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Boundary and defect CFT: open problems and applications. (English) Zbl 1519.81440

Summary: A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7–8 Sept. 2017. We attempt to provide a broad, bird’s-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof.

MSC:

81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
81T60 Supersymmetric field theories in quantum mechanics

Software:

GitHub

References:

[1] Affleck I 1995 Conformal field theory approach to the Kondo effect Acta Phys. Polon. B 26 1869 · Zbl 0966.81561
[2] Recknagel A and Schomerus V 2013 Boundary conformal field theory and the worldsheet approach to D-branes (Cambridge: : Cambridge University Press) · Zbl 1332.81005 · doi:10.1017/CBO9780511806476
[3] Herzog C P and Huang K-W 2017 Boundary conformal field theory and a boundary central charge J. High Energy Phys. JHEP10(2017)189 · Zbl 1383.81229 · doi:10.1007/jhep10(2017)189
[4] Herzog C P, Huang K-W and Jensen K 2017 Displacement operators and constraints on boundary central charges (arXiv:1709.07431 [hep-th])
[5] Cardy J 2017 Bulk renormalization group flows and boundary states in conformal field theories SciPost Phys.3 011 · doi:10.21468/scipostphys.3.2.011
[6] Cardy J L and Lewellen D C 1991 Bulk and boundary operators in conformal field theory Phys. Lett. B 259 274 · doi:10.1016/0370-2693(91)90828-e
[7] Konechny A 2017 RG boundaries and interfaces in Ising field theory J. Phys. A: Math. Theor.50 145403 · Zbl 1372.82019 · doi:10.1088/1751-8121/aa60f6
[8] Yurov V P and Zamolodchikov A B 1990 Truncated conformal space approach to scaling Lee-Yang model Int. J. Mod. Phys. A 05 3221 · doi:10.1142/s0217751x9000218x
[9] Hogervorst M, Rychkov S and van Rees B C 2015 A cheap alternative to the lattice? Phys. Rev. D 91 025005 · doi:10.1103/physrevd.91.025005
[10] Konechny A 2017 RG boundaries and interfaces in Ising field theory J. Phys. A: Math. Theor.50 14 · Zbl 1372.82019 · doi:10.1088/1751-8121/aa60f6
[11] Brunner I and Roggenkamp D 2008 Defects and bulk perturbations of boundary Landau-Ginzburg orbifolds J. High Energy Phys. JHEP04(2008)001 · Zbl 1246.81315 · doi:10.1088/1126-6708/2008/04/001
[12] Fonseca P and Zamolodchikov A 2003 Ising field theory in a magnetic field: analytic properties of the free energy J. Stat. Phys.110 527 · Zbl 1016.82014 · doi:10.1023/A:1022147532606
[13] Quella T, Runkel I and Watts G M T 2007 Reflection and transmission for conformal defects J. High Energy Phys. JHEP04(2007)095 · doi:10.1088/1126-6708/2007/04/095
[14] Klebanov I R and Witten E 1999 Ads/CFT correspondence and symmetry breaking Nucl. Phys. B 556 89 · Zbl 0958.81134 · doi:10.1016/s0550-3213(99)00387-9
[15] Melby-Thompson C M and Schmidt-Colinet C 2017 Double trace interfaces J. High Energy Phys. JHEP11(2017)110 · Zbl 1383.81248 · doi:10.1007/jhep11(2017)110
[16] Bak D, Gutperle M and Hirano S 2003 A dilatonic deformation ofAdS5and its field theory dual J. High Energy Phys. JHEP05(2003)072 · doi:10.1088/1126-6708/2003/05/072
[17] McAvity D M and Osborn H 1995 Conformal field theories near a boundary in general dimensions Nucl. Phys. B 455 522 · Zbl 0925.81295 · doi:10.1016/0550-3213(95)00476-9
[18] Aharony O, DeWolfe O, Freedman D Z and Karch A 2003 Defect conformal field theory and locally localized gravity J. High Energy Phys. JHEP07(2003)030 · doi:10.1088/1126-6708/2003/07/030
[19] Díaz D E and Dorn H 2007 Partition functions and double-trace deformations in AdS/CFT J. High Energy Phys. JHEP05(2007)046 · doi:10.1088/1126-6708/2007/05/046
[20] Gaberdiel M R and Gopakumar R 2013 Minimal model holography J. Phys. A: Math. Theor.46 214002 · Zbl 1276.81125 · doi:10.1088/1751-8113/46/21/214002
[21] Gaiotto D 2012 Domain walls for two-dimensional renormalization group flows J. High Energy Phys. JHEP12(2012)103 · Zbl 1397.81180 · doi:10.1007/jhep12(2012)103
[22] Sneddon I N 1966 Mixed Boundary Value Problems in Potential Theory (Amsterdam: North-Holland) · Zbl 0139.28801
[23] Cardy J 2004 Boundary conformal field theory (arXiv:hep-th/0411189)
[24] Cho G Y, Ludwig A W W and Ryu S 2017 Universal entanglement spectra of gapped one-dimensional field theories Phys. Rev. B 95 115122 · doi:10.1103/physrevb.95.115122
[25] Cho G Y, Shiozaki K, Ryu S and Ludwig A W W 2017 Relationship between symmetry protected topological phases and boundary conformal field theories via the entanglement spectrum J. Phys. A: Math. Gen.50 304002 · Zbl 1375.81213 · doi:10.1088/1751-8121/aa7782
