Abstract
The AdS/CFT correspondence relates the expectation value of Wilson loops in \( \mathcal{N} \) = 4 SYM to the area of minimal surfaces in AdS5. In this paper we consider minimal area surfaces in generic Euclidean AdSn+1 using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS3. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.
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References
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [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].
N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].
N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].
N. Drukker and B. Fiol, On the integrability of Wilson loops in AdS 5 × S 5 : Some periodic ansatze, JHEP 01 (2006) 056 [hep-th/0506058] [INSPIRE].
K. Zarembo, Supersymmetric Wilson loops, Nucl. Phys. B 643 (2002) 157 [hep-th/0205160] [INSPIRE].
N. Drukker, 1/4 BPS circular loops, unstable world-sheet instantons and the matrix model, JHEP 09 (2006) 004 [hep-th/0605151] [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].
D.J. Gross and H. Ooguri, Aspects of large-N gauge theory dynamics as seen by string theory, Phys. Rev. D 58 (1998) 106002 [hep-th/9805129] [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. Kruczenski and A. Tirziu, Matching the circular Wilson loop with dual open string solution at 1-loop in strong coupling, JHEP 05 (2008) 064 [arXiv:0803.0315] [INSPIRE].
A. Faraggi and L.A. Pando Zayas, The Spectrum of Excitations of Holographic Wilson Loops, JHEP 05 (2011) 018 [arXiv:1101.5145] [INSPIRE].
E.I. Buchbinder and A.A. Tseytlin, 1/N correction in the D3-brane description of a circular Wilson loop at strong coupling, Phys. Rev. D 89 (2014) 126008 [arXiv:1404.4952] [INSPIRE].
V. Forini, V. Giangreco M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Precision calculation of 1/4-BPS Wilson loops in AdS 5 × S 5, JHEP 02 (2016) 105 [arXiv:1512.00841] [INSPIRE].
A. Faraggi, L.A. Pando Zayas, G.A. Silva and D. Trancanelli, Toward precision holography with supersymmetric Wilson loops, JHEP 04 (2016) 053 [arXiv:1601.04708] [INSPIRE].
B. Fiol and G. Torrents, Exact results for Wilson loops in arbitrary representations, JHEP 01 (2014) 020 [arXiv:1311.2058] [INSPIRE].
D. Müller, H. Münkler, J. Plefka, J. Pollok and K. Zarembo, Yangian Symmetry of smooth Wilson Loops in \( \mathcal{N} \) = 4 super Yang-Mills Theory, JHEP 11 (2013) 081 [arXiv:1309.1676] [INSPIRE].
H. Münkler and J. Pollok, Minimal surfaces of the AdS 5 × S 5 superstring and the symmetries of super Wilson loops at strong coupling, J. Phys. A 48 (2015) 365402 [arXiv:1503.07553] [INSPIRE].
S. Ryang, Algebraic Curves for Long Folded and Circular Winding Strings in AdS 5 × S 5, JHEP 02 (2013) 107 [arXiv:1212.6109] [INSPIRE].
A. Dekel, Algebraic Curves for Factorized String Solutions, JHEP 04 (2013) 119 [arXiv:1302.0555] [INSPIRE].
A. Dekel and T. Klose, Correlation Function of Circular Wilson Loops at Strong Coupling, JHEP 11 (2013) 117 [arXiv:1309.3203] [INSPIRE].
A. Irrgang and M. Kruczenski, Double-helix Wilson loops: Case of two angular momenta, JHEP 12 (2009) 014 [arXiv:0908.3020] [INSPIRE].
V. Forini, V.G.M. Puletti and O. Ohlsson Sax, The generalized cusp in AdS 4 × CP 3 and more one-loop results from semiclassical strings, J. Phys. A 46 (2013) 115402 [arXiv:1204.3302] [INSPIRE].
B.A. Burrington and L.A. Pando Zayas, Phase transitions in Wilson loop correlator from integrability in global AdS, Int. J. Mod. Phys. A 27 (2012) 1250001 [arXiv:1012.1525] [INSPIRE].
