Abstract
We compute a set of correlation functions of operator insertions on the 1/8 BPS Wilson loop in \( \mathcal{N}=4 \) SYM by employing supersymmetric localization, OPE and the Gram-Schmidt orthogonalization. These correlators exhibit a simple determinant structure, are position-independent and form a topological subsector, but depend nontrivially on the ’t Hooft coupling and the rank of the gauge group. When applied to the 1/2 BPS circular (or straight) Wilson loop, our results provide an infinite family of exact defect CFT data, including the structure constants of protected defect primaries of arbitrary length inserted on the loop. At strong coupling, we show precise agreement with a direct calculation using perturbation theory around the AdS2 string worldsheet. We also explain the connection of our results to the “generalized Bremsstrahlung functions” previously computed from integrability techniques, reproducing the known results in the planar limit as well as obtaining their finite N generalization. Furthermore, we show that the correlators at large N can be recast as simple integrals of products of polynomials (known as Q-functions) that appear in the Quantum Spectral Curve approach. This suggests an interesting interplay between localization, defect CFT and integrability.
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21 November 2018
Typos in the formulae. A factor 1/?2 was missing in both equations in (6.28).
21 November 2018
Typos in the formulae. A factor 1/��2 was missing in both equations in (6.28).
References
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in proceedings of Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings (TASI 2015), Boulder, CO, U.S.A., June 1–26, 2015, pp. 1–74 (2017) DOI:https://doi.org/10.1142/9789813149441_0001 [arXiv:1602.07982] [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].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
J.M. Maldacena, Wilson loops in large N field theories, Phys. Rev. Lett. 80 (1998) 4859 [hep-th/9803002] [INSPIRE].
S.-J. Rey and J.-T. Yee, Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity, Eur. Phys. J. C 22 (2001) 379 [hep-th/9803001] [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].
N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [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].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].
V. Pestun, Localization of the four-dimensional N = 4 SYM to a two-sphere and 1/8 BPS Wilson loops, JHEP 12 (2012) 067 [arXiv:0906.0638] [INSPIRE].
S. Giombi, V. Pestun and R. Ricci, Notes on supersymmetric Wilson loops on a two-sphere, JHEP 07 (2010) 088 [arXiv:0905.0665] [INSPIRE].
S. Giombi and V. Pestun, The 1/2 BPS ’t Hooft loops in N = 4 SYM as instantons in 2d Yang-Mills, J. Phys. A 46 (2013) 095402 [arXiv:0909.4272] [INSPIRE].
A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson-loops, protected operators and matrix models in N = 4 SYM, JHEP 08 (2009) 061 [arXiv:0905.1943] [INSPIRE].
A. Bassetto, L. Griguolo, F. Pucci, D. Seminara, S. Thambyahpillai and D. Young, Correlators of supersymmetric Wilson loops at weak and strong coupling, JHEP 03 (2010) 038 [arXiv:0912.5440] [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].
S. Giombi and V. Pestun, Correlators of Wilson Loops and Local Operators from Multi-Matrix Models and Strings in AdS, JHEP 01 (2013) 101 [arXiv:1207.7083] [INSPIRE].
M. Bonini, L. Griguolo and M. Preti, Correlators of chiral primaries and 1/8 BPS Wilson loops from perturbation theory, JHEP 09 (2014) 083 [arXiv:1405.2895] [INSPIRE].
M. Bonini, L. Griguolo, M. Preti and D. Seminara, Bremsstrahlung function, leading Lüscher correction at weak coupling and localization, JHEP 02 (2016) 172 [arXiv:1511.05016] [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, An exact formula for the radiation of a moving quark in N = 4 super Yang-Mills, JHEP 06 (2012) 048 [arXiv:1202.4455] [INSPIRE].
H. Au-Yang and J.H.H. Perk, Toda lattice equation and Wronskians in the 2d Ising model, Physica D 18 (1986) 365.
H. Au-Yang and J.H.H. Perk, Critical correlations in a Z-invariant inhomogeneous Ising model, Physica A 144 (1987) 44.
N. Gromov and A. Sever, Analytic Solution of Bremsstrahlung TBA, JHEP 11 (2012) 075 [arXiv:1207.5489] [INSPIRE].
N. Gromov, F. Levkovich-Maslyuk and G. Sizov, Analytic Solution of Bremsstrahlung TBA II: Turning on the Sphere Angle, JHEP 10 (2013) 036 [arXiv:1305.1944] [INSPIRE].
N. Drukker and S. Kawamoto, Small deformations of supersymmetric Wilson loops and open spin-chains, JHEP 07 (2006) 024 [hep-th/0604124] [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].
