Wilson loops in ABJM theory reloaded

TL;DR

The work develops an exact, nonperturbative framework to compute ABJM Wilson loops by merging supersymmetric localization with a Tracy–Widom-based resolvent formalism. By recasting the ABJM matrix model as a Fermi gas and exploiting a ψ-function with Baxter-type relations, the authors derive closed expressions for Wilson-loop generating functions in terms of the partition function, valid for arbitrary N and k. The approach resolves prior discrepancies between numerical data and semiclassical predictions, reproduces weak-coupling expansions at large k, and matches high-precision large-N numerics at fixed k, while clarifying instanton structure and nonperturbative corrections. The results pave the way for broader applications, including latitude and deformed Wilson loops in ABJM theory and potential Airy-function representations in more general settings.

Abstract

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.

Figures (1)

• Figure 1: The function $\psi(x|z)$ is analytic in the complex $x$-plane within the strip $-\pi k < \mathop{\rm Im}\nolimits x < \pi k$, whose boundaries are indicated by dashed lines. The blue and red lines denote the integration contours appearing in relations (\ref{['psi-ell']}) and (\ref{['psi-ell1']}), respectively. In going from relation (\ref{['psi-ell1']}) to (\ref{['sh-1']}), the red contour is deformed so as to coincide with the blue one, thereby enclosing the poles indicated by blue crosses, whose residues are then picked up.