  A correspondence is established between the dynamics of the two-vortex system
and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super-
(chiral) and critical phases. As a result, a trivial permutation symmetry of
the point vortices induces a weak-strong coupling duality in the NCLP. We show
that quantum two-vortex systems with non-zero total vorticity can be generated
by applying conformal bridge transformation to a two-dimensional quantum free
particle or to a quantum vortex-antivortex system of zero total vorticity. The
sub- and super-critical phases of the quantum NCLP are generated in a similar
way from the 2D quantum free particle in a commutative or non-commutative
plane. The composition of the inverse and direct transformations of the
conformal bridge also makes it possible to link the non-chiral and chiral
phases in each of these two systems.
