Abstract
We use modern bootstrap techniques to study half-BPS line defects in 4d\( \mathcal{N}=4 \) superconformal theories. Specifically, we consider the 1d CFT with OSP(4∗|4) superconformal symmetry living on such a defect. Our analysis is general and based only on symmetries, it includes however important examples like Wilson and ’t Hooft lines in \( \mathcal{N}=4 \) super Yang-Mills. We present several numerical bounds on OPE coefficients and conformal dimensions. Of particular interest is a numerical island obtained from a mixed correlator bootstrap that seems to imply a unique solution to crossing. The island is obtained if some assumptions about the spectrum are made, and is consistent with Wilson lines in planar \( \mathcal{N}=4 \) super Yang-Mills at strong coupling. We further analyze the vicinity of the strong-coupling point by calculating perturbative corrections using analytic methods. This perturbative solution has the sparsest spectrum and is expected to saturate the numerical bounds, explaining some of the features of our numerical results.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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. Poland, S. Rychkov and A. Vichi, The conformal bootstrap: theory, numerical techniques and applications, arXiv:1805.04405 [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].
E. Lauria, M. Meineri and E. Trevisani, Radial coordinates for defect CFTs, arXiv:1712.07668 [INSPIRE].
S. Guha and B. Nagaraj, Correlators of mixed symmetry operators in defect CFTs, arXiv:1805.12341 [INSPIRE].
F. Gliozzi, More constraining conformal bootstrap, Phys. Rev. Lett. 111 (2013) 161602 [arXiv:1307.3111] [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].
F. Gliozzi, Truncatable bootstrap equations in algebraic form and critical surface exponents, JHEP 10 (2016) 037 [arXiv:1605.04175] [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].
L.F. Alday, A. Bissi and T. Lukowski, Large spin systematics in CFT, JHEP 11 (2015) 101 [arXiv:1502.07707] [INSPIRE].
L.F. Alday, Large spin perturbation theory for conformal field theories, Phys. Rev. Lett. 119 (2017) 111601 [arXiv:1611.01500] [INSPIRE].
S. Caron-Huot, Analyticity in spin in conformal theories, JHEP 09 (2017) 078 [arXiv:1703.00278] [INSPIRE].
M. Lemos, P. Liendo, M. Meineri and S. Sarkar, Universality at large transverse spin in defect CFT, arXiv:1712.08185 [INSPIRE].
L. Rastelli and X. Zhou, The Mellin formalism for boundary CFT d, JHEP 10 (2017) 146 [arXiv:1705.05362] [INSPIRE].
V. Goncalves and G. Itsios, A note on defect Mellin amplitudes, arXiv:1803.06721 [INSPIRE].
M. Hogervorst, Crossing kernels for boundary and crosscap CFTs, arXiv:1703.08159 [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].
N. Kobayashi and T. Nishioka, Spinning conformal defects, JHEP 09 (2018) 134 [arXiv:1805.05967] [INSPIRE].
D. Gaiotto, D. Mazac and M.F. Paulos, Bootstrapping the 3d Ising twist defect, JHEP 03 (2014) 100 [arXiv:1310.5078] [INSPIRE].
D. Mazac, Analytic bounds and emergence of AdS 2 physics from the conformal bootstrap, JHEP 04 (2017) 146 [arXiv:1611.10060] [INSPIRE].
J. Qiao and S. Rychkov, A tauberian theorem for the conformal bootstrap, JHEP 12 (2017) 119 [arXiv:1709.00008] [INSPIRE].
D. Mazac and M.F. Paulos, The analytic functional bootstrap I: 1D CFTs and 2D S-matrices, arXiv:1803.10233 [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for N = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
J.K. Erickson, G.W. Semenoff and K. Zarembo, Wilson loops in N = 4 supersymmetric Yang-Mills theory, Nucl. Phys. B 582 (2000) 155 [hep-th/0003055] [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].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [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].
S. Giombi, R. Roiban and A.A. Tseytlin, Half-BPS Wilson loop and AdS 2 /CFT 1, Nucl. Phys. B 922 (2017) 499 [arXiv:1706.00756] [INSPIRE].
N. Drukker, Integrable Wilson loops, JHEP 10 (2013) 135 [arXiv:1203.1617] [INSPIRE].
D. Correa, J. Maldacena and A. Sever, The quark anti-quark potential and the cusp anomalous dimension from a TBA equation, JHEP 08 (2012) 134 [arXiv:1203.1913] [INSPIRE].
