A Euclidean Lattice Construction of Supersymmetric Yang-Mills Theories with Sixteen Supercharges

TL;DR

This work constructs Euclidean spacetime lattices for ${\cal Q}=16$ supersymmetric Yang-Mills theories in $d\le 4$ by orbifolding a zero-dimensional ${\cal Q}=16$ mother theory with $SO(10)$ symmetry. The construction uses ${\cal A}_d^*$ lattice geometries (notably ${\cal A}_4^{*}$ in 4D) to preserve a subset of supersymmetry ($Q=1$ in 4D, $Q=2$ in 3D, $Q=4$ in 2D, and $Q=8$ in 1D) and to produce local, gauge-invariant lattice actions with moduli-space dynamics that enable a tree-level continuum limit to the target theories, including ${\cal N}=4$ SYM in $d=4$. Continuum limits are demonstrated at tree level for each dimensional case, with no boson doublers and no fermion doublers argued via supersymmetry; the six scalar fields naturally map to continuum scalars, while fermions assemble into Majorana-like multiplets aligned with the target symmetry groups. The paper discusses the potential for minimal fine-tuning due to exact lattice supersymmetry, the challenges of numerical simulation (e.g., sign problems and massless fermions), and broader implications for nonperturbative studies of gauge/string dualities and M-theory via Euclidean path integrals.

Abstract

We formulate supersymmetric Euclidean spacetime Ad* lattices whose classical continuum limits are U(N) supersymmetric Yang-Mills theories with sixteen supercharges in d=1,2,3 and 4 dimensions. This family includes the especially interesting N=4 supersymmetry in four dimensions, as well as a Euclidean path integral formulation of Matrix Theory on a one dimensional lattice.