Fathoming competing orders is one of the most fascinating subjects in modern condensed matter physics. Various order parameters such as spin and charge density waves, nematic order, and superconductivity appear in cuprates, pnictides, and heavy fermion systems. The Landau-Ginzburg-Wilson theory is generically not enough to explain such competing order physics because it does not incorporate quantum mechanical effects properly. Including the Berry phase term is one way to incorporate quantum mechanical effects, which is natural in non-linear sigma models with the Wess-Zumino-Witten term. We investigate its non-perturbative effects and apply them to competing order physics, especially in three spatial dimensions.