F.A. Dolan, L. Gallot and E. Sokatchev, On four-point functions of 1/2-BPS operators in general dimensions, JHEP 09 (2004) 056 [hep-th/0405180] [INSPIRE].
R. Doobary and P. Heslop, Superconformal partial waves in Grassmannian field theories, JHEP 12 (2015) 159 [arXiv:1508.03611] [INSPIRE].
P. Liendo, C. Meneghelli and V. Mitev, On correlation functions of BPS operators in 3d N = 6 superconformal theories,Commun. Math. Phys. 350 (2017) 387 [arXiv:1512.06072] [INSPIRE].
M. Lemos, P. Liendo, C. Meneghelli and V. Mitev, Bootstrapping N = 3 superconformal theories, JHEP 04 (2017) 032 [arXiv:1612.01536] [INSPIRE].
N. Drukker and J. Plefka, Superprotected n-point correlation functions of local operators in N = 4 super Yang-Mills, JHEP 04 (2009) 052 [arXiv:0901.3653] [INSPIRE].
S. Giombi and S. Komatsu, Exact correlators on the Wilson loop in N = 4 SYM: localization, defect CFT and integrability, JHEP 05 (2018) 109 [arXiv:1802.05201] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, W. Peelaers, L. Rastelli and B.C. van Rees, Infinite chiral symmetry in four dimensions, Commun. Math. Phys. 336 (2015) 1359 [arXiv:1312.5344] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, W symmetry in six dimensions, JHEP 05 (2015) 017 [arXiv:1404.1079] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, Exact correlators of BPS operators from the 3d superconformal bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE].
C. Beem, W. Peelaers and L. Rastelli, Deformation quantization and superconformal symmetry in three dimensions, Commun. Math. Phys. 354 (2017) 345 [arXiv:1601.05378] [INSPIRE].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Wilson loops: from four-dimensional SYM to two-dimensional YM, Phys. Rev. D 77 (2008) 047901 [arXiv:0707.2699] [INSPIRE].
S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S 2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].
B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP 01 (2014) 020 [arXiv:1311.2058] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, The N = 4 superconformal bootstrap, Phys. Rev. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE].
S.M. Chester, J. Lee, S.S. Pufu and R. Yacoby, The N = 8 superconformal bootstrap in three dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE].
C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2, 0) superconformal bootstrap, Phys. Rev. D 93 (2016) 025016 [arXiv:1507.05637] [INSPIRE].
C. Beem, M. Lemos, P. Liendo, L. Rastelli and B.C. van Rees, The N = 2 superconformal bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE].
C. Beem, L. Rastelli and B.C. van Rees, More N = 4 superconformal bootstrap, Phys. Rev. D 96 (2017) 046014 [arXiv:1612.02363] [INSPIRE].
C.-M. Chang and Y.-H. Lin, Carving out the end of the world or (superconformal bootstrap in six dimensions), JHEP 08 (2017) 128 [arXiv:1705.05392] [INSPIRE].
C.-M. Chang, M. Fluder, Y.-H. Lin and Y. Wang, Spheres, charges, instantons and bootstrap: a five-dimensional odyssey, JHEP 03 (2018) 123 [arXiv:1710.08418] [INSPIRE].
D. Simmons-Duffin, A semidefinite program solver for the conformal bootstrap, JHEP 06 (2015) 174 [arXiv:1502.02033] [INSPIRE].
C. Behan, PyCFTBoot: a flexible interface for the conformal bootstrap, Commun. Comput. Phys. 22 (2017) 1 [arXiv:1602.02810] [INSPIRE].
T. Ohtsuki, CBoot: a sage module to create (convolved) conformal block table, https://github.com/tohtsky/cboot, (2016).
M.F. Paulos, JuliBootS: a hands-on guide to the conformal bootstrap, arXiv:1412.4127 [INSPIRE].
D. Poland, D. Simmons-Duffin and A. Vichi, Carving out the space of 4D CFTs, JHEP 05 (2012) 110 [arXiv:1109.5176] [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].
S. Rychkov, EPFL lectures on conformal field theory in D ≥ 3 dimensions, SpringerBriefs in Physics. 2016, https://doi.org/10.1007/978-3-319-43626-5 [arXiv:1601.05000] [INSPIRE].
