This paper develops a new analytic method to compute small, exponentially suppressed corrections — called instantons — in ABJM theory. ABJM theory is a well‑studied three‑dimensional quantum field theory that can be reduced by supersymmetric localization to finite matrix integrals. The authors build a “bootstrap” framework that starts from that reduced form and targets the free energy and certain supersymmetric Wilson loop observables.

The authors work in the Fermi‑gas formulation of the ABJM matrix model. In that language the partition function is expressed through a grand potential J(µ,k) that depends on a chemical potential µ and the Chern–Simons level k. At large µ the grand potential splits into a perturbative part, compactly encoded by an Airy function, plus nonperturbative contributions that scale like exp[−(4 m k + 2 ℓ) µ] for integers m and ℓ. In the dual string and M‑theory pictures those terms correspond to different wrapped branes or strings, often called membrane and world‑sheet instantons.

Technically, the new work exploits exact functional relations and consistency conditions obeyed by grand‑canonical observables in the Fermi‑gas picture. The authors rewrite Wilson loop averages and the partition function in terms of a one‑particle density operator and a set of auxiliary coefficient functions fm,ℓ(µ) and wm,ℓ(µ) that encode the nonperturbative pieces. Requiring that different representations agree produces a closed system of functional constraints. Solving this system is the “bootstrap” step. Using it, the authors derive several relations for the free energy that were previously known only from conjectures or from high‑precision numerics. They also determine nonperturbative corrections to both 1/2 and 1/6 BPS (supersymmetric) Wilson loops and point out qualitative differences between them.