Abstract
Extending the methods developed in our previous works (arXiv:1110.3949, arXiv:1205.6060), we compute the three-point functions at strong coupling of the non-BPS states with large quantum numbers corresponding to the composite operators belonging to the so-called SU(2) sector in the \( \mathcal{N} \) = 4 super-Yang-Mills theory in four dimensions. This is achieved by the semi-classical evaluation of the three-point functions in the dual string theory in the AdS 3 × S 3 spacetime, using the general one-cut finite gap solutions as the external states. In spite of the complexity of the contributions from various parts in the intermediate stages, the final answer for the three-point function takes a remarkably simple form, exhibiting the structure reminiscent of the one obtained at weak coupling. In particular, in the Frolov-Tseytlin limit the result is expressed in terms of markedly similar integrals, however with different contours of integration. We discuss a natural mechanism for introducing additional singularities on the worldsheet without affecting the infinite number of conserved charges, which can modify the contours of integration.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, hep-th/9711200 [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, hep-th/9802109 [INSPIRE].
E. Witten, Anti-de Sitter space and holography, hep-th/9802150 [INSPIRE].
N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability III. Classical Tunneling, arXiv:1111.2349 [INSPIRE].
I. Kostov, Classical Limit of the Three-Point Function of N = 4 Supersymmetric Yang-Mills Theory from Integrability, arXiv:1203.6180 [INSPIRE].
I. Kostov, Three-point function of semiclassical states at weak coupling, arXiv:1205.4412 [INSPIRE].
D. Serban, A note on the eigenvectors of long-range spin chains and their scalar products, arXiv:1203.5842 [INSPIRE].
R.A. Janik and A. Wereszczynski, Correlation functions of three heavy operators: The AdS contribution, arXiv:1109.6262 [INSPIRE].
Y. Kazama and S. Komatsu, On holographic three point functions for GKP strings from integrability, arXiv:1110.3949 [Erratum ibid. 1206 (2012) 150] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, A Semiclassical limit of the gauge/string correspondence, hep-th/0204051 [INSPIRE].
Y. Kazama and S. Komatsu, Wave functions and correlation functions for GKP strings from integrability, arXiv:1205.6060 [INSPIRE].
V.A. Kazakov, A. Marshakov, J.A. Minahan and K. Zarembo, Classical/quantum integrability in AdS/CFT, hep-th/0402207 [INSPIRE].
N. Dorey and B. Vicedo, On the dynamics of finite-gap solutions in classical string theory, hep-th/0601194 [INSPIRE].
N. Dorey and B. Vicedo, A Symplectic Structure for String Theory on Integrable Backgrounds, hep-th/0606287 [INSPIRE].
B. Vicedo, The method of finite-gap integration in classical and semi-classical string theory, J. Phys. A 44 (2011) 124002 [arXiv:0810.3402] [INSPIRE].
E.K. Sklyanin, Separation of variables — new trends, Prog. Theor. Phys. Suppl. 118 (1995) 35 [solv-int/9504001] [INSPIRE].
I. Bena, J. Polchinski and R. Roiban, Hidden symmetries of the AdS 5 × S 5 superstring, hep-th/0305116 [INSPIRE].
K. Pohlmeyer, Integrable Hamiltonian Systems and Interactions Through Quadratic Constraints, [INSPIRE].
H.J. De Vega and N.G. Sanchez, Exact integrability of strings in D-Dimensional de Sitter space-time, [INSPIRE].
B. Vicedo, Semiclassical Quantisation of Finite-Gap Strings, arXiv:0803.1605 [INSPIRE].
H.M. Farkas and I. Kra, Riemann surfaces, Springer New York (1992).
J. Caetano and J. Toledo, χ-Systems for Correlation Functions, arXiv:1208.4548 [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
L.F. Alday and J. Maldacena, Null polygonal Wilson loops and minimal surfaces in Anti-de-Sitter space, arXiv:0904.0663 [INSPIRE].
L.F. Alday, D. Gaiotto and J. Maldacena, Thermodynamic Bubble Ansatz, arXiv:0911.4708 [INSPIRE].
L.F. Alday, J. Maldacena, A. Sever and P. Vieira, Y-system for Scattering Amplitudes, arXiv:1002.2459 [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability, arXiv:1012.2475 [INSPIRE].
V.E. Korepin, N.M. Bogoliubov and A.G. Izergin, Quantum inverse scattering method and correlation functions, Cambridge University Press, Cambridge, U.K. (1993).
D. Honda and S. Komatsu, Classical Liouville Three-point Functions from Riemann-Hilbert Analysis, arXiv:1311.2888 [INSPIRE].
J. Escobedo, N. Gromov, A. Sever and P. Vieira, Tailoring Three-Point Functions and Integrability II. Weak/strong coupling match, arXiv:1104.5501 [INSPIRE].
N. Gromov and P. Vieira, The AdS 5 × S 5 superstring quantum spectrum from the algebraic curve, hep-th/0703191 [INSPIRE].
M. Kruczenski, Spin chains and string theory, hep-th/0311203 [INSPIRE].
N. Beisert, V. Dippel and M. Staudacher, A Novel long range spin chain and planar N = 4 super Yang-Mills, hep-th/0405001 [INSPIRE].
N. Beisert, The SU(2|2) dynamic S-matrix, hep-th/0511082 [INSPIRE].
V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov, Higher level eigenvalues of Q operators and Schroedinger equation, hep-th/0307108 [INSPIRE].
Y. Kazama, S. Komatsu and T. Nishimura, A new integral representation for the scalar products of Bethe states for the XXX spin chain, arXiv:1304.5011 [INSPIRE].
K. Zarembo, Holographic three-point functions of semiclassical states, arXiv:1008.1059 [INSPIRE].
M.S. Costa, R. Monteiro, J.E. Santos and D. Zoakos, On three-point correlation functions in the gauge/gravity duality, arXiv:1008.1070 [INSPIRE].
R. Roiban and A.A. Tseytlin, On semiclassical computation of 3-point functions of closed string vertex operators in AdS 5 × S 5, arXiv:1008.4921 [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: 1312.3727
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
Kazama, Y., Komatsu, S. Three-point functions in the SU(2) sector at strong coupling. J. High Energ. Phys. 2014, 52 (2014). https://doi.org/10.1007/JHEP03(2014)052
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2014)052