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We investigate the two-dimensional $\mathcal{N}=(2,2)$ supersymmetricYang-Mills (SYM) theory on the discretized curved space (polyhedra). We firstrevisit that the number of supersymmetries of the continuum $\mathcal{N}=(2,2)$SYM theory on any curved manifold can be enhanced at least to two byintroducing an appropriate $U(1)$ gauge background associated with the$U(1)_{V}$ symmetry. We then show that the generalized Sugino model on thediscretized curved space, which was proposed in our previous work, can beidentified to the discretization of this SUSY enhanced theory, where one of thesupersymmetries remains and the other is broken but restored in the continuumlimit. We find that the $U(1)_{A}$ anomaly exists also in the discretizedtheory as a result of an unbalance of the number of the fermions proportionalto the Euler characteristics of the polyhedra. We then study this model byusing the numerical Monte-Carlo simulation. We propose a novel phase-quenchmethod called "anomaly-phase-quenched approximation" with respect to the$U(1)_A$ anomaly. We numerically show that the Ward-Takahashi (WT) identityassociated with the remaining supersymmetry is realized by adopting thisapproximation. We figure out the relation between the sign (phase) problem andpseudo-zero-modes of the Dirac operator. We also show that the divergentbehavior of the scalar one-point function gets milder as the genus of thebackground increases. These are the first numerical observations for thesupersymmetric lattice model on the curved space with generic topologies.

Anomaly and Sign problem in $\mathcal{N}=(2,2)$ SYM on Polyhedra : Numerical Analysis