Higgs fields of principal bundles

The goal here is to understand the setting of Higgs fields on Riemannian manifolds, in the manner of Hitchin. First we consider general topological spaces $X$ and groups $G$.

Definition: Let $X$ be a topological space and $G$ a group. A principal bundle (or principal $G$-bundle) $P$ over $X$ is a fiber bundle $\pi:P\to X$ together with a continuous, free, and transitive right action $P\times G\to P$ that preserves the fibers. That is, if $p\in \pi^{-1}(x)$, then $pg\in \pi^{-1}(x)$ for all $g\in G$ and $x\in X$.

Now suppose we have a principal bundle $\pi:P\to X$, a representation $\rho$ of $G$, and another space $Y$ on which $G$ acts on the left. Define an equivalence relation $(p,y)\sim (p',y')$ on $P\times Y$ iff there is some $g\in G$ for which $p'=pg$ and $y'=\rho(g^{-1})y$. This is an equivalence relation. We will be interested in the adjoint representation (induced by conjugation).

Proposition: The projection map $\pi':P\times_\rho Y := (P\times Y)/\sim\ \to X$, where $\pi'([p,y]) = \pi(p)$, defines a vector bundle over $X$, called the associated bundle of $P$.

Recall a Lie group $G$ is a group that is also a topological space, in the sense that there is a continuous map $G\times G\to G$, given by $(g,h)\mapsto gh^{-1}$. The Lie algebra $\mathfrak g$ of the Lie group $G$ is the tangent space $T_eG$ of $G$ at the identity $e$. We will be interested in principal $G$-bundles $P\to \R^2$ and associated bundles $P\times_\ad \mathfrak g\to \R^2$, where $\ad$ is the adjoint representation of $G$.

Next, recall we had the space $\mathcal A^k_M$ of $k$-differential forms on $M$ (see post "Smooth projective varieties as Kähler manifiolds," 2016-06-16), defined in terms of wedge products of elements in the cotangent bundle $(TM)^* = T^*M$ of $M$. Now we generalize this to get differential forms over arbitrary vector bundles.

Definition: Let $E\to M$ be a vector bundle. Let
$$\begin{align*} \mathcal A^k_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^k T^*M) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^k_M, \\ \mathcal A^{p,q}_M(E) & := \Gamma(E\otimes \textstyle\bigwedge^p (T^{1,0}M)^*\otimes \bigwedge^q (T^{0,1}M)^*) = \Gamma(E)\otimes_{\mathcal A^0_M}\mathcal A^{p,q}_M \end{align*}$$
be the spaces of $k$- and $(p,q)$-differential forms, respectively, over $M$ with values in $E$.

Equality above follows by functoriality. Now we are close to understanding where exactly the Higgs field lives, in Hitchin's context.

Definition: Given a function $f:\C\to \C$, the conjugate of $f$ is $\bar f$, defined by $\bar f(z) = \overline{f(\bar z)}$.

Hitchin denotes this as $f^*$, but we will stick to $\bar f$. Finally, let $P$ be a $G$-principal bundle over $\R^2$ and $P\times_\ad \mathfrak g$ the associated bundle of $P$. Given $f\in \mathcal A^0_{\R^2}( (P\times_\ad \mathfrak g)\otimes \C)$, set
$$\begin{align*} \theta & = \textstyle \frac12 f(dx+i\ dy) \in \mathcal A^{1,0}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C) ,\\ \theta^* & = \textstyle \frac12 \bar f(dx-i\ dy) \in \mathcal A^{0,1}_{\R^2}((P\times_\ad\mathfrak g)\otimes \C), \end{align*}$$
called a Higgs field over $\R^2$ and (presumably) a dual (or conjugate) Higgs field over $\R^2$. Note this agrees with the definition in a previous post ("Connections, curvature, and Higgs bundles," 2016-07-25).

References: Hitchin (Self-duality equations on a Riemann surface), Wikipedia (article on associated bundles, article on vector-valued differential forms)