Abstract
We show how to derive the supersymmetric orbifold lattices of Cohen et al. \cite{Cohen:2003xe,Cohen:2003qw} and Kaplan et al. \cite{Kaplan:2005ta} by direct discretization of an appropriate twisted supersymmetric Yang-Mills theory. We examine in detail the four supercharge two dimensional theory and the theory with sixteen supercharges in four dimensions. The continuum limit of the latter theory is the well known Marcus twist of ${\cal N}=4$ Yang-Mills. The lattice models are gauge invariant and possess one exact supersymmetry at non-zero lattice spacing.
Highlights
While the orbifold constructions are essentially unique [18] various approaches to discretization of the twisted theories have been advocated in [19, 20, 21, 22, 23, 24]
Recent work by Damgaard, Matsuura and Takimi has indicated that there are, strong connections between these twisted theories and the orbifold models [25, 26, 27]. This had already been anticipated by Unsal who showed that the naive continuum limit of the sixteen supercharge orbifold model in four dimensions led to the Marcus twist of N = 4 Yang-Mills [28]
This simple five dimensional structure allows us to use the geometric discretization prescription employed in two dimensions to write down a supersymmetric lattice theory corresponding to this Marcus twist of N = 4 YangMills
Summary
Paralleling the four supercharge theory we introduce an additional auxiliary bosonic scalar field d and a set of five dimensional antisymmetric tensor fields to represent the fermions Ψ = (η, ψa, χab). To make contact with a twist of N = 4 in four dimensions we must dimensionally reduce this theory along the 5th direction This will yield a complex scalar φ = A5 + iB5 and its superpartner η. We have shown how to derive this theory by dimensional reduction of a rather simple five dimensional theory employing a complex gauge field and integer spin twisted fermions. It will be the basis of our lattice formulation to which we turn
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