Let $D_i=\partial_i+A_i$ and $D_\tau=\partial_\tau+A_\tau$. In <temporal gauge>, $A_\tau=0$, so
$$
F_{\tau i}=\partial_\tau A_i.
$$
Choose the orientation $dx^1\wedge dx^2\wedge dx^3\wedge d\tau$ and the anti-self-duality convention matching the question. The three independent components of $F=-{}^\star F$ are
$$
F_{\tau1}=F_{23},
\qquad
F_{\tau2}=F_{31},
\qquad
F_{\tau3}=F_{12}.
$$
Consequently the <Anti-self-dual Yang-Mills equations in temporal gauge> are
$$
\boxed{
\frac{\partial A_i}{\partial\tau}
=\frac12\epsilon_{ijk}F_{jk},
\qquad i=1,2,3}.
$$
They identify anti-self-dual Yang-Mills fields with a first-order flow of three-dimensional gauge connections.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 313 / 3 / c / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-313/3/c/solution