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Quantum corrections to short folded superstring in AdS_3 x S^3 x M^4

We consider integrable superstring theory on AdS_3 x S^3 x M^4 where M^4=T^4 or M^4=S^3 x S^1 with generic ratio of the radii of the two 3-spheres. We compute the one-loop energy of a short folded string spinning in AdS_3 and rotating in S^3. The computation is performed by world-sheet small spin perturbation theory as well as by quantizing the classical algebraic curve characterizing the finite-gap equations. The two methods give equal results up to regularization contributions that are under control. One important byproduct of the calculation is the part of the energy which is due to the dressing phase in the Bethe Ansatz. Remarkably, this contribution E_1^{dressing} turns out to be independent on the radii ratio. In the M^4=T^4 limit, we discuss how E_1^{dressing} relates to a recent proposal for the dressing phase tested in the su(2) sector. We point out some difficulties suggesting that quantization of the AdS_3 classical finite-gap equations could be subtler than the easier AdS_5 x S^5 case.