Abstract
We present a Lorentzian inversion formula valid for any defect CFT that extracts the bulk channel CFT data as an analytic function of the spin variable. This result complements the already obtained inversion formula for the corresponding defect channel, and makes it now possible to implement the analytic bootstrap program for defect CFT, by going back and forth between bulk and defect expansions. A crucial role in our derivation is played by the Calogero-Sutherland description of defect blocks which we review. As first applications we obtain the large-spin limit of bulk CFT data necessary to reproduce the defect identity, and also calculate one-point functions of the twist defect of the 3d Ising model to first order in the ϵ-expansion.
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M. Billò, V. Gonçalves, E. Lauria and M. Meineri, Defects in conformal field theory, JHEP 04 (2016) 091 [arXiv:1601.02883] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Radial coordinates for defect CFTs, JHEP 11 (2018) 148 [arXiv:1712.07668] [INSPIRE].
M. Isachenkov, P. Liendo, Y. Linke and V. Schomerus, Calogero-Sutherland approach to defect blocks, JHEP 10 (2018) 204 [arXiv:1806.09703] [INSPIRE].
E. Lauria, M. Meineri and E. Trevisani, Spinning operators and defects in conformal field theory, JHEP 08 (2019) 066 [arXiv:1807.02522] [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. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
M. Hogervorst and B.C. van Rees, Crossing symmetry in alpha space, JHEP 11 (2017) 193 [arXiv:1702.08471] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS2/CFT1, Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping the half-BPS line defect, JHEP 10 (2018) 077 [arXiv:1806.01862] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap. Part II. Natural bases for the crossing equation, JHEP 02 (2019) 163 [arXiv:1811.10646] [INSPIRE].
D. Mazáč, A crossing-symmetric OPE inversion formula, JHEP 06 (2019) 082 [arXiv:1812.02254] [INSPIRE].
N. Kiryu and S. Komatsu, Correlation functions on the half-BPS Wilson loop: perturbation and hexagonalization, JHEP 02 (2019) 090 [arXiv:1812.04593] [INSPIRE].
N. Arkani-Hamed, Y.-T. Huang and S.-H. Shao, On the positive geometry of conformal field theory, JHEP 06 (2019) 124 [arXiv:1812.07739] [INSPIRE].
M. Beccaria, S. Giombi and A.A. Tseytlin, Correlators on non-supersymmetric Wilson line in N = 4 SYM and AdS2/CFT1, JHEP 05 (2019) 122 [arXiv:1903.04365] [INSPIRE].
A. Fitzpatrick, J. Kaplan and M.T. Walters, Universality of long-distance AdS physics from the CFT bootstrap, JHEP 08 (2014) 145 [arXiv:1403.6829] [INSPIRE].
Z. Komargodski and A. Zhiboedov, Convexity and liberation at large spin, JHEP 11 (2013) 140 [arXiv:1212.4103] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
D. Simmons-Duffin, D. Stanford and E. Witten, A spacetime derivation of the Lorentzian OPE inversion formula, JHEP 07 (2018) 085 [arXiv:1711.03816] [INSPIRE].
P. Kravchuk and D. Simmons-Duffin, Light-ray operators in conformal field theory, JHEP 11 (2018) 102 [arXiv:1805.00098] [INSPIRE].
L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin, The conformal bootstrap at finite temperature, JHEP 10 (2018) 070 [arXiv:1802.10266] [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, JHEP 09 (2018) 091 [arXiv:1712.08185] [INSPIRE].
M. Isachenkov and V. Schomerus, Superintegrability of d-dimensional conformal blocks, Phys. Rev. Lett. 117 (2016) 071602 [arXiv:1602.01858] [INSPIRE].
M. Billò, 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. Fitzpatrick, J. Kaplan, D. Poland and D. Simmons-Duffin, The analytic bootstrap and AdS superhorizon locality, JHEP 12 (2013) 004 [arXiv:1212.3616] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
M. Isachenkov and V. Schomerus, Integrability of conformal blocks. Part I. Calogero-Sutherland scattering theory, JHEP 07 (2018) 180 [arXiv:1711.06609] [INSPIRE].
F.A. Dolan and H. Osborn, Conformal partial waves: further mathematical results, arXiv:1108.6194 [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [INSPIRE].
S. El-Showk and M.F. Paulos, Extremal bootstrapping: go with the flow, JHEP 03 (2018) 148 [arXiv:1605.08087] [INSPIRE].
M. Hogervorst and S. Rychkov, Radial coordinates for conformal blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
A. Gadde, Conformal constraints on defects, JHEP 01 (2020) 038 [arXiv:1602.06354] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for N = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, unpublished notes.
