Orientation of a monopole lift ($B_i=D_i\Phi ,\mathcal{A}=A_{i}d{x}^i+\Phi d\tau$)

Use ${ϵ}_{123}=1$ and orientation $d\tau \wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$. 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 ${\sum }_{\mathrm{cyclic}}{B}_i(d{x}^j\wedge d{x}^k-d\tau \wedge d{x}^i)$. The Hodge star operator interchanges the two displayed basis forms, so $\ast \mathcal{F}=-\mathcal{F}$. With orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ 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.