Abstract
We consider the defect CFT defined by a ’t Hooft line embedded in \( \mathcal{N} \) = 4 super Yang-Mills theory. By explicitly quantizing around the given background we exactly reproduce a prediction from S-duality for the correlators between the ’t Hooft line and chiral primaries in the bulk and pave the way for higher loop analyses for non-protected operators. Furthermore, we demonstrate at the leading perturbative order that correlators between the ’t Hooft line and non-protected bulk operators can be efficiently computed using integrability. As a byproduct we find new integrable overlaps in \( \mathfrak{sl} \)(2) spin chains in different representations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. ’t Hooft, On the Phase Transition Towards Permanent Quark Confinement, Nucl. Phys. B 138 (1978) 1 [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].
J. Gomis, T. Okuda and D. Trancanelli, Quantum ’t Hooft operators and S-duality in N = 4 super Yang-Mills, Adv. Theor. Math. Phys. 13 (2009) 1941 [arXiv:0904.4486] [INSPIRE].
J. Gomis and T. Okuda, S-duality, ’t Hooft operators and the operator product expansion, JHEP 09 (2009) 072 [arXiv:0906.3011] [INSPIRE].
J. Gomis, T. Okuda and V. Pestun, Exact Results for ’t Hooft Loops in Gauge Theories on S4, JHEP 05 (2012) 141 [arXiv:1105.2568] [INSPIRE].
D.-E. Diaconescu, D-branes, monopoles and Nahm equations, Nucl. Phys. B 503 (1997) 220 [hep-th/9608163] [INSPIRE].
A. Dekel and Y. Oz, Integrability of Green-Schwarz Sigma Models with Boundaries, JHEP 08 (2011) 004 [arXiv:1106.3446] [INSPIRE].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N} \) = 4 SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
C. Montonen and D.I. Olive, Magnetic Monopoles as Gauge Particles?, Phys. Lett. B 72 (1977) 117 [INSPIRE].
E. Witten and D.I. Olive, Supersymmetry Algebras That Include Topological Charges, Phys. Lett. B 78 (1978) 97 [INSPIRE].
H. Osborn, Topological Charges for N = 4 Supersymmetric Gauge Theories and Monopoles of Spin 1, Phys. Lett. B 83 (1979) 321 [INSPIRE].
I. Buhl-Mortensen et al., Asymptotic One-Point Functions in Gauge-String Duality with Defects, Phys. Rev. Lett. 119 (2017) 261604 [arXiv:1704.07386] [INSPIRE].
S. Komatsu and Y. Wang, Non-perturbative defect one-point functions in planar \( \mathcal{N} \) = 4 super-Yang-Mills, Nucl. Phys. B 958 (2020) 115120 [arXiv:2004.09514] [INSPIRE].
T. Gombor and Z. Bajnok, Boundary states, overlaps, nesting and bootstrapping AdS/dCFT, JHEP 10 (2020) 123 [arXiv:2004.11329] [INSPIRE].
T. Gombor and Z. Bajnok, Boundary state bootstrap and asymptotic overlaps in AdS/dCFT, JHEP 03 (2021) 222 [arXiv:2006.16151] [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].
I. Buhl-Mortensen, M. de Leeuw, C. Kristjansen and K. Zarembo, One-point Functions in AdS/dCFT from Matrix Product States, JHEP 02 (2016) 052 [arXiv:1512.02532] [INSPIRE].
P.A.M. Dirac, Quantised singularities in the electromagnetic field,, Proc. Roy. Soc. Lond. A 133 (1931) 60 [INSPIRE].
M. Bianchi, M.B. Green and S. Kovacs, Instanton corrections to circular Wilson loops in N=4 supersymmetric Yang-Mills, JHEP 04 (2002) 040 [hep-th/0202003] [INSPIRE].
M.A. Shifman, Wilson Loop in Vacuum Fields, Nucl. Phys. B 173 (1980) 13 [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].
G.W. Semenoff and K. Zarembo, More exact predictions of SUSYM for string theory, Nucl. Phys. B 616 (2001) 34 [hep-th/0106015] [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].
S. Giombi and V. Pestun, Correlators of local operators and 1/8 BPS Wilson loops on S2 from 2d YM and matrix models, JHEP 10 (2010) 033 [arXiv:0906.1572] [INSPIRE].
K. Okuyama and G.W. Semenoff, Wilson loops in N = 4 SYM and fermion droplets, JHEP 06 (2006) 057 [hep-th/0604209] [INSPIRE].
N. Drukker and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [INSPIRE].
S.A. Hartnoll and S.P. Kumar, Higher rank Wilson loops from a matrix model, JHEP 08 (2006) 026 [hep-th/0605027] [INSPIRE].
S. Kawamoto, T. Kuroki and A. Miwa, Boundary condition for D-brane from Wilson loop, and gravitational interpretation of eigenvalue in matrix model in AdS/CFT correspondence, Phys. Rev. D 79 (2009) 126010 [arXiv:0812.4229] [INSPIRE].
M. Beccaria and A.A. Tseytlin, On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators, JHEP 01 (2021) 149 [arXiv:2011.02885] [INSPIRE].
