The smoothness of solutions to the Hitchin self-dual equations within a stable orbit after Sobolev completion

Question

First, let me introduce the background of the problem: While studying chapter 4 of the Hitchin's paper "THE SELF-DUALITY EQUATIONS ON A RIEMANN SURFACE", I encountered an issue. Hitchin solves the self-dual equations $$ \begin{cases} F_A + [\Phi, \Phi^*]=0\\ d_A^{\prime\prime}\Phi=0 \end{cases} $$ within a stable orbit, by completing the unitary connection space, Higgs field space, and gauge transformation space in the Sobolev space, using the method of weak convergence to obtain a solution set $(A, \Phi)$. My question is, are these solutions necessarily smooth? In fact, the challenging part of this problem is whether the obtained connection $A$ is smooth; when $A$ is smooth, $\Phi$ is naturally smooth. We know that if $(A, \Phi)$ is a solution to the equation above, and $g$ is a unitary gauge transformation, then under the action of $g$, $$ (g^{-1} A g, g^{-1} \Phi g) $$ is also a solution to this Hitchin equations. So, if $A$ is not a smooth connection, we hope to seek a unitary gauge transformation $g$ such that $g^{-1} A g$ is a smooth connection. Hitchin's original words are: “Since (from Atiyah-Bott) every $W^{2,2}$ orbit in the $W^{1,2}$ space of connections contains a smooth connection, there is no loss of generality as far as a connection is concerned.”

The specific problem setting is: Let $M$ be a Riemann surface, and $V$ be a rank 2 complex vector bundle on $M$. Let $H$ be a Hermitian metric on $V$, and denote $\mathscr{A}$ as the space of smooth unitary connections on $(V, H)$. Perform a $W^{1,2}$ Sobolev completion on $\mathscr{A}$, and denote this as $\mathscr{A}_1^2$.

Let $\mathscr{G}$ be the group of unitary gauge transformations on $(V, H)$, and $\mathscr{G}^\mathbb{C}$ be the complexification of $\mathscr{G}$, essentially the general gauge transformation group on $V$.

The action of $\mathscr{G}$ on $\mathscr{A}$ is given by: $$ g \cdot A \mapsto g^{-1} A g $$ This action is extended to $\mathscr{G}^\mathbb{C}$ acting on $\mathscr{A}$ as follows: $$ g \cdot A \mapsto A' \text{ such that } A' \text{ is the Chern connection corresponding to } g^{-1} d_A^{\prime\prime} g $$

Further, consider the $W^{2,2}$ completions of $\mathscr{G}$ and $\mathscr{G}^\mathbb{C}$, denoted as $\mathscr{G}_2^2$ and ${\mathscr{G}^\mathbb{C}}_2^2$, acting on $\mathscr{A}_1^2$.

Atiyah and Bott pointed out that in $\mathscr{A}_1^2$, on every ${\mathscr{G}^\mathbb{C}}_2^2$ orbit, there exists at least one smooth connection. Hence, it is evident that smooth connections are dense in the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits. However, for the background problem I presented, this conclusion is insufficient. What we really hope to find is that on each $\mathscr{G}_2^2$ orbit, there exists at least one smooth connection. However, the $\mathscr{G}_2^2$ orbits are a lower-dimensional subspace of the ${\mathscr{G}^\mathbb{C}}_2^2$ orbits, thus the problem is non-trivial.

My question is: Does each $\mathscr{G}_2^2$ orbit contain at least one smooth connection? If the answer is negative, are the solutions to the Hitchin self-duality equations in the Sobolev sense or in the smooth sense?

Answer 1

Yes. Denote by $D$ your $W^{1,2}$ unitary connection with respect to the (smooth) hermitian metric $H,$ and let $g$ be the (complex) $W^{2,2}$ gauge transform such that the gauged connection $\nabla:=D.g$ is smooth. Note that $W^{2,2}\to C^0$ emedds continuously on compact surfaces, and moreover $W^{2,2}$ is a Banach algebra. In particular, the hermitian metric $g^*H$ is continuous and moreover has weak first and second order derivatives. Furthermore, $g^*H$ satisfies the linear ODE $\nabla (g^*H)=0,$ from which you can deduce (using smoothness of $\nabla$ and bootstrapping) that $g^*H$ is actually a smooth hermitian metric. Thus, there is a smooth gauge $h$ such that $h^*(g^*H)=H$, and $\nabla.h=D.(gh)$ is a smooth unitary connection with respect to $H$, and $gh$ is the unitary $W^{2,2}$ gauge transformation.