In this talk I will introduce a natural elliptic equation (the
vortex equation) in two dimensional gauged linear sigma-model. It
generalizes the Cauchy-Riemann equation in Gromov-Witten theory, and the
Hermitian-Einstein equation in the theory of vector bundles. I will also
discuss the "large-area limit" of the vortex equation and its relation with
the nonlinear sigma-model. Almost everything will be discussed in the
context of line bundles over Riemann surfaces.