[26] Han B, Tiwari A, Hsieh C-T and Ryu S 2017 Boundary conformal field theory and symmetry-protected topological phases in 2 +1 dimensions Phys. Rev. B 96 125105 · doi:10.1103/physrevb.96.125105
[27] Fliss J R, Wen X, Parrikar O, Hsieh C-T, Han B, Hughes T L and Leigh R G 2017 Interface contributions to topological entanglement in abelian Chern-Simons theory J. High Energy Phys. JHEP09(2017)056 · Zbl 1382.58019 · doi:10.1007/jhep09(2017)056
[28] Wong G 2017 A note on entanglement edge modes in Chern Simons theory (arXiv:1706.04666)
[29] Wen X, Matsuura S and Ryu S 2016 Edge theory approach to topological entanglement entropy, mutual information, and entanglement negativity in Chern-Simons theories Phys. Rev. B 93 245140 · doi:10.1103/physrevb.93.245140
[30] Qi X-L, Katsura H and Ludwig A W W 2012 General relationship between the entanglement spectrum and the edge state spectrum of topological quantum states Phys. Rev. Lett.108 196402 · doi:10.1103/physrevlett.108.196402
[31] Chen X, Gu Z-C, Liu Z-X and Wen X-G 2013 Symmetry protected topological orders and the group cohomology of their symmetry group Phys. Rev. B 87 155114 · doi:10.1103/physrevb.87.155114
[32] Ryu S and Zhang S-C 2012 Interacting topological phases and modular invariance Phys. Rev. B 85 245132 · doi:10.1103/physrevb.85.245132
[33] Konechny A 2017 RG boundaries and interfaces in Ising field theory J. Phys. A: Math. Gen.50 14 · Zbl 1372.82019 · doi:10.1088/1751-8121/aa60f6
[34] Miyaji M, Ryu S, Takayanagi T and Wen X 2015 Boundary states as holographic duals of trivial spacetimes J. High Energy Phys. JHEP05(2015)152 · Zbl 1388.83305 · doi:10.1007/jhep05(2015)152
[35] Cho G Y, Hsieh C-T, Morimoto T and Ryu S 2015 Topological phases protected by reflection symmetry and cross-cap states Phys. Rev. B 91 195142 · doi:10.1103/physrevb.91.195142
[36] Kane C L and Fisher M P A 1992 Transmission through barriers and resonant tunneling in an interacting one-dimensional electron gas Phys. Rev. B 46 15233 · doi:10.1103/physrevb.46.15233
[37] Giamarchi T 2003 Quantum Physics in One Dimension (Oxford: Clarendon) · Zbl 1075.81001 · doi:10.1093/acprof:oso/9780198525004.001.0001
[38] Furusaki A and Nagaosa N 1993 Resonant tunneling in a Luttinger liquid Phys. Rev. B 47 3827 · doi:10.1103/physrevb.47.3827
[39] Komnik A and Gogolin A O 2003 Resonant tunneling between Luttinger liquids: a solvable case Phys. Rev. Lett.90 246403 · doi:10.1103/physrevlett.90.246403
[40] Furusaki A and Matveev K A 2002 Occupation of a resonant level coupled to a chiral Luttinger liquid Phys. Rev. Lett.88 226404 · doi:10.1103/physrevlett.88.226404
[41] Furusaki A 1998 Resonant tunneling through a quantum dot weakly coupled to quantum wires or quantum Hall edge states Phys. Rev. B 57 7141 · doi:10.1103/physrevb.57.7141
[42] Goldstein M, Weiss Y and Berkovits R 2009 Interacting resonant level coupled to a Luttinger liquid: Universality of thermodynamic properties Europhys. Lett.86 67012 · doi:10.1209/0295-5075/86/67012
[43] Goldstein M and Berkovits R 2010 Duality between different geometries of a resonant level in a Luttinger liquid Phys. Rev. Lett.104 106403 · doi:10.1103/physrevlett.104.106403
[44] Goldstein M and Berkovits R 2010 Capacitance of a resonant level coupled to Luttinger liquids Phys. Rev. B 82 161307 · doi:10.1103/physrevb.82.161307
[45] Kakashvili P and Johannesson H 2003 Measuring Luttinger liquid correlations from charge fluctuations in a nanoscale structure Phys. Rev. Lett.91 186403 · doi:10.1103/physrevlett.91.186403
[46] Wächter P, Menden V and Schönhammer K 2007 Charging of a quantum dot coupled to Luttinger-liquid leads Phys. Rev. B 76 129905 · doi:10.1103/physrevb.76.129905
[47] Lerner I V, Yudson V I and Yurkevich I V 2008 Quantum wire hybridized with a single-level impurity Phys. Rev. Lett.100 25 · doi:10.1103/physrevlett.100.256805
[48] Haldane F H M 1981 Effective harmonic-fluid approach to low-energy properties of one-dimensional quantum fluids Phys. Rev. Lett.47 25 · doi:10.1103/physrevlett.47.1840
[49] Bockrath M, Cobden D H, Lu J, Rinzler A G, Smalley R E, Balents L and McEuen P L 1999 Luttinger-liquid behaviour in carbon nanotubes Nature397 598 · doi:10.1038/17569
[50] Yao Z, Postma H W C, Balents L and Dekker C 1999 Carbon nanotube intramolecular junctions Nature402 273 · doi:10.1038/46241
