My question is about computing the curvature of a quotient connection, specifically for the case of the quotient of the tautological connection of a universal bundle on the moduli space of connections.

The calculation I am trying to understand shows up in a few places, for instance in the book "Geometry of Four-Manifolds" by Donaldson and Kronheimer, in section 5.2.3. I am confused on the proof of Proposition 5.2.17, which I will recount below after introducing some notation.

Let $E\to X$ be a complex smooth bundle over a smooth four-manifold $X$ with hermitian structure $h$. (Donaldson and Kronheimer assume it is an $SU(2)$-bundle, but I think this can be generalized to the structure group $U(n)$.) Let $\mathscr{A}$ be the affine space of unitary connections on $E$, and let $\mathscr{A}^*\subset \mathscr{A}$ be the space of irreducible connections. Set $\mathbb{E}:= \pi_2^*E$ where $\pi_2:\mathscr{A}^*\times X \to X$ is the projection, and let $\mathfrak{g}_\mathbb{E}$ be the associated bundle of Lie algebras. Both $\mathbb{E}$ and $\mathfrak{g}_\mathbb{E}$ carry a tautological connection $\hat\nabla$, which restricts to a connection $A$ on $E$ on every slice $\{A\}\times X\subset \mathscr{A}^*\times X$, and which is trivial in the $\mathscr{A}$-directions.

Let $\mathscr{G}$ be the group of gauge transformations of $E$, and $\mathscr{G}_0:= \mathscr{G}/U(1)$ be the group of reduced gauge transformations. We then have an orbit space $\mathscr{B}^*:= \mathscr{A}^*/\mathscr{G}_0$.

Here is Proposition 5.2.17, which involves calculating the curvature of the quotient connection $\nabla$ in the quotient $\mathfrak{g}_\mathbb{P}:=\mathfrak{g}_\mathbb{E}/\mathscr{G}_0\to \mathscr{B}^*\times X$.

Proposition (5.2.17) Let $\hat{\nabla}$ be the tautological connection on $\mathfrak g_{\mathbb{E}}$, and $\nabla$ the quotient of this connection on the quotient bundle $\mathfrak g_{\mathbb{P}}\to \mathscr{B}^*\times X$. The three components of the curvature of $\nabla$ at a point $([A], x)\in \mathscr{B}^*\times X$ are given by

- $F(\nabla)(u,v) = F(A)(u,v)$
- $F(\nabla)(a,v) = \langle a, v\rangle$
- $F(\nabla)(a,b) = - 2 G_A\{a,b\}|_x$.
Here, $u,v\in T_x X$, $a,b\in \Omega^1(\mathfrak g_E)$ satisfying $d_A^*a=d_A^*b=0$; $G_A$ is the Green's operator for the Laplacian $d_A^*d_A$ on $\Omega^0(\mathfrak g_E)$; and $\{,\}$ is the natural pairing formed from a metric on $X$ and the Lie bracket on $\operatorname{Lie}(G)$, with $G=U(n)$ being the gauge group.

I am specifically confused about how they apply equation (5.2.16) (which is marked as equation $(*)$ below) to deduce this curvature, mainly because I do not understand how they figure out what $\Phi$ "does" in the case of the qoutient of the tautological connection. So, my question is,

How do they calculate the map $\Phi$ below in order to apply equation $(*)$ below to deduce this Proposition?

Here are the relevant details from this passage in the book. Suppose a Lie group $\Gamma$ acts freely and properly on a manifold $\hat{Y}$. Also assume we have a bundle $\hat{E}\to \hat{Y}$ and an action of $\Gamma$ on $\hat{E}$ that is linear on fibers and that covers the group action on $\hat{Y}$. Let $Y := \hat{Y}/\Gamma$ and $E:= \hat{E}/\Gamma$.

We now suppose we are given two things:

- A connection $\hat{\nabla}$ in $\hat{E}$ invariant under $\Gamma$.
- A connection $H$ in the $\Gamma$-bundle $p:\hat{Y}\to Y$.

One then gets a quotient connection $\nabla$ in $E$ from this. Then, in order to compute its curvature, introduce the 1-form $B \in \Omega_{ \hat Y }^1 \otimes \operatorname{End}(\hat{E})$ given by $$ B:= \hat{\nabla} - p^* \nabla.$$ Then, because $B$ vanishes on $H$-horizontal vectors, we can write $B$ as $\Phi \circ \theta$, where $\theta$ is the connection 1-form for $H$ and $\Phi: \operatorname{Lie}(\Gamma) \to \operatorname{End}(\hat{E})$ is a linear map. One can then compute that $$(*)\quad F(\nabla)(U,V) = F(\hat{\nabla})(\hat{U},\hat{V}) - \Phi\circ \Theta(U,V)$$ where $U,V\in T_y Y$ and $\hat{U},\hat{V}$ are horizontal lifts to $T\hat{Y}$.

They apply this to $g_{\mathbb{E}}$, with :

- $\Gamma=\mathscr{G}_0$
- $\hat{Y} = \mathscr{A}^*\times X$
- $H$ is the connection on the $\mathscr{G}_0$-bundle $p:\mathscr{A}^*\to \mathscr{B}^*$ obtained from slice neighborhoods for the action of the gauge transformations
- $\hat{E} = \mathfrak{g}_\mathbb{E}$
- $\hat{\nabla}$ is the tautological connection on $\mathfrak g_{\mathbb{E}}$

They use the results that for $H$, the connection form $\theta$ and curvature form $\Theta$ are $\theta_A(a) = -G_A d_A^* a$ and $\Theta_A(a,b) = -2G_A\{a,b\}$, which I am fine with. What is really bothering me is how they figure what $\Phi$ (or $B$ for that matter) is. I can see the answer they get looks like the restriction map $\Omega^0(\mathfrak{g}_E)\to \operatorname{Lie}(G)_x$, but how they deduce this is completely opaque to me.