I want to know how does holonomy group change when we apply gauge transformation.
More explicitly, for given principal bundle $G\to P\to M$, if we have a holonomy group $G_A$ w.r.t. a connection $A$, and a gauge transformation $h$ map $A$ to $h\cdot A$, produce a holonomy group $G_{h\cdot A}$. Is there any relationship between $G_A$ and $G_{h\cdot A}$?
I hope this two group are naturally isomorphic, at first I try to show that, if $\bar{\gamma}(t)$ a horizontal lifting of loop $\gamma(t)\subset M$ w.r.t. connection $A$, then $h\bar{\gamma}(t)$ a horizontal lifting w.r.t. $h\cdot A$, which means that if we consider locally $\phi_U: P|_U\cong U\times G$ and write $(\phi^{-1})^*A=g^{-1}dg+g^{-1}ag$, if
$g^{-1}\frac{d}{dt}g+g^{-1}a(\gamma'(t))g=0.$
Then,
$g^{-1}h^{-1}\frac{d}{dt}hg+g^{-1}a(h^*\gamma'(t))g=0.$
But this seems is not correct, can anyone help me? Thanks!
I think the easiest way to see this, is by using horizontal distributions. A path $\widetilde{\gamma}(t):I\to P$ is horizontal if and only if for all $t_0\in I$, it holds that $\frac{d}{dt}|_{t_0}\widetilde{\gamma}(t)\in\mathcal{H}_{\widetilde{\gamma}(t_0)}^A$, where $\mathcal{H}^A$ denotes the horizontal distribution determined by the connection. Write a gauge transformation in local coordinates as $(x,g)\mapsto (x,g\cdot h(x))$ for some function $h:M\to G$. Then we would like to show that $$\frac{d}{dt}|_{t_0}(\widetilde{\gamma}(t)\cdot h(\widetilde{\gamma}(t)))\in\mathcal{H}^{h\cdot A}_{\widetilde{\gamma}(t_0)\cdot h(\widetilde{\gamma}(t_0))}$$ However, the horizontal distribution of the connection $h\cdot A$ is defined by $\mathcal{H}^{h\cdot A}_{p\cdot h(x)}=dR_{h(x)}\mathcal{H}^A_p$. Clearly, one has $$\frac{d}{dt}|_{t_0}(\widetilde{\gamma}(t)\cdot h(\widetilde{\gamma}(t)))=dR_{h(\widetilde{\gamma}(t_0))}\frac{d}{dt}|_{t_0}\widetilde{\gamma}(t_0)$$ which lies in $dR_{h(\widetilde{\gamma}(t_0))}\mathcal{H}^A_{\widetilde{\gamma}(t_0)}$. Therefore, $\widetilde{\gamma}\cdot h$ is indeed horizontal with respect to $h\cdot A$.