Abstract
We study the two-point function of local operators in the presence of a defect in a generic conformal field theory. We define two pairs of cross ratios, which are convenient in the analysis of the OPE in the bulk and defect channel respectively. The new coordinates have a simple geometric interpretation, which can be exploited to efficiently compute conformal blocks in a power expansion. We illustrate this fact in the case of scalar external operators. We also elucidate the convergence properties of the bulk and defect OPE decompositions of the two-point function. In particular, we remark that the expansion of the two-point function in powers of the new cross ratios converges everywhere, a property not shared by the cross ratios customarily used in defect CFT. We comment on the crucial relevance of this fact for the numerical bootstrap.
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References
M. Billó et al., Line defects in the 3d Ising model, JHEP 07 (2013) 055 [arXiv:1304.4110] [INSPIRE].
A. Söderberg, Anomalous dimensions in the WF O(N) model with a monodromy line defect, JHEP 03 (2018) 058 [arXiv:1706.02414] [INSPIRE].
M. Cooke, A. Dekel and N. Drukker, The Wilson loop CFT: insertion dimensions and structure constants from wavy lines, J. Phys. A 50 (2017) 335401 [arXiv:1703.03812] [INSPIRE].
L. Bianchi, L. Griguolo, M. Preti and D. Seminara, Wilson lines as superconformal defects in ABJM theory: a formula for the emitted radiation, JHEP 10 (2017) 050 [arXiv:1706.06590] [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].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
L. Bianchi, M. Meineri, R.C. Myers and M. Smolkin, Rényi entropy and conformal defects, JHEP 07 (2016) 076 [arXiv:1511.06713] [INSPIRE].
S. Balakrishnan, T. Faulkner, Z.U. Khandker and H. Wang, A general proof of the quantum null energy condition, arXiv:1706.09432 [INSPIRE].
J.L. Cardy, Conformal invariance and surface critical behavior, Nucl. Phys. B 240 (1984) 514 [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].
D. Simmons-Duffin, The Conformal Bootstrap, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), June 1–26, Boulder, U.S.A. (2015), arXiv:1602.07982 [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].
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. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP 10 (2016) 037 [arXiv:1605.04175] [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].
L. Rastelli and X. Zhou, The Mellin formalism for boundary CFT d, JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
A.M. Polyakov and V.S. Rychkov, Gauge field strings duality and the loop equation, Nucl. Phys. B 581 (2000) 116 [hep-th/0002106] [INSPIRE].
D.E. Berenstein, R. Corrado, W. Fischler and J.M. Maldacena, The operator product expansion for Wilson loops and surfaces in the large N limit, Phys. Rev. D 59 (1999) 105023 [hep-th/9809188] [INSPIRE].
A. Gadde, Conformal constraints on defects, arXiv:1602.06354 [INSPIRE].
M. Fukuda, N. Kobayashi and T. Nishioka, Operator product expansion for conformal defects, JHEP 01 (2018) 013 [arXiv:1710.11165] [INSPIRE].
D. Pappadopulo, S. Rychkov, J. Espin and R. Rattazzi, OPE convergence in conformal field theory, Phys. Rev. D 86 (2012) 105043 [arXiv:1208.6449] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, Springer Briefs in Physics, Springer Germany (2016) [arXiv:1601.05000].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, arXiv:1807.02522 [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Radial expansion for spinning conformal blocks, JHEP 07 (2016) 057 [arXiv:1603.05552] [INSPIRE].
F. Kos, D. Poland and D. Simmons-Duffin, Bootstrapping the O(N) vector models, JHEP 06 (2014) 091 [arXiv:1307.6856] [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].
J. Penedones, E. Trevisani and M. Yamazaki, Recursion relations for conformal blocks, JHEP 09 (2016) 070 [arXiv:1509.00428] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP 07 (2018) 180 [arXiv:1711.06609] [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, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].
B. Czech, L. Lamprou, S. McCandlish, B. Mosk and J. Sully, A stereoscopic look into the bulk, JHEP 07 (2016) 129 [arXiv:1604.03110] [INSPIRE].
J. de Boer, F.M. Haehl, M.P. Heller and R.C. Myers, Entanglement, holography and causal diamonds, JHEP 08 (2016) 162 [arXiv:1606.03307] [INSPIRE].
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Lauria, E., Meineri, M. & Trevisani, E. Radial coordinates for defect CFTs. J. High Energ. Phys. 2018, 148 (2018). https://doi.org/10.1007/JHEP11(2018)148
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DOI: https://doi.org/10.1007/JHEP11(2018)148