J. Polchinski, String theory, Cambridge University Press (1998) [https://doi.org/10.1017/CBO9780511816079].
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].
N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].
J. Ambjorn, R.A. Janik and C. Kristjansen, Wrapping interactions and a new source of corrections to the spin-chain/string duality, Nucl. Phys. B 736 (2006) 288 [hep-th/0510171] [INSPIRE].
V. Borokhov, A. Kapustin and X.-K. Wu, Topological disorder operators in three-dimensional conformal field theory, JHEP 11 (2002) 049 [hep-th/0206054] [INSPIRE].
V. Borokhov, A. Kapustin and X.-K. Wu, Monopole operators and mirror symmetry in three-dimensions, JHEP 12 (2002) 044 [hep-th/0207074] [INSPIRE].
G. Grignani, L. Griguolo, N. Mori and D. Seminara, Thermodynamics of theories with sixteen supercharges in non-trivial vacua, JHEP 10 (2007) 068 [arXiv:0707.0052] [INSPIRE].
I. Buhl-Mortensen et al., One-loop one-point functions in gauge-gravity dualities with defects, Phys. Rev. Lett. 117 (2016) 231603 [arXiv:1606.01886] [INSPIRE].
I. Buhl-Mortensen et al., A Quantum Check of AdS/dCFT, JHEP 01 (2017) 098 [arXiv:1611.04603] [INSPIRE].
I. Tamm, Die verallgemeinerten Kugelfunktionen und die Wellenfunktionen eines Elektrons im Felde eines Magnetpoles, Z. Phys. 71 (1931) 141 [INSPIRE].
M. Fierz, Zur Theorie magnetisch geladener Teilchen, Helv. Phys. Acta 17 (1944) 27.
T.T. Wu and C.N. Yang, Dirac’s Monopole Without Strings: Classical Lagrangian Theory, Phys. Rev. D 14 (1976) 437 [INSPIRE].
S.M. Roy and V. Singh, Exact Solution of Schrodinger Equation in Aharonov-Bohm Plus Dirac Monopole Potential, Phys. Rev. Lett. 51 (1983) 2069 [INSPIRE].
H.A. Olsen, P. Osland and T.T. Wu, On the Existence of Bound States for a Massive Spin 1 Particle and a Magnetic Monopole, Phys. Rev. D 42 (1990) 665 [INSPIRE].
E.J. Weinberg, Monopole vector spherical harmonics, Phys. Rev. D 49 (1994) 1086 [hep-th/9308054] [INSPIRE].
Y. Kazama, C.N. Yang and A.S. Goldhaber, Scattering of a Dirac Particle with Charge Ze by a Fixed Magnetic Monopole, Phys. Rev. D 15 (1977) 2287 [INSPIRE].
H. Liu and A.A. Tseytlin, On four point functions in the CFT/AdS correspondence, Phys. Rev. D 59 (1999) 086002 [hep-th/9807097] [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for N = 4 superYang-Mills, JHEP 03 (2003) 013 [hep-th/0212208] [INSPIRE].
M. De Leeuw et al., Spin Chain Overlaps and the Twisted Yangian, JHEP 01 (2020) 176 [arXiv:1912.09338] [INSPIRE].
S. Ghoshal and A.B. Zamolodchikov, Boundary S matrix and boundary state in two-dimensional integrable quantum field theory, Int. J. Mod. Phys. A 9 (1994) 3841 [Erratum ibid. 9 (1994) 4353] [hep-th/9306002] [INSPIRE].
B. Pozsgay, L. Piroli and E. Vernier, Integrable Matrix Product States from boundary integrability, SciPost Phys. 6 (2019) 062 [arXiv:1812.11094] [INSPIRE].
C. Kristjansen, D. Müller and K. Zarembo, Duality relations for overlaps of integrable boundary states in AdS/dCFT, JHEP 09 (2021) 004 [arXiv:2106.08116] [INSPIRE].
N. Beisert, The complete one loop dilatation operator of N = 4 superYang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].
M. Brockmann, J.D. Nardis, B. Wouters and J.-S. Caux, Néel-XXZ state overlaps: odd particle numbers and Lieb-Liniger scaling limit, arXiv:1403.7469 [https://doi.org/10.1088/1751-8113/47/34/345003].
Y. Jiang, S. Komatsu and E. Vescovi, Structure constants in \( \mathcal{N} \) = 4 SYM at finite coupling as worldsheet g-function, JHEP 07 (2020) 037 [arXiv:1906.07733] [INSPIRE].
Y. Jiang and B. Pozsgay, On exact overlaps in integrable spin chains, JHEP 06 (2020) 022 [arXiv:2002.12065] [INSPIRE].
C. Kristjansen, D. Müller and K. Zarembo, Integrable boundary states in D3-D5 dCFT: beyond scalars, JHEP 08 (2020) 103 [arXiv:2005.01392] [INSPIRE].
