Various Super Yang-Mills Theories with Exact Supersymmetry on the Lattice

TL;DR

This paper advances the construction of lattice super Yang-Mills theories with exact supersymmetry in various dimensions, addressing issues like vacuum degeneracy and surplus modes, and clarifying fine-tuning requirements for different models.

Contribution

It extends previous lattice formulations of super Yang-Mills theories to eliminate surplus modes and resolve vacuum degeneracy, providing models with exact supersymmetry in multiple dimensions.

Findings

Vacuum degeneracy problem is resolved in 4D theories.

Surplus modes are eliminated in 2D and 3D models through modifications.

Certain models require fine-tuning of parameters to reach the continuum limit.

Abstract

We continue to construct lattice super Yang-Mills theories along the line discussed in the previous papers \cite{sugino, sugino2}. In our construction of ${\cal N}=2, 4$ theories in four dimensions, the problem of degenerate vacua seen in \cite{sugino} is resolved by extending some fields and soaking up would-be zero-modes in the continuum limit, while in the weak coupling expansion some surplus modes appear both in bosonic and fermionic sectors reflecting the exact supersymmetry. A slight modification to the models is made such that all the surplus modes are eliminated in two- and three-dimensional models obtained by dimensional reduction thereof. ${\cal N}=4, 8$ models in three dimensions need fine-tuning of three and one parameters respectively to obtain the desired continuum theories, while two-dimensional models with ${\cal N}=4, 8$ do not require any fine-tuning.