New Approach to Strongly Coupled N = 4 SYM via Integrability

TL;DR

This paper introduces a new numerical framework using a novel set of Quantum Spectral Curve variables that enables stable calculations at very strong coupling in N=4 SYM, leading to new analytic results and insights into operator mixing.

Contribution

The authors develop a new set of QSC variables with a regular strong coupling expansion, enabling stable numerical solutions at high coupling and deriving new analytic results in N=4 SYM.

Findings

Numerical algorithm remains stable at coupling g ~ 1000000.

Derived new analytic predictions for conformal dimensions.

Uncovered operator mixing phenomena outside the sl(2) sector.

Abstract

Finding a systematic expansion of the spectrum of free superstrings on AdS${}_5\times $S${}^5$, or equivalently strongly coupled N = 4 SYM in the planar limit, remains an outstanding challenge. No first principle string theory methods are readily available, instead the sole tool at our disposal is the integrability-based Quantum Spectral Curve (QSC). For example, through the QSC the first five orders in the strong coupling expansion of the conformal dimension of an infinite family of short operators have been obtained. However, when using the QSC at strong coupling one must often rely on numerics, and the existing methods for solving the QSC rapidly lose precision as we approach the strong coupling regime. In this paper, we introduce a new framework that utilises a novel set of QSC variables with a regular strong coupling expansion. We demonstrate how to use this approach to construct a new numerical algorithm that remains stable even at a 't Hooft coupling as large as $10^6$ (or g ~ 100). Employing this approach, we derive new analytic results for some states in the sl(2) sector and beyond. We present a new analytic prediction for a coefficient in the strong coupling expansion of the conformal dimension for the lowest trajectory at a given twist L. For non-lowest trajectories, we uncover a novel feature of mixing with operators outside the sl(2) sector, which manifests as a new type of analytic dependence on the twist.