Infrared properties of two-dimensional $\mathrm{SU}(N)/H$ nonlinear $\sigma$ models at nonzero $\theta$ angles

A general strategy is proposed to explore the low-energy properties of two-dimensional nonlinear $\sigma$ models with $\theta$ terms. We demonstrate its application to nonlinear $\sigma$ models with the target space \text{SU(N)}/H, which include $\mathbb{C}P^{N-1}$, complex Grassmannian manifolds as well as the flag \text{SU(N)}/\text{U(1)}^{N-1} and \text{SU(N)})/\text{SO(N)} manifolds. By analyzing the symmetry and its anomaly content, we realize these nonlinear $\sigma$ models through perturbations added to the SU(N)$_1$ conformal field theory. For the flag-manifold \text{SU(N)}/\text{U(1)}^{N-1} and \text{SU(N)})/\text{SO(N)} models, those perturbations are shown to correspond to the marginal current-current operator with the specific sign which leads to a massless renormalization group flow to the SU(N)$_1$ fixed point. In contrast, a massive regime with a two-fold ground-state degeneracy is found for the $\mathbb{C}P^{N-1}$ ($N >2$) and Grassmannian nonlinear $\sigma$ models at $\theta=\pi$.