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

TL;DR

The paper introduces and develops the quantum spectral curve (QSC) as a concise, finite set of Riemann-Hilbert equations for the exact planar spectrum of N=4 SYM. It reframes the problem in terms of a rich Q-system of Baxter-like Q-functions, organized by Plücker QQ-relations, and shows complete compatibility with analytic Y-system/TBA and FiNLIE formulations. By deriving both the Pμ- and Qω- presentations and their large-volume and quasi-classical limits, it unifies finite-size spectral data with classical finite-gap and asymptotic Bethe ansatz results, while enabling a universal description for all local single-trace operators. The construction clarifies how global charges fix asymptotics, how exact Bethe equations emerge, and how the framework extends to ABJM-type theories, marking a broad, powerful toolkit for exact spectral problems in integrable AdS/CFT systems.

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.

Figures (15)

• Figure 1: Riemann sheets of the function $f$: the notation $f$ denotes the function $\hat{f}$ (left) which has infinite ladders of cuts except on the upper half plane of its main Riemann sheet. It coincides with $\check f$ (right) on this upper half plane. The red arrow indicates a path to define the tilde transformation.
• Figure 2: The AdS$_5$/CFT$_4$ T-hook: the domain for the variables $a$ and $s$ in the Y-system (circles) and the Hirota equation (all nodes of the underlying grid).
• Figure 3: Cuts structure of the ${\bf P}\mu$ and the ${\bf Q}\omega$-system: the functions ${\bf P}_a$ (resp ${\bf Q}_j$) are analytic except on a short (resp long) Zhukovsky cut on the real axis. By contrast, $\mu_{ab}$ has infinite ladder of cuts and is $i$-periodic in the mirror kinematics (hence it obeys $\tilde{\mu}_{ab}=\mu_{a,b}^{[+2]}$ in the physical kinematics). Similarly, $\omega_{jk}$ is periodic in the physical kinematics.
• Figure 4: A projection of the Hasse diagram (left), where all Q-functions having the same grading (number of bosonic and fermionic indices) are identified. A more precise picture (right) of some small portions of this diagram illustrates the "facets" (red) corresponding to the QQ-relations $Q_{13|\emptyset}Q_{1234|\emptyset}=Q_{123|\emptyset}^+Q_{134|\emptyset}^--Q_{123|\emptyset}^-Q_{134|\emptyset}^+$ and $Q_{1|\emptyset}Q_{\emptyset|2}=Q_{1|2}^+Q_{\emptyset}^--Q_{1|2}^-Q_{\emptyset}^+$.
• Figure 5: UHPA and LHPA ${\mathcal{Q}}$-systems: in their main Riemann sheet in the physical kinematics, the functions ${\bf P}_a$ are analytic except on a short Zhukovsky cut on the real axis. There are two ways to define a mirror-Q-system from these functions: one option is to identify them with Q-functions on the upper half plane (for instance ${\mathcal{Q}}_{a|\emptyset}={\bf P}_a$ on the upper half plane, hence ${\mathcal{Q}}_{a|\emptyset}=\tilde{{\bf P}}_a$ on the lower half plane). The other option is to identify them on the lower half plane (for instance ${\mathcal{Q}}_{a}|^{\emptyset}={\bf P}_a$ on the lower half plane). The first option defines the UHPA ${\mathcal{Q}}$-system, while the second one defines the LHPA ${\mathcal{Q}}$-system.

Theorems & Definitions (1)

• proof