Semiclassical origin of the O(N) model trans-series sectors
In plain words
For solvable two-dimensional theories, the full set of exponentially small corrections to the divergent series has been extracted from the exact solution. Which field configurations of the theory produce each of these corrections is unknown.
Precise statement
For the 2D $O(N)$ nonlinear sigma model with a chemical potential $h$ coupled to a conserved charge, the free energy $F(h)$ follows exactly from the Bethe ansatz, and a full analytic transseries for observables of this type in integrable field theories was derived by Wiener-Hopf methods with median resummation reproducing the exact value (Bajnok, Balog, Vona 2022). For each nonperturbative sector $\exp(-k/\alpha)$ of the $O(N)$ transseries, identify the path-integral saddles, fractional instantons or condensates that produce it, and compute the corresponding coefficients directly from the path integral.
What would settle it
A path-integral computation that reproduces at least the leading nonperturbative coefficients of the Bethe-ansatz transseries from identified semiclassical configurations or condensates.
Status in the literature
Unverified note
The full transseries was obtained from the Bethe ansatz (Bajnok, Balog, Vona 2022, arXiv:2212.09416) and extended to all conserved charges in 2025; its semiclassical interpretation remains open.