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

TL;DR

This paper develops a lattice formulation of four-dimensional ${ m N}=4$ super Yang-Mills theory that preserves a single supersymmetry, is gauge invariant, local, and avoids fermion doubling, based on a novel fermion mapping and a geometric twist.

Contribution

It introduces a new lattice action for ${ m N}=4$ super Yang-Mills that maintains a scalar supercharge and is derived from a geometric twist, extending previous ${ m N}=2$ lattice formulations.

Findings

Constructed a gauge-invariant, local lattice action with exact supersymmetry.

Mapped continuum fermions to a single Kahler-Dirac field.

Reduced to a known twist of ${ m N}=4$ super Yang-Mills on the lattice.

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.