T. Gombor, On the classification of rational K-matrices, J. Phys. A 53 (2020) 135203 [arXiv:1904.03044] [INSPIRE].
T. Gombor and B. Pozsgay, On factorized overlaps: Algebraic Bethe Ansatz, twists, and Separation of Variables, Nucl. Phys. B 967 (2021) 115390 [arXiv:2101.10354] [INSPIRE].
T. Gombor, On exact overlaps for \( \mathfrak{gl} \)(N) symmetric spin chains, Nucl. Phys. B 983 (2022) 115909 [arXiv:2110.07960] [INSPIRE].
M. Staudacher, The Factorized S-matrix of CFT/AdS, JHEP 05 (2005) 054 [hep-th/0412188] [INSPIRE].
G. ’t Hooft, Computation of the Quantum Effects Due to a Four-Dimensional Pseudoparticle, Phys. Rev. D 14 (1976) 3432 [Erratum ibid. 18 (1978) 2199] [INSPIRE].
N. Beisert and M. Staudacher, The N = 4 SYM integrable super spin chain, Nucl. Phys. B 670 (2003) 439 [hep-th/0307042] [INSPIRE].
G. Ferretti, R. Heise and K. Zarembo, New integrable structures in large-N QCD, Phys. Rev. D 70 (2004) 074024 [hep-th/0404187] [INSPIRE].
N. Beisert, G. Ferretti, R. Heise and K. Zarembo, One-loop QCD spin chain and its spectrum, Nucl. Phys. B 717 (2005) 137 [hep-th/0412029] [INSPIRE].
A.B. Zamolodchikov and V.A. Fateev, Model factorized S matrix and an integrable heisenberg chain with spin 1 (in Russian), Sov. J. Nucl. Phys. 32 (1980) 298 [INSPIRE].
P.P. Kulish, N.Y. Reshetikhin and E.K. Sklyanin, Yang-Baxter Equation and Representation Theory. I, Lett. Math. Phys. 5 (1981) 393 [INSPIRE].
N.Y. Reshetikhin, Integrable Models of Quantum One-dimensional Magnets With O(N) and Sp(2k) Symmetry, Theor. Math. Phys. 63 (1985) 555 [INSPIRE].
L.A. Takhtajan, The picture of low-lying excitations in the isotropic Heisenberg chain of arbitrary spins, Phys. Lett. A 87 (1982) 479 [INSPIRE].
H.M. Babujian, Exact solution of the one-dimensional isotropic Heisenberg chain with arbitrary spin S, Phys. Lett. A 90 (1982) 479 [INSPIRE].
H.M. Babujian, Exact solution of the isotropic Heisenberg chain with arbitrary spins: thermodynamics of the model, Nucl. Phys. B 215 (1983) 317 [INSPIRE].
W. Hao, R.I. Nepomechie and A.J. Sommese, Singular solutions, repeated roots and completeness for higher-spin chains, J. Stat. Mech. 1403 (2014) P03024 [arXiv:1312.2982] [INSPIRE].
J. Hou, Y. Jiang and R.-D. Zhu, Spin-s Rational Q-system, arXiv:2303.07640 [INSPIRE].
C. Kristjansen, D. Müller and K. Zarembo, Overlaps and fermionic dualities for integrable super spin chains, JHEP 03 (2021) 100 [arXiv:2011.12192] [INSPIRE].
M. De Leeuw, C. Kristjansen and G. Linardopoulos, Scalar one-point functions and matrix product states of AdS/dCFT, Phys. Lett. B 781 (2018) 238 [arXiv:1802.01598] [INSPIRE].
A. Bissi, C. Kristjansen, D. Young and K. Zoubos, Holographic three-point functions of giant gravitons, JHEP 06 (2011) 085 [arXiv:1103.4079] [INSPIRE].
A. Gorsky, A. Monin and A.V. Zayakin, Correlator of Wilson and t’Hooft Loops at Strong Coupling in N = 4 SYM Theory, Phys. Lett. B 679 (2009) 529 [arXiv:0904.3665] [INSPIRE].
S. Lee, S. Minwalla, M. Rangamani and N. Seiberg, Three point functions of chiral operators in D = 4, N = 4 SYM at large N, Adv. Theor. Math. Phys. 2 (1998) 697 [hep-th/9806074] [INSPIRE].
G. Arutyunov and S. Frolov, Scalar quartic couplings in type IIB supergravity on AdS5 × S5, Nucl. Phys. B 579 (2000) 117 [hep-th/9912210] [INSPIRE].
H.J. Kim, L.J. Romans and P. van Nieuwenhuizen, The Mass Spectrum of Chiral N = 2 D = 10 Supergravity on S5, Phys. Rev. D 32 (1985) 389 [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
Acknowledgments
We would like to thank T. Gombor, S. Komatsu and G. Linardopoulos for interesting discussions. This work was supported by DFF-FNU through grant number 1026-00103B (C.K.) and by VR grant 2021-04578 (K.Z.). Nordita is partially supported by Nordforsk.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2305.03649
Rights and permissions
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.
About this article
Cite this article
Kristjansen, C., Zarembo, K. ’t Hooft loops and integrability. J. High Energ. Phys. 2023, 184 (2023). https://doi.org/10.1007/JHEP08(2023)184
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2023)184