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We give a derivation of quantum spectral curve (QSC) - a finite set ofRiemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposedin our recent paper Phys.Rev.Lett. 112 (2014). We also generalize thisconstruction to all local single trace operators of the theory, in contrast tothe TBA-like approaches worked out only for a limited class of states. Wereveal a rich algebraic and analytic structure of the QSC in terms of a socalled Q-system -- a finite set of Baxter-like Q-functions. This new point ofview on the finite size spectral problem is shown to be completely compatible,though in a far from trivial way, with already known exact equations (analyticY-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system todemonstrate how the classical finite gap solutions and the asymptotic Betheansatz emerge from our formalism in appropriate limits.

Quantum spectral curve for arbitrary state/operator in AdS$_5$/CFT$_4$