Quantum Spectral Curve for the eta-deformed AdS_5xS^5 superstring

Abstract

The spectral problem for the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring and its dual planar maximally supersymmetric Yang-Mills 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 $\eta$-deformed ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring, an integrable deformation of the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the ${\mathrm{S}}^2$ sigma models for instance. We derive the quantum spectral curve for the $\eta$-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 $\mathbf{P}\mu$ system -- 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 $\eta$-deformed string and its quantum spectral curve interpolate between the ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ superstring and a superstring on "mirror" ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$, reflecting a more general relationship between the spectral and thermodynamic data of the $\eta$-deformed string. In particular, the spectral problem of the mirror ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ string, and the thermodynamics of the undeformed ${\mathrm{AdS}}_5\times {\mathrm{S}}^5$ string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.

arXiv comment32+37 pages; 6 figures. v2: added references

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