From One to Eight: Supersymmetry Restoration in Lattice 3D ${\cal N} = 4$ Super Yang--Mills

Abstract

Topological twisting provides a powerful framework for constructing lattice formulations of supersymmetric gauge theories. In three dimensions, a twisted version of ${\cal N} = 4$ super Yang--Mills theory can be discretized so that one nilpotent scalar supersymmetry is preserved exactly at nonzero lattice spacing. The remaining seven supersymmetries are broken by lattice artifacts of ${\cal O}(a)$, where $a$ is the lattice spacing. An important question is whether these supersymmetries are automatically restored in the continuum limit $a \to 0$, or whether fine-tuning of the lattice couplings is required. In this work, we derive the additional twisted supersymmetries by combining discrete $R$-symmetries of the continuum theory with the action of the scalar supercharge. This construction suggests that restoration of rotational symmetry in the continuum limit implies restoration of $R$-symmetry, leading to an automatic enhancement to the full ${\cal N} = 4$ supersymmetry without further tuning. These results may enable nonperturbative lattice studies of three-dimensional supersymmetric gauge theories relevant to string theory and mirror symmetry.