${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ Mirror Model as a String Sigma Model

Doing a double Wick rotation in the world sheet theory of the light cone ${\mathrm{AdS}}_5\times {\text{S}}^5$ superstring results in an inequivalent, so-called mirror theory that plays a central role in the field of integrability in the AdS-CFT correspondence. We show that this mirror theory can be interpreted as the light cone theory of a free string on a different background. This background is related to ${\mathrm{dS}}_5\times {\mathrm{H}}^5$ by a double $T$-duality, and has hidden supersymmetry. The geometry can also be extracted from an integrable deformation of the ${\mathrm{AdS}}_5\times {\text{S}}^5$ sigma model, and we prove the observed mirror duality of these deformed models at the bosonic level as a byproduct. While we focus on ${\mathrm{AdS}}_5\times {\text{S}}^5$, our results apply more generally.