# Charge-Vortex Duality for Bosons

Consider a (2+1)d continuum field theory with the Minkowski action $$\mathcal{L}=|\partial\phi|^2-r|\phi|^2-u|\phi|^4,$$ where $$\phi$$ is a complex field. The theory undergoes a quantum phase transition by tunning $$r$$ (this theory can be regarded as the effective description of Bose Hubbard model in the vicinity of the critical point).

In the dual picture, we have $$\mathcal{L}_d=|(\partial-a)\phi_v|^2-r_v|\phi_v|^2-u_v|\phi_v|^4+f^{\mu\nu}f_{\mu\nu}$$ where $$\phi_v$$ corresponds to the vortex field and is also complex valued. The gauge field is related with the orignal theory by $$j_\mu\sim \epsilon_{\mu\nu\lambda}\partial_\nu a_\lambda$$ where $$j_\mu$$ is the 3-current associated with the global $$U(1)$$ symmetry of the original Lagrangian.

The superfluid phase of the original theory corresponds to the gapped phase of $$\phi_v$$ in the dual theory and the superfluid mode is identified with the gapless mode: photon in the dual theory. The insulating phase of the original theory corresponds to the symmetry breaking phase of $$\phi_v$$ in the dual theory where vortices have condensed. By Anderson-Higgs mechanism, the gauge field will gain a mass.

My questions are:

1. When we say vortices condense, are we referring to 'vortex condensation' or 'antivortex condensation' or other cases? It seems there is no preference for vortex or antivortex.

2. In the insulating phase (vortices condensed in the dual theory), will the vortex superfluid if we inject a vortex into the system?