Double Wick rotating Green-Schwarz strings

Via an appropriate field redefinition of the fermions, we find a set of conditions under which light cone gauge fixed world sheet theories of strings on two different backgrounds are related by a double Wick rotation. These conditions take the form of a set of transformation laws for the background fields, complementing a set of transformation laws for the metric and B field we found previously with a set for the dilaton and RR fields, and are compatible with the supergravity equations of motion. Our results prove that at least to second order in fermions, the AdS5 × S5 mirror model which plays an important role in the field of integrability in AdS/CFT, represents a string on ‘mirror AdS5 × S5’, the background that follows from our transformations. We discuss analogous solutions for AdS3 × S3 × T4 and AdS2 × S2 × T6. The main ingredient in our derivation is the light cone gauge fixed action for a string on an (almost) completely generic background, which we explicitly derive to second order in fermions.