[51] Chang A M 2003 Chiral Luttinger liquids at the fractional quantum Hall edge Rev. Mod. Phys.75 1449 · Zbl 1205.82126 · doi:10.1103/revmodphys.75.1449
[52] Milliken F P, Umbach C P and Webb R A 1996 Indications of a Luttinger liquid in the fractional quantum Hall regimet Solid State Commun.97 309 · doi:10.1016/0038-1098(95)00181-6
[53] Maasilta I J and Goldman V J 1997 Line shape of resonant tunneling between fractional quantum Hall edges Phys. Rev. B 55 4081 · doi:10.1103/physrevb.55.4081
[54] Citro R, Orignac E, De Palo S and Chiofalo M L 2007 Evidence of Luttinger-liquid behavior in one-dimensional dipolar quantum gases Phys. Rev. A 75 051602 · doi:10.1103/physreva.75.051602
[55] Recati A, Fedichev P O, Zwerger W, von Delft J and Zoller P 2005 Atomic quantum dots coupled to a reservoir of a superfluid Bose-Einstein condensate Phys. Rev. Lett.94 040404 · doi:10.1103/physrevlett.94.040404
[56] Jiang L, Baksmaty L O, Hu H, Chen Y and Pu H 2011 Single impurity in ultracold Fermi superfluids Phys. Rev. A 83 061604 · doi:10.1103/physreva.83.061604
[57] Bloch I, Dalibard J and Zwerger W 2009 Many-body physics with ultracold gases Rev. Mod. Phys.80 885 · doi:10.1103/revmodphys.80.885
[58] Del Maestro A, Boninsegni M and Affleck I 2011 4He Phys. Rev. Lett.106 105303 · doi:10.1103/physrevlett.106.105303
[59] Duc P F, Savard M, Petrescu M, Rosenow B, Del Maestro A and Gervais G 2015 Critical flow and dissipation in a quasi-one-dimensional superfluid Sci. Adv.1 e1400222 · doi:10.1126/sciadv.1400222
[60] Safi I and Saleur H 2004 One-channel conductor in an ohmic environment: mapping to a Tomonaga-Luttinger liquid and full counting statistics Phys. Rev. Lett.93 126602 · doi:10.1103/physrevlett.93.126602
[61] Le Hur K and Li M-R 2005 Unification of electromagnetic noise and Luttinger liquid via a quantum dot Phys. Rev. B 72 073305 · doi:10.1103/physrevb.72.073305
[62] Mebrahtu H T, Brozenets I V, Liu D E, Zheng H, Bomze Y V, Smirnov A I, Baranger H U and Finkelstein G 2012 Quantum phase transition in a resonant level coupled to interacting leads Nature488 7409 · doi:10.1038/nature11265
[63] Mebrahtu H T, Borzenets I V, Zheng H, Bomze Y V, Smirnov A I, Florens S, Baranger H U and Finkelstein G 2013 Observation of Majorana quantum critical behaviour in a resonant level coupled to a dissipative environment Nat. Phys.9 732 · doi:10.1038/nphys2735
[64] Andrei N 1992 Integrable Models in Condensed Matter Physics (Singapore: World Scientific)
[65] Tsvelick A M and Wiegmann P B 1983 Exact results in the theory of magnetic alloys Adv. Phys.32 453 · doi:10.1080/00018738300101581
[66] Rylands C and Andrei N 2016 Quantum impurity in a Luttinger liquid: exact solution of the Kane-Fisher model Phys. Rev. B 94 115142 · doi:10.1103/physrevb.94.115142
[67] Zamolodchikov A B 1991 TBA equations for integrable perturbed SU(2)k×SU(2)lSU(2)k+1 coset models Nucl. Phys. B 366 122 · doi:10.1016/0550-3213(91)90054-2
[68] Fendley P, Saleur H and Warner N P 1994 Exact solution of a massless scalar field with a relevant boundary interaction Nucl. Phys. B 430 577 · Zbl 1020.81686 · doi:10.1016/0550-3213(94)90160-0
[69] Rylands C and Andrei N 2017 Quantum dot in interacting environments (arXiv:1708.07212 [cond-mat.str-el])
[70] Ghoshal S and Zamolodchikov A 1994 Boundary S matrix and boundary state in two-dimensional integrable quantum field theory Int. J. Mod. Phys. A 09 3841-86 · Zbl 0985.81714 · doi:10.1142/s0217751x94001552
[71] MacIntyre A 1995 Integrable boundary conditions for classical sine-Gordon theory J. Phys. A: Math. Gen.28 1089-100 · Zbl 0853.35107 · doi:10.1088/0305-4470/28/4/029
[72] Arthur R, Dorey P and Parini R 2016 Breaking integrability at the boundary: the sine-Gordon model with Robin boundary conditions J. Phys. A: Math. Theor.49 165205 · Zbl 1350.35167 · doi:10.1088/1751-8113/49/16/165205
[73] See the solitonscattering GitHub repository at https://github.com/rparini/solitonscattering
[74] Campbell D K, Schonfeld J F and Wingate C A 1983 Resonance structure in kink-antikink interactions in ϕ4 theory Phys. D 9 1 · doi:10.1016/0167-2789(83)90289-0
[75] Anninos P, Oliveira S and Matzner R A 1991 Fractal structure in the scalar λ(φ2−1)2 theory λ (ϕ 2 − 1) 2 theory Phys. Rev. D 44 1147 · doi:10.1103/physrevd.44.1147
[76] Goodman R and Haberman R 2007 Chaotic scattering and the n-bounce resonance in solitary wave interactions Phys. Rev. Lett.98 104103 · doi:10.1103/physrevlett.98.104103
