Quantum corrections to spinning superstrings in AdS^sub 3^ × S^sup 3^ × M ^sup 4^: determining the dressing phase

Abstract

We study the leading quantum string correction to the dressing phase in the asymptotic Bethe Ansatz system for superstring in AdS ^sub 3^ × S ^sup 3^ × T ^sup 4^ supported by RR flux. We find that the phase should be different from the phase appearing in the AdS ^sub 5^ × S ^sup 5^ case. We use the simplest example of a rigid circular string with two equal spins in S ^sup 3^ and also consider the general approach based on the algebraic curve description. We also discuss the case of the AdS ^sub 3^ × S ^sup 3^ × S ^sup 3^ × S ^sup 1^ theory and find the dependence of the 1-loop correction to the effective string tension function h([lambda]) (expected to enter the magnon dispersion relation) on the parameter [alpha] related to the ratio of the two 3-sphere radii. This correction vanishes in the AdS ^sub 3^ × S ^sup 3^ × T ^sup 4^ case.