A note on integrated correlators with a Wilson line in $\mathcal{N}=4$ SYM

TL;DR

This paper refines the analysis of integrated two-point functions involving a Wilson line in $ ext{N}=4$ SYM, providing a general solution to superconformal Ward identities and confirming results through localization and bootstrap methods.

Contribution

It introduces parity-odd terms into defect correlator parametrization and derives a simple measure for integrated correlators, aligning bootstrap and localization results.

Findings

Derived a general solution to superconformal Ward identities for defect correlators.

Provided a simple expression for the integration measure of Wilson line correlators.

Confirmed the consistency of bootstrap and localization predictions at strong coupling.

Abstract

We revisit the analysis of the integrated 2-point functions of local operators with a $\frac{1}{2}$-BPS Wilson line in $\mathcal{N}=4$ SYM. After including suitable parity-odd terms in the parametrization of the defect correlators, we are able to solve the superconformal Ward identities in terms of an unconstrained function of the cross-ratios. Exploiting this general solution, we obtain a simple expression of the integration measure for the integrated correlators with a Wilson line. We test our result by integrating the available bootstrap expression of the unintegrated correlator at strong coupling against the predictions of supersymmetric localization, finding perfect agreement.