Strong coupling results from the numerical solution of the quantum spectral curve

TL;DR

This paper numerically solves the Quantum Spectral Curve equations for certain operators in the AdS/CFT correspondence, confirming known strong coupling results and predicting new coefficients with high precision.

Contribution

It provides a detailed numerical method for solving the QSC equations and extends the understanding of strong coupling behavior of spectral data in AdS/CFT.

Findings

Confirmed analytical coefficients for twist-2 operators at strong coupling.

Predicted additional coefficients for the Konishi operator.

Proposed polynomial index dependence of coefficients at strong coupling.

Abstract

In this paper, we solved numerically the Quantum Spectral Curve (QSC) equations corresponding to some twist-2 single trace operators with even spin from the $sl(2)$ sector of $AdS_5/CFT_4$ correspondence. We describe all technical details of the numerical method which are necessary to implement it in C++ language. In the $S=2,4,6,8$ cases, our numerical results confirm the analytical results, known in the literature for the first 4 coefficients of the strong coupling expansion for the anomalous dimensions of twist-2 operators. In the case of the Konishi operator, due to the high precision of the numerical data we could give numerical predictions to the values of two further coefficients, as well. The strong coupling behaviour of the coefficients $c_{a,n}$ in the power series representation of the ${\bf P}_{\!a}$-functions is also investigated. Based on our numerical data, in the regime, where the index of the coefficients is much smaller than $\lambda^{1/4}$, we conjecture that the coefficients have polynomial index dependence at strong coupling. This allows one to propose a strong coupling series representation for the ${\bf P}$-functions being valid far enough from the real short cut. In the paper the qualitative strong coupling behaviour of the ${\bf P}$-functions at the branch points is also discussed.