Supersymmetry on a Spatial Lattice

TL;DR

This work develops a framework to realize supersymmetric gauge theories on spatial lattices by deconstructing higher-dimensional theories from a 0+1D mother theory via orbifold projections by a $Z_N^d$ symmetry. By preserving a subset of the original supercharges exactly on the lattice, the construction suppresses dangerous radiative corrections and minimizes fine tuning when taking the continuum limit, enabling target theories with 4, 8, or 16 real supercharges in 1+1D, 2+1D, and 3+1D, including the $N=4$ SYM in 3+1D. The paper provides explicit lattice actions for the simple 1+1D $(2,2)$ theory and discusses renormalization, the role of radion moduli, and the extension to higher dimensions with various lattice geometries (square, hexagonal, and BCC) and corresponding E- and J-type plaquette interactions. It also outlines a two-step, asymmetric approach to realize 3+1D $N=4$ SYM without excessive fine tuning, suggesting promising avenues for nonperturbative studies and potential connections to conformal fixed points. Overall, the framework offers a systematic path to robust lattice SUSY constructions with reduced tuning and potential Euclidean extensions for numerical exploration.

Abstract

We construct a variety of supersymmetric gauge theories on a spatial lattice, including N=4 supersymmetric Yang-Mills theory in 3+1 dimensions. Exact lattice supersymmetry greatly reduces or eliminates the need for fine tuning to arrive at the desired continuum limit in these examples.