The full Quantum Spectral Curve for $AdS_4/CFT_3$

TL;DR

This work provides the full Quantum Spectral Curve formulation for the ABJM/$AdS_4/CFT_3$ system, embedding the QSC into a novel $OSp(4|6)$ Q‑system that yields exact Bethe equations. By constructing a rich set of ${\bf P}_A$, ${\bf Q}_I$, $Q_{a|i}$, and tau functions, the authors reveal the hidden $SO(3,2)$ and $SO(6)$ structure, derive gluing conditions that quantize spin, and connect finite‑coupling equations to the familiar Asymptotic Bethe Ansatz in the large volume limit. They also clarify the analytic properties across multiple Riemann sheets, clarify the role of the phase $\mathcal{P}$, and discuss the classical limit and unitarity constraints within this framework. The results enable quantitative spectrum computations at finite coupling and point toward generalizations to other AdS/CFT pairs, potentially aiding extensions to $AdS_3/CFT_2$ and related dualities.

Abstract

The spectrum of planar N=6 superconformal Chern-Simons theory, dual to type IIA superstring theory on $AdS_4 \times CP^3$, is accessible at finite coupling using integrability. Starting from the results of [arXiv:1403.1859], we study in depth the basic integrability structure underlying the spectral problem, the Quantum Spectral Curve. The new results presented in this paper open the way to the quantitative study of the spectrum for arbitrary operators at finite coupling. Besides, we show that the Quantum Spectral Curve is embedded into a novel kind of Q-system, which reflects the OSp(4|6) symmetry of the theory and leads to exact Bethe Ansatz equations. The discovery of this algebraic structure, more intricate than the one appearing in the $AdS_5/CFT_4$ case, could be a first step towards the extension of the method to $AdS_3/CFT_2$.

Figures (7)

• Figure 1: Cut structure of the ${\bf P}_A$ functions, with a single cut on the first sheet. We denote with $\widetilde{{\bf P}}_A$ the analytic continuation to the next sheet, through the cut on the real axis.
• Figure 2: The quasi-periodicity property of $\nu_a$ functions on a sheet with long cuts corresponds to $\nu_a(u+i)= e^{-i \mathcal{P} } \, \widetilde{\nu}_a(u)$ on the defining sheet with short cuts.
• Figure 3: Cut structure of the ${\bf Q}$ functions in the physical Riemann section. On the first (second) sheet, ${\bf Q}$ is analytic in the upper (lower) half plane.
• Figure 4: Gluing the two analyticity regions from the sheets $1$ and $2$ of Figure \ref{['fig:cutQ']}, one defines the mirror sheet, with a single long cut.
• Figure 5: Chain of Q functions corresponding to the $\eta=+1$ grading of the Bethe Ansatz.