Integrated correlators with a Wilson line in $\mathcal{N}=4$ SYM

TL;DR

This paper investigates integrated correlators involving a Wilson line in  SYM, deriving explicit measures and employing localization techniques to connect superconformal symmetry, matrix models, and recursion relations.

Contribution

It introduces a novel framework linking superconformal primaries, integrated correlators, and matrix model computations in  SYM with Wilson lines.

Findings

Derived explicit integration measure from superconformal Ward identities.

Reformulated matrix model computations using recursion relations and Bessel kernels.

Established a direct link between Wilson line correlators and localization in  SYM.

Abstract

In the context of integrated correlators in $\mathcal{N}=4$ SYM, we study the 2-point functions of local operators with a superconformal line defect. Starting from the mass-deformed $\mathcal{N}=2^*$ theory in presence of a $\frac{1}{2}$-BPS Wilson line, we exploit the residual superconformal symmetry after the defect insertion, and show that the massive deformation corresponds to integrated insertions of the superconformal primaries belonging to the stress tensor multiplet with a specific integration measure which is explicitly derived after enforcing the superconformal Ward identities. Finally, we show how the Wilson line integrated correlator can be computed by the $\mathcal{N}=2^*$ Wilson loop vacuum expectation value on a 4-sphere in terms of a matrix model using supersymmetric localization. In particular, we reformulate previous matrix model computations by making use of recursion relations and Bessel kernels, providing a direct link with more general localization computations in $\mathcal{N}=2$ theories.