Pohlmeyer reduction for superstrings in AdS space

The Pohlmeyer reduced equations for strings moving only in the AdS subspace of AdS₅ × S⁵ have been used recently in the study of classical Euclidean minimal surfaces for Wilson loops and some semiclassical three-point correlation functions. We find an action that leads to these reduced superstring equations. For example, for a bosonic string in AdS_n such an action contains a Liouville scalar part plus a K/K gauged WZW model for the group K = SO(n - 2) coupled to another term depending on two additional fields transforming as vectors under K. Solving for the latter fields gives a non-Abelian Toda model coupled to the Liouville theory. For n = 5 we generalize this bosonic action to include the S⁵ contribution and fermionic terms. The corresponding reduced model for the AdS₂ × S² truncation of the full AdS₅ × S⁵ superstring turns out to be equivalent to {N}=2 super Liouville theory. Our construction is based on taking a limit of the previously found reduced theory actions for bosonic strings in AdS_n × S¹ and superstrings in AdS₅ × S⁵. This new action may be useful as a starting point for possible quantum generalizations or deformations of the classical Pohlmeyer-reduced theory. We give examples of simple extrema of this reduced superstring action which represent strings moving in the AdS₅ part of the space. Expanding near these backgrounds we compute the corresponding fluctuation spectra and show that they match the spectra found in the original superstring theory.