Lattice formulation of ${\cal N}=4$ super Yang-Mills theory

TL;DR

The paper develops a lattice formulation of four-dimensional N=4 SYM by twisting to expose a nilpotent scalar supercharge and packaging fermions into a real Kähler–Dirac field, enabling a Q-exact action. The continuum theory is shown to reduce to the Marcus twist and to be equivalent to the conventional N=4 SYM under appropriate field redefinitions. A geometric lattice discretization for p-forms preserves gauge invariance and avoids fermion doubling, at the cost of complexified fields; a real-contour prescription is proposed to recover the physical theory. The work provides a nonperturbative framework for studying N=4 SYM on the lattice and outlines crucial future tests of twisted SUSY Ward identities and continuum-limit behavior.

Abstract

We construct a lattice action for ${\cal N}=4$ super Yang-Mills theory in four dimensions which is local, gauge invariant, free of spectrum doubling and possesses a single exact supersymmetry. Our construction starts from the observation that the fermions of the continuum theory can be mapped into the component fields of a single real anticommuting Kahler-Dirac field. The original supersymmetry algebra then implies the existence of a nilpotent scalar supercharge $Q$ and a corresponding set of bosonic superpartners. Using this field content we write down a $Q$-exact action and show that, with an appropriate change of variables, it reduces to a well-known twist of ${\cal N}=4$ super Yang-Mills theory due to Marcus. Using the discretization prescription developed in an earlier paper on the ${\cal N}=2$ theory in two dimensions we are able to translate this geometrical action to the lattice.