G. Papathanasiou, Pohlmeyer reduction and Darboux transformations in Euclidean worldsheet AdS 3, JHEP 08 (2012) 105 [arXiv:1203.3460] [INSPIRE].
N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [INSPIRE].
B.A. Burrington, General Leznov-Savelev solutions for Pohlmeyer reduced AdS 5 minimal surfaces, JHEP 09 (2011) 002 [arXiv:1105.3227] [INSPIRE].
L.F. Alday and A.A. Tseytlin, On strong-coupling correlation functions of circular Wilson loops and local operators, J. Phys. A 44 (2011) 395401 [arXiv:1105.1537] [INSPIRE].
C. Kalousios and D. Young, Dressed Wilson Loops on S 2, Phys. Lett. B 702 (2011) 299 [arXiv:1104.3746] [INSPIRE].
R. Ishizeki, M. Kruczenski and A. Tirziu, New open string solutions in AdS 5, Phys. Rev. D 77 (2008) 126018 [arXiv:0804.3438] [INSPIRE].
R.A. Janik and P. Laskos-Grabowski, Surprises in the AdS algebraic curve constructions: Wilson loops and correlation functions, Nucl. Phys. B 861 (2012) 361 [arXiv:1203.4246] [INSPIRE].
M. Kruczenski, Wilson loops and minimal area surfaces in hyperbolic space, JHEP 11 (2014) 065 [arXiv:1406.4945] [INSPIRE].
R. Ishizeki, M. Kruczenski and S. Ziama, Notes on Euclidean Wilson loops and Riemann Theta functions, Phys. Rev. D 85 (2012) 106004 [arXiv:1104.3567] [INSPIRE].
A. Dekel, Wilson Loops and Minimal Surfaces Beyond the Wavy Approximation, JHEP 03 (2015) 085 [arXiv:1501.04202] [INSPIRE].
C. Huang, Y. He and M. Kruczenski, Minimal area surfaces dual to Wilson loops and the Mathieu equation, JHEP 08 (2016) 088 [arXiv:1604.00078] [INSPIRE].
Y. He and M. Kruczenski, Minimal area surfaces in AdS 3 through integrability, J. Phys. A 50 (2017) 495401 [arXiv:1705.10037] [INSPIRE].
T. Klose, F. Loebbert and H. Munkler, Master Symmetry for Holographic Wilson Loops, Phys. Rev. D 94 (2016) 066006 [arXiv:1606.04104] [INSPIRE].
T. Klose, F. Loebbert and H. Münkler, Nonlocal Symmetries, Spectral Parameter and Minimal Surfaces in AdS/CFT, Nucl. Phys. B 916 (2017) 320 [arXiv:1610.01161] [INSPIRE].
G. Cairns, R. Sharpe and L. Webb, Conformal Invariants for Curves and Surfaces in Three Dimensional Space Forms, Rocky Mountain J. Math. 24 (1994) 933.
K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, Commun. Math. Phys. 46 (1976) 207 [INSPIRE].
B. Hoare and A.A. Tseytlin, Pohlmeyer reduction for superstrings in AdS space, J. Phys. A 46 (2013) 015401 [arXiv:1209.2892] [INSPIRE].
A.M. Polyakov and V.S. Rychkov, Loop dynamics and AdS/CFT correspondence, Nucl. Phys. B 594 (2001) 272 [hep-th/0005173] [INSPIRE].
G.W. Semenoff and D. Young, Wavy Wilson line and AdS/CFT, Int. J. Mod. Phys. A 20 (2005) 2833 [hep-th/0405288] [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].
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ArXiv ePrint: 1712.06269
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He, Y., Huang, C. & Kruczenski, M. Minimal area surfaces in AdSn+1 and Wilson loops. J. High Energ. Phys. 2018, 27 (2018). https://doi.org/10.1007/JHEP02(2018)027
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DOI: https://doi.org/10.1007/JHEP02(2018)027