  In this paper, we give a microscopic derivation of Gibbs measures for the
focusing cubic nonlinear Schr\"odinger equation on the one-dimensional torus
from many-body quantum Gibbs states. Since we are not making any positivity
assumptions on the interaction, it is necessary to introduce a truncation of
the mass in the classical setting and of the rescaled particle number in the
quantum setting. Our methods are based on a perturbative expansion of the
interaction, similarly as in previous work of Fr\"ohlich, Knowles, Schlein, and
the second author. Due to the presence of the truncation, the obtained series
have infinite radius of convergence. We treat the case of bounded, integrable,
and delta function interaction potentials, without any sign assumptions. Within
this framework, we also study time-dependent correlation functions. This is the
first such known result in the focusing regime.
