Framing and localization in Chern-Simons theories with matter

TL;DR

This paper investigates the subtle framing effects in three-dimensional supersymmetric Chern-Simons theories with matter, specifically ABJ(M), revealing how framing influences Wilson loop calculations and their relation to localization and physical observables.

Contribution

It demonstrates that framing effects in ABJ(M) are complex, coupling-dependent, and can include contributions from vertex diagrams, advancing understanding of localization results in these theories.

Findings

Reproduces three-loop Wilson loop results matching localization predictions.

Identifies framing as a non-trivial, coupling-dependent phase in ABJ(M).

Links framing effects to the Bremsstrahlung function of the 1/2-BPS cusp.

Abstract

Supersymmetric localization provides exact results that should match QFT computations in some regularization scheme. The agreement is particularly subtle in three dimensions where complex answers from localization procedure sometimes arise. We investigate this problem by studying the expectation value of the 1/6 BPS Wilson loop in planar ABJ(M) theory at three loops in perturbation theory. We reproduce the corresponding term in the localization result and argue that it originates entirely from a non-trivial framing of the circular contour. Contrary to pure Chern-Simons theory, we point out that for ABJ(M) the framing phase is a non-trivial function of the couplings and that it potentially receives contributions from vertex-like diagrams. Finally, we briefly discuss the intimate link between the exact framing factor and the Bremsstrahlung function of the 1/2-BPS cusp.