  We study complex Chern-Simons theory on a Seifert manifold $M_3$ by embedding
it into string theory. We show that complex Chern-Simons theory on $M_3$ is
equivalent to a topologically twisted supersymmetric theory and its partition
function can be naturally regularized by turning on a mass parameter. We find
that the dimensional reduction of this theory to 2d gives the low energy
dynamics of vortices in four-dimensional gauge theory, the fact apparently
overlooked in the vortex literature. We also generalize the relations between
1) the Verlinde algebra, 2) quantum cohomology of the Grassmannian, 3)
Chern-Simons theory on $\Sigma\times S^1$ and 4) index of a spin$^c$ Dirac
operator on the moduli space of flat connections to a new set of relations
between 1) the "equivariant Verlinde algebra" for a complex group, 2) the
equivariant quantum K-theory of the vortex moduli space, 3) complex
Chern-Simons theory on $\Sigma \times S^1$ and 4) the equivariant index of a
spin$^c$ Dirac operator on the moduli space of Higgs bundles.
