Journal of Mathematical Sciences

, Volume 151, Issue 2, pp 2840–2847 | Cite as

On string solutions of the Bethe equations in the \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory



The Bethe equations which arise in description of the spectrum of the dilatation operator for the su(2) sector of the \( \mathcal{N} = 4 \) supersymmetric Yang-Mills theory are considered in the anti-ferromagnetic regime. These equations are a deformation of those for the Heisenberg XXX magnet. We prove that in the thermodynamic limit, roots of the deformed equations group into strings. We prove that the corresponding Yang's action is convex, which implies the uniqueness of a solution for centers of the strings. The state formed by strings of length (2n+1) is considered, and the density of their distribution is found. It is shown that the energy of such a state decreases as n grows. It is observed that the nonanalyticity of the left-hand sides of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. We conclude that the structure of the anti-ferromagnetic vacuum is determined by the behavior of exponential corrections to string solutions in the thermodynamic limit and, possibly, involves strings of length 2. Bibliography: 14 titles.


Russia Physic Department Additional Contribution Thermodynamic Limit Equation Group 


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© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.St.Petersburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia
  2. 2.Physics Department of St.Petersburg State UniversitySt.PetersburgRussia

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