Circular Wilson loops in defect ${\cal N}$=4 SYM: phase transitions, double-scaling limits and OPE expansion

TL;DR

This paper investigates phase transitions and operator expansions of circular Wilson loops in a defect $ ext{N}=4$ SYM theory, revealing different behaviors for BPS and non-BPS cases at strong coupling and large flux.

Contribution

It provides a detailed analysis of Wilson loop phase transitions, double-scaling limits, and OPE expansions in a defect $ ext{N}=4$ SYM, highlighting differences between BPS and non-BPS configurations.

Findings

Non-BPS loops exhibit a Gross-Ooguri-like phase transition.

BPS loops lack cylindrical solutions, indicating saturation by supergravity modes.

The double-scaling limit is consistent for non-BPS loops but fails for BPS loops.

Abstract

We consider circular Wilson loops in a defect version of $\mathcal{N}=4$ super-Yang-Mills theory which is dual to the D3-D5 brane system with $k$ units of flux. When the loops are parallel to the defect, we can construct both BPS and non-BPS operators, depending on the orientation of the scalar couplings in the R-symmetry directions. At strong 't Hooft coupling we observe, in the non supersymmetric case, a Gross-Ooguri-like phase transition in the dual gravitational theory: the familiar disk solution dominates, as expected, when the operator is far from the defect while a cylindrical string worldsheet, connecting the boundary loop with the probe D5-brane, is favourite below a certain distance (or equivalently for large radii of the circles). In the BPS case, instead, the cylindrical solution does not exist for any choice of the physical parameters, suggesting that the exchange of light supergravity modes always saturate the expectation value at strong coupling. We study the double-scaling limit for large $k$ and large 't Hooft coupling, finding full consistency in the non-BPS case between the string solution and the one-loop perturbative result. Finally we discuss, in the BPS case, the failure of the double-scaling limit and the OPE expansion of the Wilson loop, finding consistency with the known results for the one-point functions of scalar composite operators.