Spinning strings in AdS_5 x S^5: one-loop correction to energy in SL(2) sector

We consider a circular string with spin $S$ in $AdS_5$ wrapped around big circle of $S^5$ and carrying also momentum $J$. The corresponding N=4 SYM operator belongs to the SL(2) sector, i.e. has tr$(D^S Z^J)+...$ structure. The leading large $J$ term in its 1-loop anomalous dimension can be computed using Bethe ansatz for the SL(2) spin chain and was previously found to match the leading term in the classical string energy. The string solution is stable at large $J$, and the Lagrangian for string fluctuations has constant coefficients, so that the 1-loop string correction to the energy $E_1$ is given simply by the sum of characteristic frequencies. Curiously, we find that the leading term in the zero-mode part of $E_1$ is the same as a 1/J correction to the one-loop anomalous dimension on the gauge theory (spin chain) side that was found in hep-th/0410105. However, the contribution of non-zero string modes does not vanish. We also discuss the ``fast string'' expansion of the classical string action which coincides with the coherent state action of the SL(2) spin chain at the first order in $\l$, and extend this expansion to higher orders clarifying the role of the $S^5$ winding number.