On string solutions of Bethe equations in $\mathcal{N}=4$ supersymmetric Yang–Mills theory

The Bethe equations, arising in the 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 antiferromagnetic regime. These equations are deformation of those for the Heisenberg XXX magnet. It is proven that in the thermodynamic limit the roots of the deformed equations group into strings. It is proven that the corresponding Yang's action is convex, which implies uniqueness of solution for centers of the strings. The state formed of 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 nonanalyticity of the left hand side of the Bethe equations leads to an additional contribution to the density and energy of strings of even length. Whence it is concluded that the antiferromagnetic vacuum is formed of either strings of length 1 or strings of length 2 depending on behaviour of the exponential corrections to string solutions in the thermodynamic limit.