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

Let $D_i={\partial }_i+A_i$ and $D_{\tau }={\partial }_{\tau }+A_{\tau }$. In temporal gauge, $A_{\tau }=0$, soChoose the orientation $d{x}^1\wedge d{x}^2\wedge d{x}^3\wedge d\tau$ and the anti-self-duality convention matching the question. The three independent components of $F=-{}^{\star }F$ areConsequently the Anti-self-dual Yang-Mills equations in temporal gauge areThey identify anti-self-dual Yang-Mills fields with a first-order flow of three-dimensional gauge connections.