Let $M$ be compact riemannian manifold without boundary, and $G$ be a compact Lie group. A \\emph{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 $g\\in G$.<\/p>\n
Now for any other manifold $F$ on which $G$ acts on the left $G\\times F\\ni(g,f)\\mapsto gf\\in F$, the \\emph{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
\n\\begin{align*}
\n\\tilde \\Phi:\\tilde\\pi^{-1}(U)&\\to U_\\alpha\\times F\\\\
\n[p,f]&\\mapsto(\\pi(p),\\phi(p)f).
\n\\end{align*}
\nWe 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.<\/p>\n
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 \\emph{$G$-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 $S^{-1}\\in\\Aut_M(P)$.<\/p>\n
Secondly, a gauge transformation can be viewed as a smooth map $u:P\\to G$ which is equivariant, i.e.
\n\\[
\nu(pg)=g^{-1}u(p)g,\\quad\\forall p\\in P, g\\in G.
\n\\]
\nIn 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.<\/p>\n
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 $p\\in P$.<\/p>\n
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
\n\\begin{enumerate}
\n\\item for every $p\\in P$,
\n\\[
\nT_pP=H_p\\oplus V_p,
\n\\]
\nwhere $V_p=\\ker(\\pi_*)\\subset T_pP$ is the vertical subspace. What’s more $\\pi_*|_p:H_p\\to T_{\\pi(p)}M$ is a linear isomorphism;
\n\\item for every $p\\in P$ and for every $g\\in G$,
\n\\[
\n(R_g)_{*p}(H_p)=H_{pg},
\n\\]
\ni.e., $H$ is $G$-invariant under the right action.
\n\\end{enumerate}<\/p>\n
Secondly, a \\emph{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
\n\\begin{align*}
\nA_{pg}(vg)=g^{-1}A_p(v)g,\\quad\\forall v\\in T_pP,g\\in G\\\\
\nA_p(p\\xi)=\\xi,\\quad\\forall p\\in P,\\xi\\in\\g.
\n\\end{align*}
\nIt related with the horizontal distribution $H$ as $H_p=\\ker A_p$.<\/p>\n
Lastly, in terms of an associated bundle $\\g_P$, we can view a connection as a linear map
\n\\[
\n\\nabla:\\Omega_M^0(\\g_P)\\to\\Omega_M^1(\\g_P),
\n\\]
\nsatisfying the \\emph{Leibnitz rule}.<\/p>\n
The set of smooth connections is denoted by $\\A(P)$.<\/p>\n
[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.<\/p>\n
[2]: Wehrheim K. Uhlenbeck compactness[M]. 2004.<\/p>\n","protected":false},"excerpt":{"rendered":"
Let $M$ be compact riemannian manifold w …<\/p>\n