Semiclassical origin of the O(N) model trans-series sectors

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.

A path-integral computation that reproduces at least the leading nonperturbative coefficients of the Bethe-ansatz transseries from identified semiclassical configurations or condensates.