Abstract
We revisit the computation of the 1-loop string correction to the “latitude” minimal surface in AdS 5 × S 5 representing 1/4 BPS Wilson loop in planar \( \mathcal{N}=4 \) SYM theory previously addressed in arXiv:1512.00841 and arXiv:1601.04708. We resolve the problem of matching with the subleading term in the strong coupling expansion of the exact gauge theory result (derived previously from localization) using a different method to compute determinants of 2d string fluctuation operators. We apply perturbation theory in a small parameter (angle of the latitude) corresponding to an expansion near the AdS 2 minimal surface representing 1/2 BPS circular Wilson loop. This allows us to compute the corrections to the heat kernels and zeta-functions of the operators in terms of the known heat kernels on AdS 2. We apply the same method also to two other examples of Wilson loop surfaces: generalized cusp and k-wound circle.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
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].
R. Kallosh and A.A. Tseytlin, Simplifying superstring action on AdS 5 × S 5, JHEP 10 (1998) 016 [hep-th/9808088] [INSPIRE].
S. Förste, D. Ghoshal and S. Theisen, Stringy corrections to the Wilson loop in N = 4 super Yang-Mills theory, JHEP 08 (1999) 013 [hep-th/9903042] [INSPIRE].
N. Drukker, D.J. Gross and A.A. Tseytlin, Green-Schwarz string in AdS 5 × S 5 : Semiclassical partition function, JHEP 04 (2000) 021 [hep-th/0001204] [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].
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].
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].
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, Supersymmetric Wilson loops on S 3, JHEP 05 (2008) 017 [arXiv:0711.3226] [INSPIRE].
V. Forini, V.G.M. Puletti, L. Griguolo, D. Seminara and E. Vescovi, Remarks on the geometrical properties of semiclassically quantized strings, J. Phys. A 48 (2015) 475401 [arXiv:1507.01883] [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].
R. Bergamin and A.A. Tseytlin, Heat kernels on cone of AdS 2 and k-wound circular Wilson loop in AdS 5 × S 5 superstring, J. Phys. A 49 (2016) 14LT01 [arXiv:1510.06894] [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].
G.W. Semenoff and K. Zarembo, Wilson loops in SYM theory: From weak to strong coupling, Nucl. Phys. Proc. Suppl. 108 (2002) 106, in Light cone physics: Particles and strings. Proceedings, International Workshop, Trento, Italy, 3-11 September 2001, pp. 106-112 [hep-th/0202156] [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 and B. Fiol, All-genus calculation of Wilson loops using D-branes, JHEP 02 (2005) 010 [hep-th/0501109] [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].
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].
R.E. Gamboa Saravi, M.A. Muschietti and J.E. Solomin, On Perturbation Theory for Regularized Determinants of Differential Operators, Commun. Math. Phys. 89 (1983) 363 [INSPIRE].
V. Mukhanov and S. Winitzki, Introduction to Quantum Effects in Gravity, Cambridge University Press (2007).
R. Camporesi, Harmonic analysis and propagators on homogeneous spaces, Phys. Rept. 196 (1990) 1 [INSPIRE].
R. Camporesi and A. Higuchi, Spectral functions and zeta functions in hyperbolic spaces, J. Math. Phys. 35 (1994) 4217 [INSPIRE].
R. Camporesi, The Spinor heat kernel in maximally symmetric spaces, Commun. Math. Phys. 148 (1992) 283 [INSPIRE].
R. Camporesi and A. Higuchi, On the Eigen functions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996) 1 [gr-qc/9505009] [INSPIRE].
I.M. Gelfand and A.M. Yaglom, Integration in functional spaces and it applications in quantum physics, J. Math. Phys. 1 (1960) 48. [INSPIRE].
R. Forman, Functional determinants and geometry, Invent. Math. 88 (1987) 447.
R. Forman, Functional determinants and geometry (Erratum), Invent. Math. 108 (1992) 453.
A.J. McKane and M.B. Tarlie, Regularization of functional determinants using boundary perturbations, J. Phys. A 28 (1995) 6931 [cond-mat/9509126] [INSPIRE].
K. Kirsten and A.J. McKane, Functional determinants by contour integration methods, Annals Phys. 308 (2003) 502 [math-ph/0305010] [INSPIRE].
