Framing fermionic Wilson loops in ABJ(M)

TL;DR

This paper computes the perturbative expectation value of the 1/2 BPS fermionic Wilson loop in ABJ(M) theory at generic framing, confirming localization predictions and extending to multiply wound loops.

Contribution

It provides the first direct perturbative verification of localization for the fermionic Wilson loop at framing one in ABJ(M) theory.

Findings

Exact match with matrix model predictions at framing 1

Extension of results to multiply wound Wilson loops

First perturbative check of localization for fermionic loops

Abstract

Framing plays a central role in the evaluation of Wilson loops in theories with Chern-Simons actions. In pure Chern-Simons theory, it guarantees topological invariance, while in theories with matter like ABJ(M), our theory of interest, it is essential to enforce the cohomological equivalence of different BPS Wilson loops. This is the case for the 1/6 BPS bosonic and the 1/2 BPS fermionic Wilson loops, which have the same expectation value when computed as matrix model averages from localization. This equivalence holds at framing $\mathfrak{f}=1$, which has so far been a challenge to implement in perturbative evaluations. In this paper, we compute the expectation value of the 1/2 BPS fermionic circle of ABJ(M) theory up to two loops in perturbation theory at generic framing. This is achieved by a careful analysis of fermionic Feynman diagrams, isolating their framing dependent contributions and evaluating them in point-splitting regularization using framed contours. Specializing our result to $\mathfrak{f}=1$ we recover exactly the matrix model prediction, thus realizing for the first time a direct perturbative check of localization for this operator. We also generalize our computation to the case of a multiply wound circle, again matching the corresponding matrix model prediction.