= Orientation of a monopole lift
{title2=$B_i=D_i\Phi,\quad\mathcal A=A_i\,dx^i+\Phi\,d\tau$}
Use $\epsilon_{123}=1$ and orientation $d\tau\wedge dx^1\wedge dx^2\wedge dx^3$. A $\tau$-independent <gauge curvature> has mixed component $\mathcal F_{i\tau}=D_i\Phi$. If $B_i=D_i\Phi$, its two-form is $\sum_{\rm cyclic}B_i(dx^j\wedge dx^k-d\tau\wedge dx^i)$. The <Hodge star operator> interchanges the two displayed basis forms, so $*\mathcal F=-\mathcal F$. With orientation $dx^1\wedge dx^2\wedge dx^3\wedge d\tau$ the same lift is self-dual; changing the sign of its $d\tau$ component instead recovers anti-self-duality. Orientation must therefore accompany any stated sign in this reduction.
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