[77] Dorey P, Mersh K, Romanczukiewicz T and Shnir Y 2011 Kink-antikink collisions in the ϕ6 model Phys. Rev. Lett.107 091602 · doi:10.1103/physrevlett.107.091602
[78] Dorey P, Halavanau A, Mercer J, Romanczukiewicz T and Shnir Y 2017 Boundary scattering in the ϕ4 model J. High Energy Phys. JHEP05(2017)107 · Zbl 1380.81310 · doi:10.1007/JHEP05(2017)107
[79] Pelloni B 2015 Advances in the study of boundary value problems for nonlinear integrable PDEs Nonlinearity28 2 · Zbl 1315.35007 · doi:10.1088/0951-7715/28/2/r1
[80] Delfino G, Mussardo G and Simonetti P 1996 Non-integrable quantum field theories as perturbations of certain integrable models Nucl. Phys. B 473 469 · Zbl 0925.81296 · doi:10.1016/0550-3213(96)00265-9
[81] Petkova V B and Zuber J-B 2001 Generalized twisted partition functions Phys. Lett. B 504 157 · Zbl 0977.81128 · doi:10.1016/S0370-2693(01)00276-3
[82] Gang D and Yamaguchi S 2008 Superconformal defects in the tricritical Ising model J. High Energy Phys. JHEP12(2008)076 · Zbl 1329.81329 · doi:10.1088/1126-6708/2008/12/076
[83] Makabe I and Watts G M T 2017 Defects in the Tri-critical Ising model J. High Energy Phys. JHEP09(2017)013 · Zbl 1382.81189 · doi:10.1007/JHEP09(2017)013
[84] Novak S and Runkel I 2015 Spin from defects in two-dimensional quantum field theory (arXiv:1506.07547)
[85] Komargodski S and Seiberg N 2010 Comments on supercurrent multiplets, supersymmetric field theories and supergravity J. High Energy Phys. JHEP07(2010)017 · Zbl 1290.81073 · doi:10.1007/jhep07(2010)017
[86] Drukker N, Martelli D and Shamir I 2017 The energy-momentum multiplet of supersymmetric defect field theories J. High Energy Phys. JHEP08(2017)010 · Zbl 1381.81136 · doi:10.1007/jhep08(2017)010
[87] Karch A and Randall L 2001 Locally localized gravity J. High Energy Phys. JHEP05(2001)008 · Zbl 1047.81062 · doi:10.1088/1126-6708/2001/05/008
[88] DeWolfe O, Freedman D Z and Ooguri H 2002 Holography and defect conformal field theories Phys. Rev. D 66 025009 · doi:10.1103/physrevd.66.025009
[89] Erdmenger J, Guralnik Z and Kirsch I 2002 Four-dimensional superconformal theories with interacting boundaries or defects Phys. Rev. D 66 025020 · doi:10.1103/physrevd.66.025020
[90] Constable N R, Myers R C and Tafjord O 2000 The noncommutative bion core Phys. Rev. D 61 106009 · doi:10.1103/physrevd.61.106009
[91] de Leeuw M, Kristjansen C and Zarembo K 2015 One-point functions in defect CFT and integrability J. High Energy Phys. JHEP08(2015)098 · Zbl 1388.81228 · doi:10.1007/jhep08(2015)098
[92] Buhl-Mortensen I, de Leeuw M, Kristjansen C and Zarembo K 2016 One-point functions in AdS/dCFT from matrix product states J. High Energy Phys. JHEP02(2016)052 · Zbl 1388.81499 · doi:10.1007/jhep02(2016)052
[93] Brockmann M 2014 Overlaps of q-raised Néel states with XXZ Bethe states and their relation to the Lieb-Liniger Bose gas J. Stat. Mech. P05006 · Zbl 1456.82232 · doi:10.1088/1742-5468/2014/05/p05006
[94] de Leeuw M, Kristjansen C and Mori S 2016 AdS/dCFT one-point functions of the SU(3) sector Phys. Lett. B 763 197 · Zbl 1370.81150 · doi:10.1016/j.physletb.2016.10.044
[95] De Leeuw M, Kristjansen C and Linardopoulos G 2018 Scalar one-point functions and matrix product states of AdS/dCFT Phys. Lett. B 781 238 · Zbl 1398.81226 · doi:10.1016/j.physletb.2018.03.083
[96] Buhl-Mortensen I, de Leeuw M, Ipsen A C, Kristjansen C and Wilhelm M 2017 Asymptotic one-point functions in gauge-string duality with defects Phys. Rev. Lett.119 261604 · doi:10.1103/physrevlett.119.261604
[97] Piroli L, Pozsgay B and Vernier E 2017 What is an integrable quench? Nucl. Phys. B 925 362 · Zbl 1375.81191 · doi:10.1016/j.nuclphysb.2017.10.012
[98] Nagasaki K, Tanida H and Yamaguchi S 2012 Holographic interface-particle potential J. High Energy Phys. JHEP01(2012)139 · Zbl 1306.81262 · doi:10.1007/jhep01(2012)139
[99] Nagasaki K and Yamaguchi S 2012 Expectation values of chiral primary operators in holographic interface CFT Phys. Rev. D 86 086004 · doi:10.1103/physrevd.86.086004
[100] Buhl-Mortensen I, de Leeuw M, Ipsen A C, Kristjansen C and Wilhelm M 2016 One-loop one-point functions in gauge-gravity dualities with defects Phys. Rev. Lett.117 231603 · doi:10.1103/physrevlett.117.231603
[101] Buhl-Mortensen I, de Leeuw M, Ipsen A C, Kristjansen C and Wilhelm M 2017 A quantum check of AdS/dCFT J. High Energy Phys. JHEP01(2017)098 · Zbl 1373.81316 · doi:10.1007/jhep01(2017)098
