Mirror symmetry in 2+1 and 1+1 dimensions

We study the Coulomb-Higgs duality of N = 2 supersymmetric abelian Chern-Simons theories in 2 + 1 dimensions, by compactifying dual pairs on a circle of radius R and comparing the resulting N = (2;2) theories in 1 + 1 dimensions. Below the compactification scale, the theory on the Higgs branch reduces to the non-linear sigma model on a toric manifold. In the dual theory on the Coulomb branch, the Kaluza-Klein modes generate an infinite tower of contributions to the superpotential. After resummation, in the limit R --> 0 the superpotential becomes that of the Landau-Ginzburg model which is the two-dimensional mirror of the toric sigma model. We further examine the conjecture of all-scale three-dimensional mirror symmetry and observe that it is consistent with mirror symmetry in 1 + 1 dimensions.