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.
Highlights
The discovery and exploration of integrability in the planar AdS/CFT correspondence has a long and largely successful history [1]
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
The computations were efficiently done in various limits and numerically, with sufficiently high precision [3,4,5], by means of an explicit but immensely complicated Thermodynamic Bethe Ansatz (TBA) formalism [6,7,8]
Summary
The discovery and exploration of integrability in the planar AdS/CFT correspondence has a long and largely successful history [1]. To a great extent this truth was unveiled in [15] where we proposed a simple finite set of non-linear Riemann-Hilbert equations which we called the quantum spectral curve (QSC) of the AdS/CFT correspondence Due to their simplicity, the equations have found numerous applications in the practical calculations: they were successfully applied to the analysis of weak coupling expansion in the sl(2) sector (Konishi up to 9 loops!) [16, 17] as well as at strong coupling [15], for the slope and curvature functions for twist-2 operators at any coupling and pomeron intercept at strong coupling [18].
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