[102] Liendo P and Meneghelli C 2017 Bootstrap equations for J. High Energy Phys. JHEP01(2017)122 · Zbl 1373.81334 · doi:10.1007/jhep01(2017)122
[103] de Leeuw M, Ipsen A C, Kristjansen C, Vardinghus K E and Wilhelm M 2017 Two-point functions in AdS/dCFT and the boundary conformal bootstrap equations J. High Energy Phys. JHEP08(2017)020 · Zbl 1381.81124 · doi:10.1007/JHEP08(2017)020
[104] de Leeuw M, Kristjansen C and Linardopoulos G 2017 One-point functions of non-protected operators in the SO(5) symmetric D3/D7 dCFT J. Phys. A: Math. Theor.50 25 · Zbl 1370.81152 · doi:10.1088/1751-8121/aa714b
[105] Kristjansen C, Semenoff G W and Young D 2013 Chiral primary one-point functions in the D3-D7 defect conformal field theory J. High Energy Phys. JHEP01(2013)117 · Zbl 1342.81503 · doi:10.1007/jhep01(2013)117
[106] Kondo J 1964 Resistance minimum in dilute magnetic alloys Prog. Theor. Phys.32 1 · doi:10.1143/ptp.32.37
[107] Erdmenger J, Hoyos C, O’Bannon A and Wu J 2013 A holographic model of the Kondo effect J. High Energy Phys. JHEP12(2013)086 · doi:10.1007/jhep12(2013)086
[108] Hartnoll S A, Herzog C P and Horowitz G T 2008 Building a holographic superconductor Phys. Rev. Lett.101 031601 · Zbl 1404.82086 · doi:10.1103/physrevlett.101.031601
[109] Read N and Newns D M 1983 On the solution of the Coqblin-Schrieffer Hamiltonian by the large-N expansion technique J. Phys. C 16 17 · doi:10.1088/0022-3719/16/17/014
[110] Coleman P and Andrei N 1986 Diagonalisation of the generalised Anderson model J. Phys. C: Solid State Phys.19 17 · doi:10.1088/0022-3719/19/17/017
[111] Ryu S and Takayanagi T 2006 Holographic derivation of entanglement entropy from AdS/CFT Phys. Rev. Lett.96 181602 · Zbl 1228.83110 · doi:10.1103/physrevlett.96.181602
[112] Calabrese P and Cardy J 2004 Entanglement entropy and quantum field theory J. Stat. Mech. P06002 · Zbl 1082.82002 · doi:10.1088/1742-5468/2004/06/p06002
[113] Erdmenger J, Flory M and Newrzella M-N 2015 Bending branes for DCFT in two dimensions J. High Energy Phys. JHEP01(2015)058 · Zbl 1388.83654 · doi:10.1007/jhep01(2015)058
[114] Erdmenger J, Flory M, Hoyos C, Newrzella M-N and Wu J M S 2016 Entanglement entropy in a holographic Kondo model Fortschr. Phys.64 109-30 · Zbl 1339.82017 · doi:10.1002/prop.201500099
[115] Israel W 1966 Singular hypersurfaces and thin shells in general relativity Nuovo Cimento B 44 1 · doi:10.1007/bf02710419
[116] Sørensen E S, Chang M-S, Laflorencie N and Affleck I 2007 Quantum impurity entanglement J. Stat. Mech. P08003 · Zbl 1456.82189 · doi:10.1088/1742-5468/2007/08/p08003
[117] Erdmenger J 2017 Introduction to gauge/gravity duality (TASI lectures 2017) PoS TASI2017 001
[118] Erdmenger J, Flory M, Newrzella M-N, Strydom M and Wu J M S 2017 Quantum quenches in a holographic Kondo model J. High Energy Phys. JHEP04(2017)045 · Zbl 1378.81114 · doi:10.1007/jhep04(2017)045
[119] Erdmenger J, Hoyos C, O’Bannon A, Papadimitriou I, Probst J and Wu J M S 2017 Holographic Kondo and Fano resonances Phys. Rev. D 96 021901 · doi:10.1103/physrevd.96.021901
[120] Erdmenger J, Hoyos C, O’Bannon A, Papadimitriou I, Probst J and Wu J M S 2017 Two-point functions in a holographic Kondo model J. High Energy Phys. JHEP03(2017)039 · Zbl 1377.81167 · doi:10.1007/jhep03(2017)039
[121] Maldacena J M 1999 The large N limit of superconformal field theories and supergravity Int. J. Theor. Phys.38 1113-33 · Zbl 0969.81047 · doi:10.1023/A:1026654312961
[122] Gubser S S, Klebanov I R and Polyakov A M 1998 Gauge theory correlators from noncritical string theory Phys. Lett. B 428 105 · Zbl 1355.81126 · doi:10.1016/s0370-2693(98)00377-3
[123] Witten E 1998 Anti-de Sitter space and holography Adv. Theor. Math. Phys.2 253 · Zbl 0914.53048 · doi:10.4310/atmp.1998.v2.n2.a2
[124] Freedman D Z, Mathur S D, Matusis A and Rastelli L 1999 Correlation functions in the CFT(d) / AdS(d+1) correspondence Nucl. Phys. B 546 96 · Zbl 0944.81041 · doi:10.1016/S0550-3213(99)00053-X
[125] D’Hoker E, Freedman D Z and Skiba W 1999 Field theory tests for correlators in the AdS/CFT correspondence Phys. Rev. D 59 045008 · doi:10.1103/physrevd.59.045008
[126] Rastelli L and Zhou X 2017 Mellin amplitudes for AdS5 × S5 Phys. Rev. Lett.118 9 · doi:10.1103/physrevlett.118.091602
