The Quantum Spectral Curve of the ABJM theory

TL;DR

The paper derives a finite ${\mathbf P}\mu$-system for the ABJM theory, recasting the spectral problem into a nonlinear Riemann-Hilbert framework akin to the Quantum Spectral Curve for $\mathcal{N}=4$ SYM. By exploiting Y-/T-system structure and two gauges, the authors express the spectrum in terms of ${\bf P}_i$ and $\mu_{ab}$ with explicit analytic properties, and reveal a formal mapping to the $\mathcal{N}=4$ SYM QSC in the symmetric sector, while highlighting a fundamental difference in analytic structure (single-cut vs. $i$-periodic). A weak-coupling check reproduces the two-loop Baxter equation, providing a nontrivial validation and illustrating how $\Delta$ is obtained from the nonlinear equations for generic coupling through the $h(\lambda)$ function. The construction lays groundwork for nonperturbative studies, potential numerical implementations, and insights into AdS$_4$/CFT$_3$ integrability and possible $q$-deformations, with the aim of fixing the $h(\lambda)$ relation and exploring cusp/Wilson-line observables. Overall, the work proposes a powerful, finite-strength reformulation of ABJM integrability that parallels the $\mathcal{N}=4$ case and broadens the toolkit for exact spectral computations in AdS/CFT contexts.

Abstract

Recently, it was shown that the spectrum of anomalous dimensions and other important observables in N = 4 SYM are encoded into a simple nonlinear Riemann-Hilbert problem: the Pμ-system or Quantum Spectral Curve. In this letter we present the Pμ-system for the spectrum of the ABJM theory. This may be an important step towards the exact determination of the interpolating function h(λ) characterising the integrability of the ABJM model. We also discuss a surprising symmetry between the Pμ-system equations for N = 4 SYM and ABJM.

Figures (2)

• Figure 1: T-hook for the ABJM T-system
• Figure 2: Analytic structure for the two types of variables in the Quantum Spectral Curve.