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

Let $\zeta \in {\mathbb{CP}}^1$ be the spectral parameter. Define the two covariant differential operatorsThe auxiliary systemis compatible exactly when $[L(\zeta ),M(\zeta )]=0$ for every $\zeta$. The constant and quadratic coefficients giveand their equivalent conjugate equations, while the linear coefficient givesSince $F_{i\tau }=-F_{\tau i}$, these are preciselyThusis a Lax pair for the anti-self-dual Yang-Mills equations. In temporal gauge one simply sets $D_{\tau }={\partial }_{\tau }$.