J. Cardy, Scaling and renormalization in statistical physics, Cambridge University Press, Cambridge, U.K. (1996).
P. Liendo, L. Rastelli and B.C. van Rees, The bootstrap program for boundary CFTd, 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].
L. Rastelli and X. Zhou, The Mellin formalism for boundary CFTd, JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
A. Bissi, T. Hansen and A. Söderberg, Analytic bootstrap for boundary CFT, JHEP 01 (2019) 010 [arXiv:1808.08155] [INSPIRE].
A. Kaviraj and M.F. Paulos, The functional bootstrap for boundary CFT, JHEP 04 (2020) 135 [arXiv:1812.04034] [INSPIRE].
D. Mazáč, L. Rastelli and X. Zhou, An analytic approach to BCFTd, JHEP 12 (2019) 004 [arXiv:1812.09314] [INSPIRE].
H. Bateman, Higher transcendental functions. 1, Krieger Pub. Co., U.S.A. (1981).
A. Söderberg, Anomalous dimensions in the WF O(N) model with a monodromy line defect, JHEP 03 (2018) 058 [arXiv:1706.02414] [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].
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].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
F. Kos, D. Poland, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
M.S. Costa, V. Goncalves and J. Penedones, Conformal Regge theory, JHEP 12 (2012) 091 [arXiv:1209.4355] [INSPIRE].
L.F. Alday, J. Henriksson and M. van Loon, Taming the ϵ-expansion with large spin perturbation theory, JHEP 07 (2018) 131 [arXiv:1712.02314] [INSPIRE].
J. Henriksson and M. Van Loon, Critical O(N) model to order ϵ4 from analytic bootstrap, J. Phys. A 52 (2019) 025401 [arXiv:1801.03512] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, Conformal bootstrap in Mellin space, Phys. Rev. Lett. 118 (2017) 081601 [arXiv:1609.00572] [INSPIRE].
R. Gopakumar, A. Kaviraj, K. Sen and A. Sinha, A Mellin space approach to the conformal bootstrap, JHEP 05 (2017) 027 [arXiv:1611.08407] [INSPIRE].
P. Dey, A. Kaviraj and A. Sinha, Mellin space bootstrap for global symmetry, JHEP 07 (2017) 019 [arXiv:1612.05032] [INSPIRE].
F. Gliozzi and A. Rago, Critical exponents of the 3d Ising and related models from conformal bootstrap, JHEP 10 (2014) 042 [arXiv:1403.6003] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal blocks, JHEP 11 (2011) 154 [arXiv:1109.6321] [INSPIRE].
M.S. Costa, J. Penedones, D. Poland and S. Rychkov, Spinning conformal correlators, JHEP 11 (2011) 071 [arXiv:1107.3554] [INSPIRE].
D. Simmons-Duffin, Projectors, shadows, and conformal blocks, JHEP 04 (2014) 146 [arXiv:1204.3894] [INSPIRE].
M.S. Costa and T. Hansen, Conformal correlators of mixed-symmetry tensors, JHEP 02 (2015) 151 [arXiv:1411.7351] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Deconstructing conformal blocks in 4D CFT, JHEP 08 (2015) 101 [arXiv:1505.03750] [INSPIRE].
A. Castedo Echeverri, E. Elkhidir, D. Karateev and M. Serone, Seed conformal blocks in 4D CFT, JHEP 02 (2016) 183 [arXiv:1601.05325] [INSPIRE].
M.S. Costa, T. Hansen, J. Penedones and E. Trevisani, Projectors and seed conformal blocks for traceless mixed-symmetry tensors, JHEP 07 (2016) 018 [arXiv:1603.05551] [INSPIRE].
G.F. Cuomo, D. Karateev and P. Kravchuk, General bootstrap equations in 4D CFTs, JHEP 01 (2018) 130 [arXiv:1705.05401] [INSPIRE].
V. Schomerus, E. Sobko and M. Isachenkov, Harmony of spinning conformal blocks, JHEP 03 (2017) 085 [arXiv:1612.02479] [INSPIRE].
V. Schomerus and E. Sobko, From spinning conformal blocks to matrix Calogero-Sutherland models, JHEP 04 (2018) 052 [arXiv:1711.02022] [INSPIRE].
D. Karateev, P. Kravchuk and D. Simmons-Duffin, Weight shifting operators and conformal blocks, JHEP 02 (2018) 081 [arXiv:1706.07813] [INSPIRE].
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Liendo, P., Linke, Y. & Schomerus, V. A Lorentzian inversion formula for defect CFT. J. High Energ. Phys. 2020, 163 (2020). https://doi.org/10.1007/JHEP08(2020)163
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DOI: https://doi.org/10.1007/JHEP08(2020)163