We define supersymmetric Yang-Mills theory on an arbitrary 2D lattice (polygon decomposition) while preserving one supercharge. When a smooth Riemann surface ∑g with genus g emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes a topologically twisted N = (2, 2) supersymmetric Yang-Mills theory on ∑g. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino's lattice model. Although the tuning of parameters is generally required while taking the continuum limit, the number of necessary parameters is at most two because of the gauge symmetry and the supersymmetry. In particular, we do not need any fine-tuning if we arrange the theory so as to possess an extra global U(1) symmetry (U(1)R symmetry), which rotates the scalar fields.
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