N=4supersymmetric Yang-Mills multiplet in nonadjoint representations

Abstract

We formulate a theory for an $N=4$ supersymmetric Yang-Mills multiplet in a nonadjoint representation $R$ of $SO(\mathcal{N})$, as an important application of our recently proposed model for $N=1$ supersymmetry. This system is obtained by dimensional reduction from an $N=1$ supersymmetric Yang-Mills multiplet in a nonadjoint representation in ten dimensions. The consistency with supersymmetry requires that a nonadjoint representation $R$ with the indices $i,j,\dots{}$ satisfy the three conditions ${\ensuremath{\eta}}^{ij}={\ensuremath{\delta}}^{ij}$, $({T}^{I}{)}^{ij}=\ensuremath{-}({T}^{I}{)}^{ji}$, and $({T}^{I}{)}^{[ij|}({T}^{I}{)}^{|k]l}=0$ for the metric ${\ensuremath{\eta}}^{ij}$ and the generators ${T}^{I}$, which are the same as the $N=1$ case.