Two-dimensional N=(2,2) super Yang-Mills theory on computer

TL;DR

This paper presents a preliminary numerical study of two-dimensional N=(2,2) super Yang-Mills theory using Sugino's lattice formulation, confirming supersymmetric identities and observing fermionic effects with manageable lattice artifacts.

Contribution

First numerical investigation of 2d N=(2,2) SYM on the lattice using Sugino's formulation, including dynamical fermions and verifying supersymmetric Ward identities.

Findings

Supersymmetric Ward identities are confirmed with fair accuracy.

Dynamical fermion effects are observed in the lattice simulation.

Complex phase of the pfaffian does not hinder Monte Carlo sampling.

Abstract

We carry out preliminary numerical study of Sugino's lattice formulation \cite{Sugino:2004qd,Sugino:2004qdf} of the two-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory (2d $\mathcal{N}=(2,2)$ SYM) with the gauge group $\SU(2)$. The effect of dynamical fermions is included by re-weighting a quenched ensemble by the pfaffian factor. It appears that the complex phase of the pfaffian due to lattice artifacts and flat directions of the classical potential are not problematic in Monte Carlo simulation. Various one-point supersymmetric Ward-Takahashi (WT) identities are examined for lattice spacings up to $a=0.5/g$ with the fixed physical lattice size $L=4.0/g$, where $g$ denotes the gauge coupling constant in two dimensions. WT identities implied by an exact fermionic symmetry of the formulation are confirmed in fair accuracy and, for most of these identities, the quantum effect of dynamical fermions is clearly observed. For WT identities expected only in the continuum limit, the results seem to be consistent with the behavior expected from supersymmetry, although we do not see clear distintion from the quenched simulation. We measure also the expectation values of renormalized gauge-invariant bi-linear operators of scalar fields.