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

Two-dimensional ${\cal N} = (2, 2)$ Lattice Gauge Theories with Matter in Higher Representations