Equations on Riemann surfaces

Recall that a Riemann surface is a complex 1-manifold $M$ with a complex structure $\Sigma$ (a class of analytically equivalent atlases on $X$). Here we consider equations that relate connections and Higgs fields with solutions on Riemann surfaces. Let $G=SU(2)$ (complex 2-matrices with determinant 1) or $SO(3)$ (real orthogonal 3-matrices with determinant 1), $\theta$ a Higgs field over $M$, and  $P$ a principal $G$-bundle over $M$.

Definition: The curvature of a principal $G$-bundle $P$ is the map
\[
\begin{array}{r c l}
F_\nabla\ :\ \mathcal A^0_M(P) & \to & \mathcal A^2_M(P), \\
\omega s & \mapsto & (d_\nabla \circ \nabla)(\omega s),
\end{array}
\]
where the extension $d_\nabla:\mathcal A^k_M(P)\to \mathcal A^{k+1}_M(P)$ is defined by the Leibniz rule, that is $d_\nabla (\omega\otimes s) = (d\omega)\otimes s +(-1)^k\omega \wedge \nabla s$, for $\omega$ a $k$-form and $s$ a smooth section of $P$.

Since we may write $\mathcal A^1 = \mathcal A^{1,0}\oplus\mathcal  A^{0,1}$ as the sum of its holomorphic and anti-holomorphic parts, respectively (see post "Smooth projective varieties as Kähler manifolds," 2016-06-16), we may consider the restriction of $d_\nabla$ to either of these summands.

Definition: For a vector space $V$, define the Hodge star $*$ by
\[
\begin{array}{r c l}
*\ :\ \bigwedge^k(V^*) & \to & \bigwedge^{n-k}(V^*), \\
e^{i_1}\wedge \cdots \wedge e^{i_k} & \mapsto & e^{j_1}\wedge\cdots \wedge e^{j_{n-k}},
\end{array}
\]
so that $e^{i_1}\wedge\cdots \wedge e^{i_k}\wedge e^{j_1}\wedge\cdots \wedge e^{j_{n-k}} = e^1\wedge \cdots \wedge e^n$. Extend by linearity from the chosen basis.

The dual of the generalized connection $d_\nabla$ is written $d_\nabla^* = (-1)^{m+mk+1}*d_\nabla *$, where $\dim(M)=m$ and the argument of $d_\nabla^*$ is in $\mathcal A^k_M$ (this holds for manifolds $M$ that are not necessarily Riemann surfaces as well).

Now we may understand some equations on Riemann surfaces. They all deal with the connection $\nabla$, its generalization $d_\nabla$, its curvature $F_\nabla$, and the Higgs field $\theta$. Below we indicate their names and where they are mentioned (and described in further detail).
\begin{align*}
\text{Hitchin equations} && \left.d_\nabla\right|_{A^{0,1}}\theta & = 0 && [2],\ \text{Introduction}\\
&& F_\nabla + [\theta,\theta^*] & = 0\\[10pt]
\text{Yang-Mills equations} && d^*_\nabla d_\nabla \theta + *[*F_\nabla,\theta] & = 0 && [1],\ \text{Section 4} \\
&& d_\nabla^*\theta & = 0 \\[10pt]
\text{self-dual Yang-Mills equation} && F_\nabla  - *F_\nabla & = 0 && [2],\ \text{Section 1}\\[10pt]
\text{Yang-Mills-Higgs equations} && d_\nabla *F_\nabla + [\theta,d_\nabla \theta] & = 0 && [4],\ \text{equation (1)} \\
&& d_\nabla * d_\nabla \theta & = 0
\end{align*}