N. Kiryu and S. Komatsu, Correlation Functions on the Half-BPS Wilson Loop: Perturbation and Hexagonalization, to appear.
M. Kim, N. Kiryu, S. Komatsu and T. Nishimura, Structure Constants of Defect Changing Operators on the 1/2 BPS Wilson Loop, JHEP 12 (2017) 055 [arXiv:1710.07325] [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].
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].
P. Liendo and C. Meneghelli, Bootstrap equations for \( \mathcal{N}=4 \) SYM with defects, JHEP 01 (2017) 122 [arXiv:1608.05126] [INSPIRE].
N. Gromov, V. Kazakov, S. Leurent and D. Volin, Quantum Spectral Curve for Planar \( \mathcal{N}=4 \) Super-Yang-Mills Theory, Phys. Rev. Lett. 112 (2014) 011602 [arXiv:1305.1939] [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.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].
E. Gerchkovitz, J. Gomis, N. Ishtiaque, A. Karasik, Z. Komargodski and S.S. Pufu, Correlation Functions of Coulomb Branch Operators, JHEP 01 (2017) 103 [arXiv:1602.05971] [INSPIRE].
M. Baggio, V. Niarchos, K. Papadodimas and G. Vos, Large-N correlation functions in \( \mathcal{N}=2 \) superconformal QCD, JHEP 01 (2017) 101 [arXiv:1610.07612] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Large N Correlation Functions in Superconformal Field Theories, JHEP 06 (2016) 109 [arXiv:1604.07416] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Operator mixing in large N superconformal field theories on S 4 and correlators with Wilson loops, JHEP 12 (2016) 120 [arXiv:1607.07878] [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].
G. Sizov and S. Valatka, Algebraic Curve for a Cusped Wilson Line, JHEP 05 (2014) 149 [arXiv:1306.2527] [INSPIRE].
N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve for a cusped Wilson line in \( \mathcal{N}=4 \) SYM, JHEP 04 (2016) 134 [arXiv:1510.02098] [INSPIRE].
B. Fiol, B. Garolera and G. Torrents, Exact momentum fluctuations of an accelerated quark in N = 4 super Yang-Mills, JHEP 06 (2013) 011 [arXiv:1302.6991] [INSPIRE].
M. Sakaguchi and K. Yoshida, A semiclassical string description of Wilson loop with local operators, Nucl. Phys. B 798 (2008) 72 [arXiv:0709.4187] [INSPIRE].
M. Sakaguchi and K. Yoshida, Holography of Non-relativistic String on AdS 5 × S 5, JHEP 02 (2008) 092 [arXiv:0712.4112] [INSPIRE].
S. Giombi and S. Komatsu, in progress.
B. Fiol, E. Gerchkovitz and Z. Komargodski, Exact Bremsstrahlung Function in N = 2 Superconformal Field Theories, Phys. Rev. Lett. 116 (2016) 081601 [arXiv:1510.01332] [INSPIRE].
A. Gadde, E. Pomoni and L. Rastelli, The Veneziano Limit of N = 2 Superconformal QCD: Towards the String Dual of N = 2 SU(N c) SYM with N f = 2N c, arXiv:0912.4918 [INSPIRE].
B. Basso, S. Komatsu and P. Vieira, Structure Constants and Integrable Bootstrap in Planar N = 4 SYM Theory, arXiv:1505.06745 [INSPIRE].
M. Kim and N. Kiryu, Structure constants of operators on the Wilson loop from integrability, JHEP 11 (2017) 116 [arXiv:1706.02989] [INSPIRE].
S. Komatsu, unpublished.
T. Bargheer, J. Caetano, T. Fleury, S. Komatsu and P. Vieira, Handling Handles I: Nonplanar Integrability, arXiv:1711.05326 [INSPIRE].
B. Eden, Y. Jiang, D. le Plat and A. Sfondrini, Colour-dressed hexagon tessellations for correlation functions and non-planar corrections, JHEP 02 (2018) 170 [arXiv:1710.10212] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, Quantum Spectral Curve and Structure Constants in N = 4 SYM: Cusps in the Ladder Limit, arXiv:1802.04237 [INSPIRE].
D. Correa, J. Henn, J. Maldacena and A. Sever, The cusp anomalous dimension at three loops and beyond, JHEP 05 (2012) 098 [arXiv:1203.1019] [INSPIRE].
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Giombi, S., Komatsu, S. Exact correlators on the Wilson loop in \( \mathcal{N}=4 \) SYM: localization, defect CFT, and integrability. J. High Energ. Phys. 2018, 109 (2018). https://doi.org/10.1007/JHEP05(2018)109
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DOI: https://doi.org/10.1007/JHEP05(2018)109