Abstract
We discuss consequences of the breaking of conformal symmetry by a flat or spherical extended operator. We adapt the embedding formalism to the study of correlation functions of symmetric traceless tensors in the presence of the defect. Two-point functions of a bulk and a defect primary are fixed by conformal invariance up to a set of OPE coefficients, and we identify the allowed tensor structures. A correlator of two bulk primaries depends on two cross-ratios, and we study its conformal block decomposition in the case of external scalars. The Casimir equation in the defect channel reduces to a hypergeometric equation, while the bulk channel blocks are recursively determined in the light-cone limit. In the special case of a defect of codimension two, we map the Casimir equation in the bulk channel to the one of a four-point function without defect. Finally, we analyze the contact terms of the stress-tensor with the extended operator, and we deduce constraints on the CFT data. In two dimensions, we relate the displacement operator, which appears among the contact terms, to the reflection coefficient of a conformal interface, and we find unitarity bounds for the latter.
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References
J.M. Maldacena, The large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
P.A.M. Dirac, Wave equations in conformal space, Annals Math. 37 (1936) 429.
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
S. Weinberg, Six-dimensional Methods for Four-dimensional Conformal Field Theories, Phys. Rev. D 82 (2010) 045031 [arXiv:1006.3480] [INSPIRE].
L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, arXiv:1511.06713 [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].
S.N. Solodukhin, Boundary terms of conformal anomaly, Phys. Lett. B 752 (2016) 131 [arXiv:1510.04566] [INSPIRE].
F. Gliozzi, P. Liendo, M. Meineri and A. Rago, Boundary and Interface CFTs from the Conformal Bootstrap, JHEP 05 (2015) 036 [arXiv:1502.07217] [INSPIRE].
D.M. McAvity and H. Osborn, Energy momentum tensor in conformal field theories near a boundary, Nucl. Phys. B 406 (1993) 655 [hep-th/9302068] [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].
J.L. Cardy, Conformal Invariance and Surface Critical Behavior, Nucl. Phys. B 240 (1984) 514 [INSPIRE].
J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde Formula, Nucl. Phys. B 324 (1989) 581 [INSPIRE].
J.L. Cardy and D.C. Lewellen, Bulk and boundary operators in conformal field theory, Phys. Lett. B 259 (1991) 274 [INSPIRE].
R.E. Behrend, P.A. Pearce, V.B. Petkova and J.-B. Zuber, Boundary conditions in rational conformal field theories, Nucl. Phys. B 570 (2000) 525 [hep-th/9908036] [INSPIRE].
C. Bachas, J. de Boer, R. Dijkgraaf and H. Ooguri, Permeable conformal walls and holography, JHEP 06 (2002) 027 [hep-th/0111210] [INSPIRE].
J. Fröhlich, J. Fuchs, I. Runkel and C. Schweigert, Duality and defects in rational conformal field theory, Nucl. Phys. B 763 (2007) 354 [hep-th/0607247] [INSPIRE].
M. Hasenbusch, Thermodynamic Casimir force: A Monte Carlo study of the crossover between the ordinary and the normal surface universality class, Phys. Rev. B 83 (2011) 134425 [arXiv:1012.4986].
H.W. Diehl and S. Dietrich, Field-theoretical approach to multicritical behavior near free surfaces, Phys. Rev. B 24 (1981) 2878 [INSPIRE].
C. Cosme, J.M. V.P. Lopes and J. Penedones, Conformal symmetry of the critical 3D Ising model inside a sphere, JHEP 08 (2015) 022 [arXiv:1503.02011] [INSPIRE].
Y. Nakayama, Is boundary conformal in CFT?, Phys. Rev. D 87 (2013) 046005 [arXiv:1210.6439] [INSPIRE].
A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063 [hep-th/0105132] [INSPIRE].
O. DeWolfe, D.Z. Freedman and H. Ooguri, Holography and defect conformal field theories, Phys. Rev. D 66 (2002) 025009 [hep-th/0111135] [INSPIRE].
J. Erdmenger, Z. Guralnik and I. Kirsch, Four-dimensional superconformal theories with interacting boundaries or defects, Phys. Rev. D 66 (2002) 025020 [hep-th/0203020] [INSPIRE].
O. Aharony, O. DeWolfe, D.Z. Freedman and A. Karch, Defect conformal field theory and locally localized gravity, JHEP 07 (2003) 030 [hep-th/0303249] [INSPIRE].
D. Gaiotto and E. Witten, Supersymmetric Boundary Conditions in N = 4 Super Yang-Mills Theory, J. Statist. Phys. 135 (2009) 789 [arXiv:0804.2902] [INSPIRE].
