= Commuting-scalar vacua of four-dimensional N=4 Yang-Mills theory
{title2=$\mathcal M_{\mathrm{cl}}=(\mathbb R^6)^r/\mathcal W$}
The <scalar potential> is a positive sum of squared commutators of the six adjoint <scalar fields>. It vanishes precisely when they commute. For a compact gauge group, simultaneous conjugation puts a commuting tuple in a Cartan subalgebra. Its arbitrary $6r$ real coordinates are classical <flat directions of a scalar potential>, with residual identification by the <Weyl group>. For $SU(N)$ the simultaneous eigenvalue vectors also obey the tracelessness constraint. Generic values break the gauge group to its maximal torus.
Back to article page