[127] Rastelli L and Zhou X 2017 How to succeed at holographic correlators without really trying J. High Energy Phys. JHEP04(2018)014 · Zbl 1390.83420 · doi:10.1007/JHEP04(2018)014
[128] Dolan F A, Nirschl M and Osborn H 2006 Conjectures for large N superconformal N=4 chiral primary four point functions Nucl. Phys. B 749 109 · Zbl 1214.81270 · doi:10.1016/j.nuclphysb.2006.05.009
[129] Uruchurtu L I 2011 Next-next-to-extremal four point functions of N= 4 1/2 BPS operators in the AdS/CFT correspondence J. High Energy Phys. JHEP08(2011)133 · Zbl 1298.81335 · doi:10.1007/jhep08(2011)133
[130] Alday L F and Bissi A 2017 Loop corrections to supergravity on AdS5 × S5 Phys. Rev. Lett.119 171601 · doi:10.1103/physrevlett.119.171601
[131] Komargodski Z and Zhivoedov A 2013 Convexity and liberation at large spin J. High Energy Phys. JHEP11(2013)140 · doi:10.1007/jhep11(2013)140
[132] Fitzpatrick A L, Kaplan J, Poland D and Simmons-Duffin D 2012 The analytic bootstrap and AdS superhorizon locality J. High Energy Phys. JHEP12(2013)004 · Zbl 1342.83239 · doi:10.1007/jhep12(2013)004
[133] Alday L F, Bissi A and Lukowski T 2015 Lessons from crossing symmetry at large N J. High Energy Phys. JHEP06(2015)074 · Zbl 1387.81041 · doi:10.1007/jhep06(2015)074
[134] Alday L F and Zhivoedov A 2017 An algebraic approach to the analytic bootstrap J. High Energy Phys. JHEP04(2017)157 · Zbl 1378.81097 · doi:10.1007/jhep04(2017)157
[135] Alday L F 2017 Large spin perturbation theory for conformal field theories Phys. Rev. Lett.119 111601 · doi:10.1103/physrevlett.119.111601
[136] Caron-Huot S 2017 Analyticity in spin in conformal theories J. High Energy Phys. JHEP09(2017)078 · Zbl 1382.81173 · doi:10.1007/jhep09(2017)078
[137] Alday L F, Bissi A and Lukowski T 2015 Large spin systematics in CFT J. High Energy Phys. JHEP11(2015)101 · Zbl 1388.81752 · doi:10.1007/jhep11(2015)101
[138] Beem C, Rastelli L and van Rees B C 2013 The N = 4 superconformal bootstrap Phys. Rev. Lett.111 071601 · doi:10.1103/physrevlett.111.071601
[139] Beem C, Rastelli L and van Rees B C 2017 More N = 4 superconformal bootstrap Phys. Rev. D 96 046014 · doi:10.1103/physrevd.96.046014
[140] Arutyunov G, Frolov S and Petkou A C 1998 Operator product expansion of the lowest weight CPOs in N = 4 S Y M 4 at strong coupling Nucl. Phys. B 609 539 · doi:10.1016/s0550-3213(01)00266-8
[141] D’Hoker E, Mathur S D, Matusis A and Rastelli L 2000 The operator product expansion of N=4 SYM and the 4 point functions of supergravity Nucl. Phys. B 589 38 · Zbl 1060.81600 · doi:10.1016/s0550-3213(00)00523-x
[142] Aharony O, Alday L F, Bissi A and Perlmutter E 2017 Loops in AdS from conformal field theory J. High Energy Phys. JHEP07(2017)036 · Zbl 1380.81280 · doi:10.1007/jhep07(2017)036
[143] Metsaev R R and Tseytlin A A 1988 On loop corrections to string theory effective actions Nucl. Phys. B 298 109 · doi:10.1016/0550-3213(88)90306-9
[144] Alday L F and Caron-Huot S 2017 Gravitational S-matrix from CFT dispersion relations J. High Energy Phys. JHEP12(2018)017 · Zbl 1405.81114 · doi:10.1007/jhep12(2018)017
[145] Aprile F, Drummond J M, Heslop P and Paul H 2017 Unmixing supergravity J. High Energy Phys. JHEP02(2018)133 · Zbl 1387.83093 · doi:10.1007/jhep02(2018)133
[146] Aprile F, Drummond J M, Heslop P and Paul H 2017 Loop corrections for Kaluza-Klein AdS amplitudes J. High Energy Phys. JHEP05(2018)056 · Zbl 1391.83099 · doi:10.1007/jhep05(2018)056
[147] Aprile F, Drummond J M, Heslop P and Paul H 2017 Quantum gravity from conformal field theory J. High Energy Phys. JHEP01(2017)035 · Zbl 1384.83011 · doi:10.1007/jhep01(2018)035
[148] Goçalves V 2015 Four point function of N = 4 stress-tensor multiplet at strong coupling J. High Energy Phys. JHEP04(2015)150 · doi:10.1007/jhep04(2015)150
[149] Alday L F, Bissi A and Perlmutter E 2017 Holographic reconstruction of AdS exchanges from crossing symmetry J. High Energy Phys. JHEP08(2017)147 · Zbl 1381.81103 · doi:10.1007/jhep08(2017)147
[150] Yuan E Y 2017 Loops in the bulk
[151] Bargheer T, Caetano J, Fleury T, Komatsu S and Vieira P 2018 Handling handles I. Nonplanar integrability Phys. Rev. Lett.121 231602 · Zbl 1404.81216 · doi:10.1103/PhysRevLett.121.231602
[152] Dowker J S and Schofield J P 1990 Conformal transformations and the effective action in the presence of boundaries J. Math. Phys.31 808 · Zbl 0702.53059 · doi:10.1063/1.528814
