From One to Eight: Supersymmetry Restoration in Lattice 3D $\mathcal{N}=4$ 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 $\mathcal{N}=4$ 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 $\mathcal{O}(a)$, 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 $\mathcal{N}=4$ supersymmetry without further tuning. These results may enable nonperturbative lattice studies of three-dimensional supersymmetric gauge theories relevant to string theory and mirror symmetry.

Comments: 6 pages, 1 figure. Contribution to the 70th Annual Conference of the South African Institute of Physics (SAIP2026), 6-10 July 2026