. Jensen, Universal entanglement and boundary geometry in conformal field theory, JHEP 01 (2016) 162 [arXiv:1510.00021] [INSPIRE].
D. Fursaev, Conformal anomalies of CFT’s with boundaries, JHEP 12 (2015) 112 [arXiv:1510.01427] [INSPIRE].
D.V. Fursaev and S.N. Solodukhin, Anomalies, entropy and boundaries, Phys. Rev. D 93 (2016) 084021 [arXiv:1601.06418] [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].
L. Fei, S. Giombi, I.R. Klebanov and G. Tarnopolsky, Generalized F-theorem and the ϵ expansion, JHEP 12 (2015) 155 [arXiv:1507.01960] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of holographic entanglement entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
S.N. Solodukhin, Entanglement entropy, conformal invariance and extrinsic geometry, Phys. Lett. B 665 (2008) 305 [arXiv:0802.3117] [INSPIRE].
J. Gomis and F. Passerini, Holographic Wilson loops, JHEP 08 (2006) 074 [hep-th/0604007] [INSPIRE].
D. Tong and K. Wong, Instantons, Wilson lines and D-branes, Phys. Rev. D 91 (2015) 026007 [arXiv:1410.8523] [INSPIRE].
C. Hoyos, A defect action for Wilson loops, JHEP 07 (2018) 045 [arXiv:1803.09809] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact results for Wilson loops in superconformal Chern-Simons theories with matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
C. Beasley, Localization for Wilson loops in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013) 1 [arXiv:0911.2687] [INSPIRE].
R. Dijkgraaf and E.P. Verlinde, Modular invariance and the fusion algebra, Nucl. Phys. Proc. Suppl. 5 (1988) 87 [INSPIRE].
B. Shi, Seeing topological entanglement through the information convex, arXiv:1810.01986 [INSPIRE].
S. Dong, E. Fradkin, R.G. Leigh and S. Nowling, Topological entanglement entropy in Chern-Simons theories and quantum Hall fluids, JHEP 05 (2008) 016 [arXiv:0802.3231] [INSPIRE].
V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-boundary entanglement in Chern-Simons theory and link invariants, JHEP 04 (2017) 061 [arXiv:1611.05460] [INSPIRE].
G. Wong, A note on entanglement edge modes in Chern Simons theory, JHEP 08 (2018) 020 [arXiv:1706.04666] [INSPIRE].
N. Drukker and D.J. Gross, An exact prediction of N = 4 SUSYM theory for string theory, J. Math. Phys. 42 (2001) 2896 [hep-th/0010274] [INSPIRE].
J. Polchinski and J. Sully, Wilson loop renormalization group flows, JHEP 10 (2011) 059 [arXiv:1104.5077] [INSPIRE].
M. Beccaria, S. Giombi and A. Tseytlin, Non-supersymmetric Wilson loop in \( \mathcal{N} \) = 4 SYM and defect 1d CFT, JHEP 03 (2018) 131 [arXiv:1712.06874] [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].
M. Mariño and P. Putrov, Exact results in ABJM theory from topological strings, JHEP 06 (2010) 011 [arXiv:0912.3074] [INSPIRE].
L. Bianchi, M. Preti and E. Vescovi, Exact Bremsstrahlung functions in ABJM theory, JHEP 07 (2018) 060 [arXiv:1802.07726] [INSPIRE].
M. Tierz, Wilson loops and free energies in 3d \( \mathcal{N} \) = 4 SYM: exact results, exponential asymptotics and duality, arXiv:1804.10845 [INSPIRE].
I. Yaakov, Redeeming bad theories, JHEP 11 (2013) 189 [arXiv:1303.2769] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
L.-Y. Hung, R.C. Myers, M. Smolkin and A. Yale, Holographic calculations of Renyi entropy, JHEP 12 (2011) 047 [arXiv:1110.1084] [INSPIRE].
D. Bak, M. Gutperle and S. Hirano, A Dilatonic deformation of AdS 5 and its field theory dual, JHEP 05 (2003) 072 [hep-th/0304129] [INSPIRE].
H.-C. Chang and A. Karch, Entanglement entropy for probe branes, JHEP 01 (2014) 180 [arXiv:1307.5325] [INSPIRE].
D.Z. Freedman, C. Núñez, M. Schnabl and K. Skenderis, Fake supergravity and domain wall stability, Phys. Rev. D 69 (2004) 104027 [hep-th/0312055] [INSPIRE].
C. Closset et al., Contact terms, unitarity and F-maximization in three-dimensional superconformal theories, JHEP 10 (2012) 053 [arXiv:1205.4142] [INSPIRE].
H. Casini, E. Teste and G. Torroba, Relative entropy and the RG flow, JHEP 03 (2017) 089 [arXiv:1611.00016] [INSPIRE].
T. Faulkner, A. Lewkowycz and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074 [arXiv:1307.2892] [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: 1810.06995
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
Kobayashi, N., Nishioka, T., Sato, Y. et al. Towards a C-theorem in defect CFT. J. High Energ. Phys. 2019, 39 (2019). https://doi.org/10.1007/JHEP01(2019)039
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2019)039