M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in Defect CFT and Integrability, JHEP 08 (2015) 098 [arXiv:1506.06958] [INSPIRE].
M. Rapcak, Nonintegrability of NS5-like Interface in \( \mathcal{N} \) = 4 Supersymmetric Yang-Mills, arXiv:1511.02243 [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].
E.I. Buchbinder, J. Gomis and F. Passerini, Holographic gauge theories in background fields and surface operators, JHEP 12 (2007) 101 [arXiv:0710.5170] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
N.R. Constable, J. Erdmenger, Z. Guralnik and I. Kirsch, Intersecting D-3 branes and holography, Phys. Rev. D 68 (2003) 106007 [hep-th/0211222] [INSPIRE].
Ò.J.C. Dias, G.T. Horowitz, N. Iqbal and J.E. Santos, Vortices in holographic superfluids and superconductors as conformal defects, JHEP 04 (2014) 096 [arXiv:1311.3673] [INSPIRE].
M. Billó, M. Caselle, D. Gaiotto, F. Gliozzi, M. Meineri and R. Pellegrini, Line defects in the 3d Ising model, JHEP 07 (2013) 055 [arXiv:1304.4110] [INSPIRE].
A. Allais and S. Sachdev, Spectral function of a localized fermion coupled to the Wilson-Fisher conformal field theory, Phys. Rev. B 90 (2014) 035131 [arXiv:1406.3022] [INSPIRE].
M.F. Paulos, S. Rychkov, B.C. van Rees and B. Zan, Conformal Invariance in the Long-Range Ising Model, Nucl. Phys. B 902 (2016) 246 [arXiv:1509.00008] [INSPIRE].
R. Rattazzi, V.S. Rychkov, E. Tonni and A. Vichi, Bounding scalar operator dimensions in 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3D Ising Model with the Conformal Bootstrap, Phys. Rev. D 86 (2012) 025022 [arXiv:1203.6064] [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
S. El-Showk, M.F. Paulos, D. Poland, S. Rychkov, D. Simmons-Duffin and A. Vichi, Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping Mixed Correlators in the 3D Ising Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and Liberation at Large Spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
A.L. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The Analytic Bootstrap and AdS Superhorizon Locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [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].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
V. Gonçalves, E. Lauria, M. Meineri and E. Trevisani, work in progress.
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
M.S. Costa, V. Gonçalves and J. Penedones, Spinning AdS Propagators, JHEP 09 (2014) 064 [arXiv:1404.5625] [INSPIRE].
V.K. Dobrev, V.B. Petkova, S.G. Petrova and I.T. Todorov, Dynamical Derivation of Vacuum Operator Product Expansion in Euclidean Conformal Quantum Field Theory, Phys. Rev. D 13 (1976) 887 [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal four point functions and the operator product expansion, Nucl. Phys. B 599 (2001) 459 [hep-th/0011040] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal Partial Waves: Further Mathematical Results, arXiv:1108.6194 [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves and the operator product expansion, Nucl. Phys. B 678 (2004) 491 [hep-th/0309180] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning Conformal Blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion Relations for Conformal Blocks, arXiv:1509.00428 [INSPIRE].
S. Rychkov and P. Yvernay, Remarks on the Convergence Properties of the Conformal Block Expansion, Phys. Lett. B 753 (2016) 682 [arXiv:1510.08486] [INSPIRE].
D. Simmons-Duffin, Projectors, Shadows and Conformal Blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
E. Eisenriegler, Polymers Near Surfaces, World Scientific, Singapore (1993).
T. Quella, I. Runkel and G.M.T. Watts, Reflection and transmission for conformal defects, JHEP 04 (2007) 095 [hep-th/0611296] [INSPIRE].
J. Fuchs, C. Schweigert and C. Stigner, The Classifying algebra for defects, Nucl. Phys. B 843 (2011) 673 [arXiv:1007.0401] [INSPIRE].
M. Meineri, work in progress.
S. Ferrara, A.F. Grillo and R. Gatto, Tensor representations of conformal algebra and conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
T. Dimofte and D. Gaiotto, An E7 Surprise, JHEP 10 (2012) 129 [arXiv:1209.1404] [INSPIRE].
L.P. Eisenhart, Riemannian geometry, Princeton University Press, Princeton U.S.A. (1997).
Y. Aminov, The geometry of submanifolds, Gordon and Breach Science Publishers, Philadelphia U.S.A. (2001).
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Billò, M., Gonçalves, V., Lauria, E. et al. Defects in conformal field theory. J. High Energ. Phys. 2016, 91 (2016). https://doi.org/10.1007/JHEP04(2016)091
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DOI: https://doi.org/10.1007/JHEP04(2016)091