Large charges on the Wilson loop in $$ \mathcal{N} $$ = 4 SYM. Part II. Quantum fluctuations, OPE, and spectral curve

Abstract

<jats:title>A<jats:sc>bstract</jats:sc> </jats:title><jats:p>We continue our study of large charge limits of the defect CFT defined by the half-BPS Wilson loop in planar <jats:inline-formula><jats:alternatives><jats:tex-math>$$ \mathcal{N} $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> = 4 supersymmetric Yang-Mills theory. In this paper, we compute 1<jats:italic>/J</jats:italic> corrections to the correlation function of two heavy insertions of charge <jats:italic>J</jats:italic> and two light insertions, in the double scaling limit where the charge <jats:italic>J</jats:italic> and the ’t Hooft coupling <jats:italic>λ</jats:italic> are sent to infinity with the ratio <jats:italic>J/</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$ \sqrt{\lambda } $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msqrt> <mml:mi>λ</mml:mi> </mml:msqrt> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>