= Hermitian commutator potential in D-brane Yang-Mills theory
{c}
{title2=$V=-\frac{1}{4g_{\mathrm{YM}}^2}\sum_{I,J}\operatorname{Tr}[\phi^I,\phi^J]^2\geq0$}
For Hermitian scalar matrices, their <commutator> is anti-Hermitian, so $-\operatorname{Tr}[\phi^I,\phi^J]^2=\operatorname{Tr}[\phi^I,\phi^J]^\dagger[\phi^I,\phi^J]\geq0$. Dimensional reduction gives $F_{IJ}=i[\phi^I,\phi^J]$, fixing the minus sign before the literal <commutator> square inside the conventional overall-negative <Yang-Mills action>. Commuting scalar expectation values minimize this potential and describe ordinary separated branes. A plus sign instead requires interpreting the squared symbol as the positive Hermitian <norm>, not literal matrix multiplication.
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