Gauge Covariant Link Formulation of Twisted N=D=4 and N=4 D=5 Super Yang-Mills on a Lattice
Alessandro D'Adda, Noboru Kawamoto, Jun Saito, Kazuhiro Nagata
TL;DR
The paper develops a gauge covariant lattice formulation of twisted N=D=4 and N=4 D=5 super Yang-Mills using a Dirac-Kähler framework. By assigning twisted supercharges to lattice links and employing Grassmann link parameters, it achieves exact lattice SUSY with manifest gauge covariance and constructs closed-loop lattice actions that are invariant under all supercharges, up to equations of motion. A key advance is the interpretation of gauge covariant link (anti-)commutators as exponentials of bosonic covariant derivatives, which unifies noncommutativity and link structure and resolves previous ordering ambiguities. The work further provides a consistent five-dimensional lift to N=4 D=5 SYM, including lattice embeddings and on-shell closure, and discusses addressing criticisms while outlining potential numerical implementations and the implications for lattice SUSY and semi-local spacetime structure.
Abstract
We propose a lattice formulation of four dimensional super Yang-Mills model with a twisted N=4 supersymmetry in a manifestly gauge covariant manner. The formulation we employ here is a four dimensional extension of the manifestly gauge covariant method which was developed in our proposals of Dirac-Kahler twisted N=D=2 and N=4 D=3 super Yang-Mills on a lattice. Twisted N=4 supersymmetry algebra is geometrically realized on a four dimensional lattice with link supercharges and the use of link (anti-)commutators. Employing Grassmann parameters with link nature, we explicitly show that the resulting super Yang-Mills action is invariant under all the supercharges on a lattice without chiral fermion problems. As a group and algebraic interpretation of the link approach, we show that promoting bosonic supercovariant derivatives to their exponentials consistently with the lattice Leibniz rule naturally gives rise to the notion of gauge covariant link (anti-)commutators. This can be regarded as a fermionic decomposition of a Lie group element, which may provide a new methodology in super Lie group and super Lie algebra. We also provide a five dimensional lift-up of the formulation with exact N=4 SUSY invariance on a five dimensional lattice.