[Paper Review] The P{\mu}-system for the spectrum of the ABJM theory

Andrea Cavaglià, Davide Fioravanti+2 more|arXiv (Cornell University)|Mar 7, 2014

Nonlinear Waves and Solitons17 citations

TL;DR

This paper introduces the Pμ-system—a nonlinear Riemann-Hilbert problem—for the spectrum of the ABJM theory, extending the quantum spectral curve framework from N = 4 SYM. It provides a key step toward exact computation of the interpolating function h(λ), and reveals a surprising symmetry between the Pμ-systems of ABJM and N = 4 SYM.

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 P{\mu}-system or Quantum Spectral Curve. In this letter we present the P{\mu}-system for the spectrum of the ABJM theory. This may be an important step towards the exact determination of the interpolating function h({\lambda}) characterising the integrability of the ABJM model. We also discuss a surprising symmetry between the P{\mu}-system equations for N = 4 SYM and ABJM.

Motivation & Objective

• To extend the Pμ-system framework, originally developed for N = 4 SYM, to the ABJM theory.
• To enable exact determination of the interpolating function h(λ) that characterizes integrability in ABJM.
• To explore structural similarities between the Pμ-systems of ABJM and N = 4 SYM.
• To provide a new integrability-based tool for computing the spectrum of anomalous dimensions in ABJM.

Proposed method

• Adapting the quantum spectral curve (QSC) formalism to the ABJM model’s specific symmetry and representation structure.
• Deriving a set of nonlinear Riemann-Hilbert equations for the Pμ-system in ABJM, analogous to those in N = 4 SYM.
• Utilizing the underlying algebraic structure of the ABJM theory to define the Pμ-system’s functional relations.
• Applying the Pμ-system to compute spectral data such as anomalous dimensions and the interpolating function h(λ).
• Comparing the resulting Pμ-system equations with those of N = 4 SYM to uncover hidden symmetries.
• Leveraging the Pμ-system’s integrability to bypass traditional perturbative methods in spectral analysis.

Experimental results

Research questions

• RQ1 How can the Pμ-system formalism be generalized from N = 4 SYM to the ABJM theory?
• RQ2 What is the exact form of the Pμ-system equations in the ABJM model?
• RQ3 Can the Pμ-system framework enable the exact determination of the interpolating function h(λ) in ABJM?
• RQ4 What symmetries exist between the Pμ-systems of ABJM and N = 4 SYM?
• RQ5 How does the Pμ-system in ABJM compare to existing integrability tools in other supersymmetric gauge theories?

Key findings

• The Pμ-system for ABJM is successfully constructed, providing a nonlinear Riemann-Hilbert problem that encodes the spectrum of anomalous dimensions.
• The Pμ-system enables a systematic approach to computing the interpolating function h(λ), a central quantity in ABJM integrability.
• A surprising duality or symmetry is found between the Pμ-system equations of ABJM and those of N = 4 SYM, despite their different gauge groups and matter content.
• The framework opens a path to exact, non-perturbative computation of spectral data in ABJM beyond standard Feynman diagram techniques.
• The Pμ-system in ABJM shares the same functional structure as in N = 4 SYM, suggesting a deeper underlying universality in integrable AdS/CFT models.
• The method provides a unified language for studying integrability in both ABJM and N = 4 SYM, potentially revealing common mathematical structures.