Wilson loops in ABJM theory reloaded

We present a new technique for computing supersymmetric Wilson loops in the ABJM theory via supersymmetric localization, valid for arbitrary values of the rank of the gauge group $N$ and the Chern-Simons level $k$. The approach relies on an operator representation of the Wilson loops within the Fermi gas formalism in terms of the resolvent of a certain integral operator previously encountered in the computation of the ABJM partition function on the round three-sphere. By deriving a set of nontrivial relations for this resolvent, we obtain exact expressions for the generating functions of Wilson loops in terms of the partition function. For large $k$, these expressions reproduce the weak-coupling expansion of the Wilson loops, and in the large-$N$ limit at fixed $k$ they match previously obtained high-precision numerical results. This analysis also resolves the longstanding discrepancy between numerical data and the semiclassical expression for the $1/6$ BPS Wilson loop.

💡 Research Summary

The paper introduces a novel, exact method for computing supersymmetric Wilson loops in the three‑dimensional N = 6 ABJM theory, overcoming long‑standing discrepancies between numerical data and the semiclassical (WKB) predictions for the 1/6‑BPS Wilson loops. The authors work within the Fermi‑gas formulation of the ABJM matrix model, where the partition function on the three‑sphere is expressed as a Fredholm determinant Ξ(z,k)=det(1+z ρ) of an integral operator ρ with kernel ρ(x,y)=1/(8πk cosh(x/2) cosh((x−y)/2k)). The same operator appears in the expectation value of Wilson loops.

By defining the holonomy operator U = e^{(x+p)/k} (with canonical commutation