Numerical Analysis of Discretized ${\cal N}=(2,2)$ SYM on Polyhedra

TL;DR

This paper numerically investigates the ${ m N}=(2,2)$ supersymmetric Yang-Mills theory on discretized curved spaces, introducing a new phase-quenched approximation to handle fermionic anomalies and analyze supersymmetry preservation.

Contribution

It proposes the anomaly-phase-quenched (APQ) method to address fermionic phase issues and demonstrates its effectiveness in preserving supersymmetry on discretized curved spaces.

Findings

APQ method successfully cancels $U(1)_A$ phase in the partition function.

The Ward-Takahashi identity for SUSY is estimated on the lattice.

Pseudo zero-modes significantly contribute to the pfaffian phase.

Abstract

We perform a numerical simulation of the two-dimensional ${\cal N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory on the discretized curved space. The $U(1)_{A}$ anomaly of the continuum theory is maintained also in the discretized theory as an unbalance of the number of the fermions. In the process, we propose a new phase-quenched approximation, which we call the "anomaly-phase-quenched (APQ) method", to make the partition function and observables well-defined by $U(1)_{A}$ phase cancellation. By adopting APQ method, we estimate the Ward-Takahashi identity for exact SUSY on lattice and clarify contribution of the pseudo zero-modes to the pfaffian phase.