The quantum 1/2 BPS Wilson loop in ${\cal N}=4$ Chern-Simons-matter theories

TL;DR

This paper computes the three-loop expectation value of fermionic Wilson loops in 3D ${ m N}=4$ Chern-Simons-matter theories, showing the quantum correction lifts classical degeneracy and identifying the 1/2 BPS loop as their average.

Contribution

It provides the first three-loop calculation of fermionic Wilson loops in these theories and demonstrates the quantum uplift of classical degeneracy, supporting the average conjecture for the 1/2 BPS loop.

Findings

Quantum corrections lift classical degeneracy of Wilson loops.

The 1/2 BPS Wilson loop is the average of two fermionic operators.

Localization results include matter interaction-induced framing corrections.

Abstract

In three dimensional ${\cal N}=4$ Chern-Simons-matter theories two independent fermionic Wilson loop operators can be defined, which preserve half of the supersymmetry charges and are cohomologically equivalent at classical level. We compute their three-loop expectation value in a convenient color sector and prove that the degeneracy is uplifted by quantum corrections. We expand the matrix model prediction in the same regime and by comparison we conclude that the quantum 1/2 BPS Wilson loop is the average of the two operators. We provide an all-loop argument to support this claim at any order. As a by-product, we identify the localization result at three loops as a correction to the framing factor induced by matter interactions. Finally, we comment on the quantum properties of the non-1/2 BPS Wilson loop operator defined as the difference of the two fermionic ones.