Anomaly and sign problem in N = (2, 2) SYM on polyhedra: Numerical analysis

We investigate two-dimensional N = (2, 2) supersymmetric Yang–Mills (SYM) theory on discretized curved space (polyhedra). We first revisit that the number of supersymmetries of the continuum N = (2, 2) SYM theory on any curved manifold can be enhanced at least to two by introducing an appropriate U(1) gauge background associated with the U(1)_V symmetry. We then show that the generalized Sugino model on discretized curved space, which was proposed in our previous work, can be identified with the discretization of this SUSY enhanced theory, where one of the supersymmetries remains, and the other is broken but restored in the continuum limit. We find that the U(1)_A anomaly exists also in discretized theory as a result of an unbalance in the number of fermions proportional to the Euler characteristic of the polyhedra. We then study this model by using the numerical Monte Carlo simulation. We propose a novel phase-quench method called the “anomaly-phase-quenched approximation” with respect to the U(1)_A anomaly. We show numerically that the Ward–Takahashi identity associated with the remaining supersymmetry is realized by adopting this approximation. We work out the relation between the sign (phase) problem and pseudo-zero-modes of the Dirac operator. We also show that the divergent behavior of the scalar one-point function gets milder as the genus of the background increases. These are the first numerical observations for the supersymmetric lattice model on curved space with generic topologies.