The exact spectrum and mirror duality of the $(\text{AdS}_5{\times}S^5)_\eta$ superstring

We discuss the spectrum of a string propagating on $\eta$-deformed AdS$_5\times S^5$ by treating its worldsheet theory as an integrable quantum field theory. The exact $S$-matrix of this field theory is given by a $q$-deformation of the AdS$_5{\times}S^5$ worldsheet $S$-matrix with a real deformation parameter. By considering mirror (double Wick-rotated) versions of these worldsheet theories, we give the thermodynamic Bethe ansatz description of their exact finite-size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the string level, this seems to indicate that the light-cone worldsheet theories of strings on particular pairs of backgrounds are related by a double Wick rotation, a feature we call “mirror duality”. We provide a partial verification of these statements at the level of a sigma model by considering reduced actions and their corresponding (mirror) giant magnon solutions.