  We construct two-dimensional ${\cal N} = (2, 2)$ supersymmetric gauge
theories on a Euclidean spacetime lattice with matter in the two-index
symmetric and anti-symmetric representations of SU($N_c$) color group. These
lattice theories preserve a subset of the supercharges exact at finite lattice
spacing. The method of topological twisting is used to construct such theories
in the continuum and then the geometric discretization scheme is used to
formulate them on the lattice. The lattice theories obtained this way are
gauge-invariant, free from fermion doubling problem and exact supersymmetric at
finite lattice spacing. We hope that these lattice constructions further
motivate the nonperturbative explorations of models inspired by technicolor,
orbifolding and orientifolding in string theories and the Corrigan-Ramond
limit.
