2D O(3) sigma model with a theta-term: construction of a positive Boltzmann weight

The continuum limit of the dual formulation of the two-dimensional lattice SU(N) principal chiral model is constructed. The continuum action is in general complex and appears to be a functional of an (N^2−1)-component non-compact scalar field. As an application of this construction we establish a relation between the dual of the SU(2) principal chiral model and the O(3) non-linear sigma model with a theta-term in the continuum limit. This relation is exact when the radial part of the scalar dual field is taken to be a constant. Therefore, the dual formulation of the lattice SU(2) model with constant radial part can be regarded as a non-perturbative regularization of the O(3) model with theta term. Furthermore, under certain conditions one could construct a positive definite dual Boltzmann weight. This property enables us to prospect Monte Carlo simulations of O(3) with a theta term at real values of theta.