Fast QSC Solver: tool for systematic study of N=4 Super-Yang-Mills spectrum

TL;DR

This paper introduces a fast, efficient algorithm and C++ implementation for solving the Quantum Spectral Curve in N=4 Super-Yang-Mills theory, enabling systematic study of a large number of states across various couplings.

Contribution

The authors developed a new, high-performance algorithm and C++ code for the QSC, allowing rapid analysis of many excited states in N=4 SYM, surpassing previous computational limitations.

Findings

Successfully studied 219 states with Δ₀ ≤ 6 across couplings.

Achieved approximately 8-fold efficiency improvement over previous implementations.

Enabled smooth integration of perturbative data with numerical solutions.

Abstract

Integrability methods give us access to a number of observables in the planar N=4 SYM. Among them, the Quantum Spectral Curve (QSC) governs the spectrum of anomalous dimensions. Low lying states were successfully studied in the past using the QSC. However, with the increased demand for a systematic study of a large number of states for various applications, there is a clear need for a fast QSC solver which can easily access a large number of excited states. Here, we fill this gap by developing a new algorithm and applied it to study all 219 states with the bare dimension $\Delta_0 \leq 6$ in a wide range of couplings. The new algorithm has an improved performance at weak coupling and allows to glue numerics smoothly the available perturbative data, resolving the previous obstruction. Further ~ 8-fold efficiency gain comes from C++ implementation over the best available Mathematica implementation. We have made the code and the data to be available via a GitHub repository. The method is generalisable for non-local observables as well as for other theories such as deformations of N=4 SYM and ABJM. It may find applications in the separation of variables and bootstrability approaches to the correlation functions. Some applications to correlators at strong coupling are also presented.