Hermitian commutator potential in D-brane Yang-Mills theory ($V=-\frac{1}{4g_{\mathrm{YM}}^2}{\sum }_{I,J}\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2\ge 0$)

For Hermitian scalar matrices, their commutator is anti-Hermitian, so $-\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^2=\mathrm{Tr}[{\varphi }^I,{\varphi }^J{]}^{†}[{\varphi }^I,{\varphi }^J]\ge 0$. Dimensional reduction gives $F_{IJ}=i[{\varphi }^I,{\varphi }^J]$, 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.