[153] Herzog C P, Huang K-W and Jensen K 2016 Universal entanglement and boundary geometry in conformal field theory J. High Energy Phys. JHEP01(2016)162 · Zbl 1388.81082 · doi:10.1007/jhep01(2016)162
[154] Fursaev D V 2015 Conformal anomalies of CFTs with boundaries J. High Energy Phys. JHEP12(2015)112 · Zbl 1388.81977 · doi:10.1007/jhep12(2015)112
[155] Tseytlin A A and Beccaria M 2017 CT for conformal higher spin fields from partition function on conically deformed sphere J. High Energy Phys. JHEP09(2017)123 · Zbl 1382.83115 · doi:10.1007/jhep09(2017)123
[156] Crardy J 1988 Is there a c theorem in four-dimensions? Phys. Lett. B 215 749 · doi:10.1016/0370-2693(88)90054-8
[157] Komargodski Z and Schwimmer A 2011 On renormalization group flows in four dimensions J. High Energy Phys. JHEP12(2011)099 · Zbl 1306.81140 · doi:10.1007/jhep12(2011)099
[158] Solodukhin S N 2016 Boundary terms of conformal anomaly Phys. Lett. B 752 131 · doi:10.1016/j.physletb.2015.11.036
[159] Osborn H and Petkou A 1994 Implications of conformal invariance in field theories for general dimensions Ann. Phys., NY231 311-62 · Zbl 0795.53073 · doi:10.1006/aphy.1994.1045
[160] Deutsch D and Candelas P 1979 Boundary effects in quantum field theory Phys. Rev. D 20 3063 · doi:10.1103/physrevd.20.3063
[161] Rong-Xin M and Chong-Sun C 2017 Universality for shape dependence of Casimir effects from Weyl anomaly (arXiv:1706.09652 [hep-th])
[162] Fursaev D V 2006 Entanglement entropy in critical phenomena and analogue models of quantum gravity Phys. Rev. D 73 124025 · doi:10.1103/physrevd.73.124025
[163] Fursaev D V 2013 Quantum entanglement on boundaries J. High Energy Phys. JHEP07(2013)119 · Zbl 1342.81527 · doi:10.1007/jhep07(2013)119
[164] Fursaev D V, Patrushev A and Solodukhin S N 2013 Distributional geometry of squashed cones Phys. Rev. D 88 044054 · doi:10.1103/physrevd.88.044054
[165] Amin F A, Berthiere C, Fursaev D V and Solodukhin S N 2017 Holographic calculation of entanglement entropy in the presence of boundaries Phys. Rev. D 95 106013 · doi:10.1103/physrevd.95.106013
[166] Takayanagi T 2011 Holographic dual of BCFT Phys. Rev. Lett.107 101602 · doi:10.1103/physrevlett.107.101602
[167] Rong-Xin M, Chong-Sun C and Wu-Zhong G 2017 New proposal for a holographic boundary conformal field theory Phys. Rev. D 96 046005 · doi:10.1103/physrevd.96.046005
[168] Astaneh F A and Solodukhin S N 2017 Holographic calculation of boundary terms in conformal anomaly Phys. Lett. B 769 25 · Zbl 1370.81134 · doi:10.1016/j.physletb.2017.03.026
[169] Schwimmer A and Theisen S 2008 Entanglement entropy, trace anomalies and holography Nucl. Phys. B 801 1 · Zbl 1189.83036 · doi:10.1016/j.nuclphysb.2008.04.015
[170] Runkel I, Fjelstad J, Fuchs J and Schweigert C 2007 Topological and conformal field theory as Frobenius algebras Contemp. Math.431 225 · Zbl 1154.18006 · doi:10.1090/conm/431/08275
[171] Polyakov A M 1974 Nonhamiltonian approach to conformal quantum field theory Zh. Eksp. Teor. Fiz.66 23
[172] Mack G 1973 Conformal invariant quantum field theory J. Phys. Colloq.34 99-106 · doi:10.1051/jphyscol:1973108
[173] Ferrara S, Grillo A F, Parisi G and Gatto R 1972 Covariant expansion of the conformal four -point function Nucl. Phys. B 49 77 · doi:10.1016/0550-3213(72)90587-1
[174] Dolan F A and Osborn H 2001 Conformal four point functions and the operator product expansion Nucl. Phys. B 599 459 · Zbl 1097.81734 · doi:10.1016/s0550-3213(01)00013-x
[175] Dolan F A and Osborn H 2004 Conformal partial waves and the operator product expansion Nucl. Phys. B 678 491 · Zbl 1097.81735 · doi:10.1016/j.nuclphysb.2003.11.016
[176] Dolan F A and Osborn H 2011 Conformal partial waves: further mathematical results (arXiv:1108.6194)
[177] Billo M, Goncalves V, Lauria E and Meineri M 2016 Defects in conformal field theory J. High Energy Phys. JHEP04(2016)091 · Zbl 1388.81029 · doi:10.1007/jhep04(2016)091
[178] Liendo P, Rastelli L and van Rees B C 2013 The Bootstrap program for boundary CFTd J. High Energy Phys. JHEP07(2013)113 · Zbl 1342.81504 · doi:10.1007/jhep07(2013)113
[179] Gadde A 2016 Conformal constraints on defects (arXiv:1602.06354)
[180] Fukuda M, Kobayashi N and Nishioka T 2017 Operator product expansion for conformal defects (arXiv:1710.11165)
[181] Isachenkov M and Schomerus V 2016 Superintegrability of d-dimensional conformal blocks Phys. Rev. Lett.117 071602 · doi:10.1103/physrevlett.117.071602
