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

TL;DR

This paper derives a quantum spectral curve (QSC) framework for exactly determining the spectrum of all local operators in planar N=4 SYM theory, unifying and extending previous approaches with a new algebraic structure.

Contribution

It introduces a Q-system-based formulation of the QSC that applies to all local single trace operators, generalizing prior state-specific methods.

Findings

QSC provides exact spectral equations for all operators.

Q-system reveals algebraic and analytic structure of the spectrum.

Classical solutions and asymptotic Bethe ansatz are derived from the formalism.

Abstract

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