Topologically Twisted $N=(2,2)$ Supersymmetric Yang-Mills Theory on Arbitrary Discretized Riemann Surface

TL;DR

This paper constructs a lattice formulation of topologically twisted $ =(2,2)$ supersymmetric Yang-Mills theory on arbitrary discretized Riemann surfaces, preserving one supercharge and avoiding fine-tuning under certain symmetries.

Contribution

It introduces a general lattice approach to discretize $ =(2,2)$ SYM on arbitrary surfaces, extending previous models and reducing parameter tuning needed for continuum limits.

Findings

Lattice model preserves one supercharge on arbitrary discretizations.

Continuum limit yields topologically twisted $ =(2,2)$ SYM on Riemann surfaces.

No fine-tuning required if extra $U(1)_R$ symmetry is present.

Abstract

We define supersymmetric Yang-Mills theory on an arbitrary two-dimensional lattice (polygon decomposition) with preserving one supercharge. When a smooth Riemann surface $\Sigma_g$ with genus $g$ emerges as an appropriate continuum limit of the generic lattice, the discretized theory becomes topologically twisted $\mathcal{N}=(2,2)$ supersymmetric Yang-Mills theory on $\Sigma_g$. If we adopt the usual square lattice as a special case of the discretization, our formulation is identical with Sugino's lattice model. Although the tuning of parameters is generally required while taking the continuum limit, the number of the necessary parameters is at most two because of the gauge symmetry and the supersymmetry. In particular, we do not need any fine-tuning if we arrange the theory so as to possess an extra global U(1) symmetry ($U(1)_{R}$ symmetry) which rotates the scalar fields.