Integrability for the Full Spectrum of Planar AdS/CFT II
Nikolay Gromov, Vladimir Kazakov, Andrii Kozak, Pedro Vieira
TL;DR
The paper develops an exact, finite-volume framework for the full planar AdS/CFT spectrum by deriving a comprehensive set of non-linear integral TBA equations. These equations yield the Y-system conjectured for all operators, and are specialized to the sl(2) sector, with a focus on real, nicely behaved kernels and a mirror-dressing factor representation. The approach starts from the Beisert–Staudacher Bethe ansatz in mirror form, constructs bound-state densities across the fat-hook structure, and uses analytic continuation to access excited states, enabling potential numerical computation of anomalous dimensions at arbitrary coupling. Overall, the work provides a practical route to compute the full spectrum of planar N=4 SYM using integrability, strengthening the link between Y-systems and TBA in AdS/CFT.
Abstract
Using the thermodynamical Bethe ansatz method we derive an infinite set of integral non-linear equations for the spectrum of states/operators in AdS/CFT. The Y-system conjectured in arXiv:0901.3753 for the spectrum of all operators in planar N=4 SYM theory follows from these equations. In particular, we present the integral equations for the spectrum of all operators within the sl(2) sector. We prove that all the kernels and free terms entering these TBA equations are real and have nice fusion properties in the relevant mirror kinematics. We find the analogue of DHM formula for the dressing kernel in the mirror kinematics.