  This thesis studies the algebro-geometric aspects of supersymmetric abelian
gauge theories in three dimensions. The supersymmetric vacua are demonstrated
to exhibit a window phenomenon in Chern-Simons levels, which is analogous to
the window phenomenon in quantum K-theory with level structures. This
correspondence between three-dimensional gauge theories and quantum K-theory is
investigated from the perspectives of semi-classical vacua, twisted chiral
rings, and twisted indices. In particular, the twisted index admits an
algebro-geometric interpretation as the supersymmetric index of an effective
quantum mechanics. Via supersymmetric localisation, the contributions from both
topological and vortex saddle points are shown to agree with the Jeffrey-Kirwan
contour integral formula. The algebro-geometric construction of Chern-Simons
contributions to the twisted index from determinant line bundles provides a
natural connection with quantum K-theory.