Recall the definitions of $\theta$ and $\theta*$ from a previous post ("Higgs fields of principal bundles," 2016-08-24). Now we look at these equations in more detail. The first of the Hitchin equations says that $\theta$ has no anti-holomorphic component, or in other words, that $\theta$ is holomorphic. In the second equation, the Lie bracket $[\cdot,\cdot]$ of the two 1-forms is
\begin{align*}
[\theta,\theta^*] & = \left[\textstyle\frac12f(dz+i\ dy), \frac12\bar f(dz - i\ dy) \right] \\
& = \textstyle -\frac i4f\bar f\ dx\wedge dy + \frac i4 f\bar f\ dy \wedge dx -\frac i4 f\bar f\ dx\wedge dy +\frac i4 f\bar f\ dy\wedge dx \\
& = -i|f|^2\ dx\wedge dy.
\end{align*}
In the Yang-Mills and Yang-Mills-Higgs equations, we can simplify some parts by noting that, for a section $s$ of the complexification of $P\times_\ad \mathfrak g$,
\begin{align*}
d_\nabla (\theta\otimes s) & = \textstyle \frac12d_\nabla (fdx\otimes s) + \frac i2 d_\nabla (fdy \otimes s) \\
& = \textstyle \frac12 (df\wedge dx \otimes s - fdx \wedge \nabla s) +\frac i2 (df\wedge dy - fdy \wedge \nabla s) \\
& = \left(\frac i2\frac{\dy f}{\dy x} - \frac 12 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s - \underbrace{\textstyle \frac12f(dx+idy)}_{\theta}\wedge \nabla s.
\end{align*}
The Hodge star of $\theta$ is $*\theta = \frac 12f(dy -idx)$, so
\begin{align*}
d_\nabla *(\theta\otimes s) & = \textstyle \frac12d_\nabla (fdy\otimes s) - \frac i2 d_\nabla (fdx \otimes s) \\
& = \textstyle \frac12 (df\wedge dy \otimes s - fdy \wedge \nabla s) -\frac i2 (df\wedge dx - fdx \wedge \nabla s) \\
& = \left(\frac 12\frac{\dy f}{\dy x} + \frac i2 \frac{\dy f}{\dy y}\right)dx\wedge dy \otimes s + \underbrace{\textstyle \frac12f(idx-dy)}_{i\theta}\wedge \nabla s.
\end{align*}
We could express $\nabla s = (s_1dx + s_2dy)\otimes s^1$, but that would not be too enlightening. Next, note the self-dual Yang-Mills equation only makes sense over a (real) 4-dimensional space, since the degrees of the forms have to match up. In that case, with a basis $dz_1=dx_1+idy_1, dz_2 = dx_2+idy_2$ of $\mathcal A^1$, we have
\begin{align*}
F_\nabla & = F_{12} dx_1\wedge dy_1 + F_{13} dx_1 \wedge dx_2 + F_{14} dx_1\wedge dy_2 + F_{23} dy_1\wedge dx_2 + F_{24} dy_1\wedge dy_2 + F_{34} dx_2\wedge dy_2, \\
*F_\nabla & = F_{12} dx_2\wedge dy_2 - F_{13} dy_1 \wedge dy_2 + F_{14} dy_1\wedge dx_2 + F_{23} dx_1\wedge dy_2 - F_{24} dx_1\wedge dx_2 + F_{34} dx_1\wedge dy_1.
\end{align*}
Then the self-dual equation simply claims that
\[
F_{12} = F_{34}
\hspace{1cm},\hspace{1cm}
F_{13} = -F_{24}
\hspace{1cm},\hspace{1cm}
F_{14} = F_{23}.
\]

Remark: This title of this post promises to talk about equations on Riemann surfaces, yet all the differential forms are valued in a principal $G$-bundle over $\R^2$ (or $\R^4$). However, since the given equations are conformally invariant (this is not obvious), and as a Riemann surface locally looks like $\R^2$, we may consider the solutions to the equations as living on a Riemann surface.

References:
[1] Atiyah and Bott (The Yang-Mills equations over Riemann surfaces)
[2] Hitchin (Self-duality equations on a Riemann surface)
[3] Huybrechts (Complex Geometry, Chapter 4.3)
[4] Taubes (On the Yang-Mills-Higgs equations)