K. Kirsten and A.J. McKane, Functional determinants for general Sturm-Liouville problems, J. Phys. A 37 (2004) 4649 [math-ph/0403050] [INSPIRE].
K. Kirsten and P. Loya, Computation of determinants using contour integrals, Am. J. Phys. 76 (2008) 60 [arXiv:0707.3755] [INSPIRE].
G.V. Dunne, Functional determinants in quantum field theory, J. Phys. A 41 (2008) 304006, in Proceedings, 5th International Symposium on Quantum theory and symmetries (QTS5) [arXiv:0711.1178] [INSPIRE].
S.A. Frolov, I.Y. Park and A.A. Tseytlin, On one-loop correction to energy of spinning strings in S 5, Phys. Rev. D 71 (2005) 026006 [hep-th/0408187] [INSPIRE].
A. Dekel and T. Klose, Correlation Function of Circular Wilson Loops at Strong Coupling, JHEP 11 (2013) 117 [arXiv:1309.3203] [INSPIRE].
G.V. Dunne and K. Kirsten, Functional determinants for radial operators, J. Phys. A 39 (2006) 11915 [hep-th/0607066] [INSPIRE].
N. Drukker and V. Forini, Generalized quark-antiquark potential at weak and strong coupling, JHEP 06 (2011) 131 [arXiv:1105.5144] [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].
M. Beccaria, G.V. Dunne, V. Forini, M. Pawellek and A.A. Tseytlin, Exact computation of one-loop correction to energy of spinning folded string in AdS 5 × S 5, J. Phys. A 43 (2010) 165402 [arXiv:1001.4018] [INSPIRE].
V. Forini, Quark-antiquark potential in AdS at one loop, JHEP 11 (2010) 079 [arXiv:1009.3939] [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].
V. Forini, V.G.M. Puletti, M. Pawellek and E. Vescovi, One-loop spectroscopy of semiclassically quantized strings: bosonic sector, J. Phys. A 48 (2015) 085401 [arXiv:1409.8674] [INSPIRE].
N. Drukker, D.J. Gross and H. Ooguri, Wilson loops and minimal surfaces, Phys. Rev. D 60 (1999) 125006 [hep-th/9904191] [INSPIRE].
D.V. Vassilevich, Heat kernel expansion: User’s manual, Phys. Rept. 388 (2003) 279 [hep-th/0306138] [INSPIRE].
D. Fursaev and D. Vassilevich, Operators, Geometry and Quanta: Methods of Spectral Geometry in Quantum Field Theory, Springer Verlag (2011).
P.B. Gilkey, Invariance Theory, the Heat Equation and the Atiyah-Singer Index Theorem, CRC Press (1995).
K. Zarembo, Supersymmetric Wilson loops, Nucl. Phys. B 643 (2002) 157 [hep-th/0205160] [INSPIRE].
J. Aguilera-Damia, D.H. Correa and G.A. Silva, Semiclassical partition function for strings dual to Wilson loops with small cusps in ABJM, JHEP 03 (2015) 002 [arXiv:1412.4084] [INSPIRE].
M. Bordag, K. Kirsten and J.S. Dowker, Heat kernels and functional determinants on the generalized cone, Commun. Math. Phys. 182 (1996) 371 [hep-th/9602089] [INSPIRE].
E. Vescovi, Perturbative and non-perturbative approaches to string sigma-models in AdS/CFT, Ph.D. Thesis (2016), https://edoc.hu-berlin.de/docviews/abstract.php?id=42898.
I. Chavel, Eigenvalues in Riemannian Geometry, Academic Press (1984), pp. 359-366.
T. Jones and D. Kucerovsky, Heat Kernel for Simply-Connected Riemann Surfaces, arXiv:1007.5467.
T. Jones, The heat kernel on noncompact Riemann surface, Ph.D. Thesis (2008).
R. Bergamin, The AdS 5 xS 5 string and the one-loop correction to the circular Wilson loop, MSc Thesis (2015), http://tesi.cab.unipd.it/49974/.
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: 1702.02164
Also at Lebedev Institute, Moscow (A.A. Tseytlin).
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Forini, V., Tseytlin, A. & Vescovi, E. Perturbative computation of string one-loop corrections to Wilson loop minimal surfaces in AdS 5 × S 5 . J. High Energ. Phys. 2017, 3 (2017). https://doi.org/10.1007/JHEP03(2017)003
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2017)003