  We study various aspects of N=(2,2) supersymmetric non-Abelian gauge theories
in two dimensions, with applications to string vacua. We compute the Witten
index of SU(k) SQCD with N>0 flavors with twisted masses; the result is
presented as the solution to a simple combinatoric problem. We further claim
that the infra-red fixed point of SU(k) gauge theory with N massless flavors is
non-singular if (k,N) passes a related combinatoric criterion. These results
are applied to the study of a class of U(k) linear sigma models which, in one
phase, reduce to sigma models on Calabi-Yau manifolds in Grassmannians. We show
that there are multiple singularities in the middle of the one-dimensional
Kahler moduli space, in contrast to the Abelian models. This result precisely
matches the complex structure singularities of the proposed mirrors. In one
specific example, we study the physics in the other phase of the Kahler moduli
space and find that it reduces to a sigma model for a second Calabi-Yau
manifold which is not birationally equivalent to the first. This proves a
mathematical conjecture of Rodland.