[182] Calogero F 1971 Solution of the one‐dimensional N‐body problems with quadratic and/or inversely quadratic pair potentials J. Math. Phys.12 419 · Zbl 1002.70558 · doi:10.1063/1.1665604
[183] Sutherland B 1972 Exact results for a quantum many-body problem in one dimension. II Phys. Rev. A 5 1372 · doi:10.1103/physreva.5.1372
[184] Heckman G J and Opdam E M 1987 Root systems and hypergeometric functions I. Compositio Mathematica64 3 · Zbl 0656.17006
[185] Isachenkov M and Schomerus V 2017 Integrability of conformal blocks I. Calogero-Sutherland scattering theory (arXiv:1711.06609)
[186] Schomerus V, Sobko E and Isachenkov M 2017 Harmony of spinning conformal blocks J. High Energy Phys. JHEP03(2017)085 · Zbl 1377.81182 · doi:10.1007/jhep03(2017)085
[187] Schomerus V and Sobko E 2017 From spinning conformal blocks to matrix Calogero-Sutherland models (arXiv:1711.02022)
[188] Olshanetsky M A and Perelomov A M 1983 Quantum integrable systems related to Lie algebras Phys. Rep.94 313 · doi:10.1016/0370-1573(83)90018-2
[189] Fehér L and Pusztai B G 2010 Derivations of the trigonometric BCn Sutherland model by quantum Hamiltonian reduction Rev. Math. Phys.22 699 · Zbl 1194.22024 · doi:10.1142/s0129055x10004065
[190] Isachenkov M, Liendo P, Linke Y and Schomerus V 2018 Calogero-Sutherland approach to defect blocks (arXiv:1806.09703 [hep-th]) · Zbl 1402.81227
[191] Nakayama Y 2016 Bootstrapping critical Ising model on three-dimensional real projective space Phys. Rev. Lett.116 141602 · doi:10.1103/physrevlett.116.141602
[192] Hasegawa C and Nakayama Y 2017 ϵ-Expansion in critical ϕ3-theory on real projective space from conformal field theory Mod. Phys. Lett. A 32 1750045 · Zbl 1360.81266 · doi:10.1142/s0217732317500456
[193] Schweigert C, Fuchs J and Runkel I 2006 Categorification and correlation functions in conformal field theory Int. Congress of Mathematicians 2006 ed M Sanz-Solé et al (arXiv:math.CT/0602079)
[194] Kapustin A and Saulina N 2011 Surface operators in 3d topological field theory and 2d rational conformal field theory Mathematical Foundations of Quantum Field Theory and Perturbative String Theory ed H Sati et al (arXiv:1012.0911) · Zbl 1248.81206 · doi:10.1090/pspum/083/2742429
[195] Fuchs J, Schweigert C and Valentino A 2013 Bicategories for boundary conditions and for surface defects in 3-d TFT Commun. Math. Phys.321 543 · Zbl 1269.81169 · doi:10.1007/s00220-013-1723-0
[196] Lyubashenko V V 1995 Invariants of three manifolds and projective representations of mapping class groups via quantum groups at roots of unity Commun. Math. Phys.172 467 · Zbl 0844.57016 · doi:10.1007/bf02101805
[197] Bakalov B and Kirillov A N 2000 On the Lego-Teichmüller game Transform. Groups5 207 · Zbl 0999.57021 · doi:10.1007/bf01679714
[198] Fuchs J and Schweigert C 2017 Consistent systems of correlators in non-semisimple conformal field theory Adv. Math.307 598 · Zbl 1355.81034 · doi:10.1016/j.aim.2016.11.020
[199] Fuchs J, Schweigert C and Stigner C 2012 Modular invariant Frobenius algebras from ribbon Hopf algebra automorphisms J. Algebra363 29 · Zbl 1296.16029 · doi:10.1016/j.jalgebra.2012.04.008
[200] Fuchs J, Schweigert C and Stigner C 2012 The Cardy-Cartan modular invariant Strings, Gauge Fields, and the Geometry Behind ed A Rebhan et al (arXiv:1201.4267 [hep-th]) · doi:10.1142/9789814412551_0013
[201] Friedan D 2016 Quantum field theories of extended objects (arXiv:1605.03279 [hep-th])
[202] Friedan D 2017 A new kind of quantum field theory of (n-1)-dimensional defects in 2n dimensions (arXiv:1711.05049 [hep-th])
[203] Friedan D 2017 Homepage of Daniel Friedan , http://www.physics.rutgers.edu/ friedan/#res
[204] Federer H and Fleming W H 1960 Normal and integral currents The Annals of Mathematics72 458-520 · Zbl 0187.31301 · doi:10.2307/1970227
[205] Ambrosio L and Kirchheim B 2000 Currents in metric spaces Acta Math.185 1-80 · Zbl 0984.49025 · doi:10.1007/bf02392711
[206] Friedan D and Konechny A 2004 On the boundary entropy of one-dimensional quantum systems at low temperature Phys. Rev. Lett.93 030402 · doi:10.1103/physrevlett.93.030402
[207] Casini H, Landea I S and Torroba G 2016 The g-theorem and quantum information theory J. High Energy Phys. JHEP10(2016)140 · Zbl 1390.81094 · doi:10.1007/JHEP10(2016)140
[208] Bianchi L, Lemos M and Meineri M 2018 Line defects and radiation in N = 2 theories (arXiv:1805.04111 [hep-th])
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