Finite-size corrections to the rotating string and the winding state

We compute higher order finite size corrections to the energies of the circular rotating string on AdS^5 × S_5, of its orbifolded generalization on AdS_5 × S^5/Z^M and of
the winding state which is obtained as the limit of the orbifolded circular string solution when J → ∞ and J/M^2 is kept fixed. We solve, at the first order in λ′ = λ/J^2, where λ
is the ’t Hooft coupling, the Bethe equations that describe the anomalous dimensions of
the corresponding gauge dual operators in an expansion in m/K, where m is the winding number and K is the “magnon number”, and to all orders in the angular momentum J. The solution for the circular rotating string and for the winding state can be matched to the energy computed from an effective quantum Landau-Lifshitz model beyond the first order correction in 1/J. For the leading 1/J corrections to the circular rotating string in
m2 and m4 and for the subleading 1/J^2 corrections to the m^2 term, we find agreement. For the winding state we match the energy completely up to, and including, the order 1/J^2
finite-size corrections. The solution of the Bethe equations corresponding to the spinning closed string is also
provided in an expansion in m/K and to all orders in J.