| Summary: | The spectral problem for the AdS5ÃS5 superstring and its dual planar maximally supersymmetric YangâMills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the η-deformed AdS5ÃS5 superstring, an integrable deformation of the AdS5ÃS5 superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the AdS5ÃS5 superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the S2 sigma models for instance. We derive the quantum spectral curve for the η-deformed string by reformulating the corresponding ground-state thermodynamic Bethe ansatz equations as an analytic Y system, and map this to an analytic T system which upon suitable gauge fixing leads to a Pμ system â the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the η-deformed string and its quantum spectral curve interpolate between the AdS5ÃS5 superstring and a superstring on âmirrorâ AdS5ÃS5, reflecting a more general relationship between the spectral and thermodynamic data of the η-deformed string. In particular, the spectral problem of the mirror AdS5ÃS5 string, and the thermodynamics of the undeformed AdS5ÃS5 string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.
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