Mass spectrum of $2$-dimensional $\mathcal{N}=(2,2)$ super Yang-Mills theory on the lattice

TL;DR

This study uses lattice simulations to analyze the mass spectrum of 2D $ ext{N}=(2,2)$ super Yang-Mills theory, demonstrating supersymmetry restoration and identifying the massless supermultiplet in the continuum limit.

Contribution

The paper provides the first lattice simulation evidence that $ ext{N}=(2,2)$ SYM in two dimensions has no sign problem and confirms supersymmetry restoration in the continuum limit.

Findings

The theory has no sign problem in lattice simulations.

Flat directions are stabilized by quantum corrections.

The Veneziano-Yankielowicz supermultiplet becomes massless in the continuum limit.

Abstract

In the present work we analyse $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills (SYM) theory in two dimensions by means of lattice simulations. The theory arises as dimensional reduction of $\mathcal{N}=1$ SYM theory in four dimensions. As in other gauge theories with extended supersymmetry, the classical scalar potential has flat directions which may destabilize numerical simulations. In addition, the fermion determinant need not be positive and this sign-problem may cause further problems in a stochastic treatment. We demonstrate that $\mathcal{N}=(2,2)$ super Yang-Mills theory has actually no sign problem and that the flat directions are lifted and thus stabilized by quantum corrections. Only the bare mass of the scalars experience a finite additive renormalization in this finite theory. On various lattices with different lattice constants we determine the scalar masses and hopping parameters for which the supersymmetry violating terms are minimal. By studying four Ward identities and by monitoring the $\pi$-mass we show that supersymmetry is indeed restored in the continuum limit. In the second part we calculate the masses of the low-lying bound states. We find that in the infinite-volume and supersymmetric continuum limit the Veneziano-Yankielowicz super-multiplet becomes massless and the Farrar-Gabadadze-Schwetz super-multiplet decouples from the theory. In addition, we estimate the masses of the excited mesons in the Veneziano-Yankielowicz multiplet. We observe that the gluino-glueballs have comparable masses to the excited mesons.