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

On an oriented Euclidean four-space with its metric volume form, the Hodge star operator is defined byfor forms $\alpha$ and $\beta$ of the same degree. In $n$ Euclidean dimensions it satisfiesHence on two-forms in four dimensions,The normalization of $\mathrm{vol}$ as the metric volume form is essential; reversing its orientation reverses a single Hodge star but leaves its square unchanged.

Because ${}^{\star }{}^{\star }=1$, the exterior algebra of two-forms has the orthogonal decompositionwhere ${}^{\star }H=H$ for a self-dual differential form and ${}^{\star }G=-G$ for an anti-self-dual differential form. The Hodge star is self-adjoint, soIt follows that

For an $SU(n)$ connection, use the positive normand define the Second Chern number byOrthogonality givesTherefore the Euclidean Yang-Mills actionobeys the Yang-Mills instanton Bogomolny boundEquality holds precisely when $F_{-}=0$ or $F_{+}=0$, according to the sign of $k$; these are the self-dual and anti-self-dual Yang-Mills instantons. Other trace and orientation conventions may reverse $k$ but leave the absolute-value bound unchanged.

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.

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 $D_{\tau }={\partial }_{\tau }$.