  In this paper, we are going to construct the classical field theory on the
boundary of the embedding of $\mathbb{R} \times S^{1}$ into the manifold $M$ by
the Jacobi sigma model. By applying the poissonization procedure and by
generalizing the known method for Poisson sigma models, we express the fields
of the model as perturbative expansions in terms of the reduced phase space of
the boundary. We calculate these fields up to the second order and illustrate
the procedure for contact manifolds.
