Classical integrability for three-point functions: cognate structure at weak and strong couplings

  • Yoichi Kazama
  • Shota Komatsu
  • Takuya NishimuraEmail author
Regular Article - Theoretical Physics


In this paper, we develop a new method of computing three-point functions in the SU(2) sector of the \( \mathcal{N}=4 \) super Yang-Mills theory in the semi-classical regime at weak coupling, which closely parallels the strong coupling analysis. The structure threading two disparate regimes is the so-called monodromy relation, an identity connecting the three-point functions with and without the insertion of the monodromy matrix. We shall show that this relation can be put to use directly for the semi-classical regime, where the dynamics is governed by the classical Landau-Lifshitz sigma model. Specifically, it reduces the problem to a set of functional equations, which can be solved once the analyticity in the spectral parameter space is specified. To determine the analyticity, we develop a new universal logic applicable at both weak and strong couplings. As a result, compact semi-classical formulas are obtained for a general class of three-point functions at weak coupling including the ones whose semi-classical behaviors were not known before. In addition, the new analyticity argument applied to the strong coupling analysis leads to a modification of the integration contour, producing the results consistent with the recent hexagon bootstrap approach. This modification also makes the Frolov-Tseytlin limit perfectly agree with the weak coupling form.


AdS-CFT Correspondence Integrable Field Theories 


Open Access

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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.

Authors and Affiliations

  • Yoichi Kazama
    • 1
    • 2
    • 3
  • Shota Komatsu
    • 4
  • Takuya Nishimura
    • 3
    Email author
  1. 1.Research Center for Mathematical PhysicsRikkyo UniversityTokyoJapan
  2. 2.Quantum Hadron Physics Laboratory, RIKEN Nishina CenterWakoJapan
  3. 3.Institute of PhysicsUniversity of TokyoTokyoJapan
  4. 4.Perimeter Institute for Theoretical PhysicsWaterlooCanada

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