D. Simmons-Duffin, The conformal bootstrap, in Proceedings, Theoretical Advanced Study Institute in Elementary Particle Physics: new frontiers in fields and strings (TASI 2015), Boulder, CO, U.S.A., 1-26 June 2015, World Scientific, Singapore, (2017), pg. 1 [arXiv:1602.07982] [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, D. Simmons-Duffin and A. Vichi, Precision islands in the Ising and O(N) models, JHEP 08 (2016) 036 [arXiv:1603.04436] [INSPIRE].
L.F. Alday and J. Maldacena, Comments on gluon scattering amplitudes via AdS/CFT, JHEP 11 (2007) 068 [arXiv:0710.1060] [INSPIRE].
S. El-Showk and M.F. Paulos, Bootstrapping conformal field theories with the extremal functional method, Phys. Rev. Lett. 111 (2013) 241601 [arXiv:1211.2810] [INSPIRE].
A. Dymarsky, F. Kos, P. Kravchuk, D. Poland and D. Simmons-Duffin, The 3d stress-tensor bootstrap, JHEP 02 (2018) 164 [arXiv:1708.05718] [INSPIRE].
D. Simmons-Duffin, The lightcone bootstrap and the spectrum of the 3d Ising CFT, JHEP 03 (2017) 086 [arXiv:1612.08471] [INSPIRE].
O. Aharony, L.F. Alday, A. Bissi and R. Yacoby, The analytic bootstrap for large N Chern-Simons vector models, JHEP 08 (2018) 166 [arXiv:1805.04377] [INSPIRE].
I. Heemskerk, J. Penedones, J. Polchinski and J. Sully, Holography from conformal field theory, JHEP 10 (2009) 079 [arXiv:0907.0151] [INSPIRE].
A.L. Fitzpatrick, E. Katz, D. Poland and D. Simmons-Duffin, Effective conformal theory and the flat-space limit of AdS, JHEP 07 (2011) 023 [arXiv:1007.2412] [INSPIRE].
L.F. Alday and A. Bissi, Loop corrections to supergravity on AdS 5 × S 5, Phys. Rev. Lett. 119 (2017) 171601 [arXiv:1706.02388] [INSPIRE].
F. Aprile, J.M. Drummond, P. Heslop and H. Paul, Quantum gravity from conformal field theory, JHEP 01 (2018) 035 [arXiv:1706.02822] [INSPIRE].
M. Cornagliotto, M. Lemos and V. Schomerus, Long multiplet bootstrap, JHEP 10 (2017) 119 [arXiv:1702.05101] [INSPIRE].
L. Bianchi, M. Lemos and M. Meineri, Line defects and radiation in N = 2 theories, arXiv:1805.04111 [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].
F. Passerini and K. Zarembo, Wilson loops in N = 2 super-Yang-Mills from matrix model, JHEP 09 (2011) 102 [Erratum ibid. 10 (2011) 065] [arXiv:1106.5763] [INSPIRE].
V. Mitev and E. Pomoni, Exact effective couplings of four dimensional gauge theories with N = 2 supersymmetry, Phys. Rev. D 92 (2015) 125034 [arXiv:1406.3629] [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 N = 4 SYM and defect 1d CFT, JHEP 03 (2018) 131 [arXiv:1712.06874] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On non-supersymmetric generalizations of the Wilson-Maldacena loops in N = 4 SYM, Nucl. Phys. B 934 (2018) 466 [arXiv:1804.02179] [INSPIRE].
D.J. Gross and V. Rosenhaus, A line of CFTs: from generalized free fields to SYK, JHEP 07 (2017) 086 [arXiv:1706.07015] [INSPIRE].
L. Iliesiu, M. Koloğlu, R. Mahajan, E. Perlmutter and D. Simmons-Duffin, The conformal bootstrap at finite temperature, arXiv:1802.10266 [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].
R. Feger and T.W. Kephart, LieART — a Mathematica application for Lie algebras and representation theory, Comput. Phys. Commun. 192 (2015) 166 [arXiv:1206.6379] [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: 1806.01862
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
Liendo, P., Meneghelli, C. & Mitev, V. Bootstrapping the half-BPS line defect. J. High Energ. Phys. 2018, 77 (2018). https://doi.org/10.1007/JHEP10(2018)077
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2018)077