Principle Bundle, Associated bundle, Gauge Group and Connections

Let $M$ be compact riemannian manifold without boundary, and $G$ be a compact Lie group. A principle $G$-bundle over $M$, denoted as $P(M,G)$ is a manifold $P$ with a free right action $P\times G\ni(p,g)\mapsto pg\in P$ of $G$ such that $M=P/G$, and $P$ is locally trivial, i.e., for every point $x\in M$, there is a neighbourhood $U$ such that the primage $\pi^{-1}(U)$ of the canonical projection is isomorphic to $U\times G$ in the sense that it preserver the fiber and $G$-equivariant, more precisely, there is a diffeomorphism $\Phi:\pi^{-1}(U)\to U\times G$ such that $\Phi(p)=(\pi(p),\phi(p))$ and $\phi:\pi^{-1}(U)\to G$ satisfying $\phi(pg)=(\phi(p))g$ for all $p\in\pi^{-1}(U)$ and $g\in G$.

Now for any other manifold $F$ on which $G$ acts on the left $G\times F\ni(g,f)\mapsto gf\in F$, the associated fiber bundle $P\times_\rho F$ is the quotient space $P\times F/\sim$, where $[p,f]\sim[pg,g^{-1}f]$, for all $g\in G$. With the projection $\tilde\pi:[p,g]\mapsto\pi(p)$ this is a principle bundle over $M$ with typical fiber $F$. The local trivialization is induced by the one $\Phi:\pi^{-1}(U)\to U\times G$ as
\begin{align*}
\tilde \Phi:\tilde\pi^{-1}(U)&\to U_\alpha\times F\\
[p,f]&\mapsto(\pi(p),\phi(p)f).
\end{align*}
We will mainly encounter the associated bundle $\Ad(P):=P\times_c G$ and $\ad(P):=\g_P:=P\times_{\ad}\g$, where $c$ is the conjugate action defined by $c:G\ni g\mapsto c_g\in\Aut(G)$, $c_g(h)=ghg^{-1}$ for $h\in G$, and $\ad=d_{\id}(\Ad)$, and $\Ad_g=d_e(c_g)$, $e, \id$ are the unit elements in $G$ and $\g$ respectively.

The are three descriptions of gauge group (the group of gauge transformation) $\G(P)$ of a principle $G$-bundle $P(M,G)$ in the literature [^1], [^2]. Firstly, a gauge transformation can be viewed as $G$-bundle automorphisms, i.e., $S:P\to P$ is a diffeomorphism such that are preserve the fiber, $\pi\comp S=S$ and that are equivariant, $S(pg)=(S(p))g$ for all $p\in P$, $g\in G$. It can be verified that $S$ has an inverse $S^{-1}\in\Aut_M(P)$.

Secondly, a gauge transformation can be viewed as a smooth map $u:P\to G$ which is equivariant, i.e.
\[
u(pg)=g^{-1}u(p)g,\quad\forall p\in P, g\in G.
\]
In fact, given $S\in\Aut_M(P)$, it correspond (bijective) to $u$ by the relation $S(p)=pu(p)$, for any $p\in P$. It is clearly that it preserve the fiber and the equivariant follows from the action is free.

Lastly, the gauge group is isomorphic to the group of sections of $\Ad(P)$. In fact, given $u\in \G(P)$, the corresponding section $\bar u:M\to \Ad(P)$ is given by $\bar u(\pi(p))=[p,u(p)]$, for all $p\in P$.

There are also three different definitions of connections on a principle $G$-bundle $P=P(M,G)$, firstly as a horizontal distribution” $H\subset TP$ such that

  1. for every $p\in P$,
    \[
    T_pP=H_p\oplus V_p,
    \]
    where $V_p=\ker(\pi_*)\subset T_pP$ is the vertical subspace. What’s more $\pi_*|_p:H_p\to T_{\pi(p)}M$ is a linear isomorphism;
  2. for every $p\in P$ and for every $g\in G$,
    \[
    (R_g)_{*p}(H_p)=H_{pg},
    \]
    i.e., $H$ is $G$-invariant under the right action.

Secondly, a connection on $P(M,G)$ can be viewed as an equivariant $\g$-valued 1-form with fixed values in the vertical direction, i.e. $A\in\Omega_P^1(\g)$, satisfies
\begin{align*}
A_{pg}(vg)=g^{-1}A_p(v)g,\quad\forall v\in T_pP,g\in G\\
A_p(p\xi)=\xi,\quad\forall p\in P,\xi\in\g.
\end{align*}
It related with the horizontal distribution $H$ as $H_p=\ker A_p$.

Lastly, in terms of an associated bundle $\g_P$, we can view a connection as a linear map
\[
\nabla:\Omega_M^0(\g_P)\to\Omega_M^1(\g_P),
\]
satisfying the Leibnitz rule.

The set of smooth connections is denoted by $\A(P)$.

[1]: Atiyah, M. F., & Bott, R. (1983). The yang-mills equations over riemann surfaces. Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 523-615.

[2]: Wehrheim K. Uhlenbeck compactness[M]. 2004.

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