Large Charges on the Wilson Loop in $\mathcal{N}=4$ SYM: II. Quantum Fluctuations, OPE, and Spectral Curve

TL;DR

This paper analyzes quantum fluctuations around a classical string solution in the AdS/CFT correspondence, deriving Green's functions and spectral curve relations to compute corrections to correlation functions in large charge limits of $ ext{N}=4$ SYM.

Contribution

It introduces a novel method to compute $1/J$ corrections using Green's functions and spectral curves, linking integrability with holographic correlator calculations.

Findings

Derived a sum-over-residues representation of Green's functions.

Connected Green's function poles to spectral curve quantization conditions.

Expressed scaling dimensions and structure constants in terms of the spectral curve.

Abstract

We continue our study of large charge limits of the defect CFT defined by the half-BPS Wilson loop in planar $\mathcal{N}=4$ supersymmetric Yang-Mills theory. In this paper, we compute $1/J$ corrections to the correlation function of two heavy insertions of charge $J$ and two light insertions, in the double scaling limit where the charge $J$ and the 't Hooft coupling $\lambda$ are sent to infinity with the ratio $J/\sqrt{\lambda}$ fixed. Holographically, they correspond to quantum fluctuations around a classical string solution with large angular momentum, and can be computed by evaluating Green's functions on the worldsheet. We derive a representation of the Green's functions in terms of a sum over residues in the complexified Fourier space, and show that it gives rise to the conformal block expansion in the heavy-light channel. This allows us to extract the scaling dimensions and structure constants for an infinite tower of non-protected dCFT operators. We also find a close connection between our results and the semi-classical integrability of the string sigma model. The series of poles of the Green's functions in Fourier space corresponds to points on the spectral curve where the so-called quasi-momentum satisfies a quantization condition, and both the scaling dimensions and the structure constants in the heavy-light channel take simple forms when written in terms of the spectral curve. These observations suggest extensions of the results by Gromov, Schafer-Nameki and Vieira on the semiclassical energy of closed strings, and in particular hint at the possibility of determining the structure constants directly from the spectral curve.