I'm relatively new to the concept of holonomy so I'm currently trying to understand holonomy in the context of specific examples, i.e., Hopf fibration.
Let $\hat{n}(t) \in \mathbb{S}^2$ denote a curve and consider the (loosely defined) infinitesimal transition from $t\mapsto t+\delta t$, so that $\hat{n}\mapsto \hat{n}+\delta \hat{n}$. We can then consider a horizontal lift, which I denote as $\psi(t) \in \mathbb{C}^2$ with $|\psi| =1$ (I shall regard $\mathbb{S}^3$ and the unit sphere in $\mathbb{C}^2$ as equivalent spaces), and consider the infinitesimal holonomy $\psi \mapsto \psi +\delta\psi$. Based on this article, the change in $\psi$ can be expressed as a change in phase, plus a rotation, i.e., $$ \psi +\delta \psi=e^{i\delta A} U^{\delta t}\psi $$ Where $iA$ is the conventional connection 1-form so that $i\delta A =\int_t^{t+\delta t} iA$ and $U(t)\in SU(2)$ is defined by $$ U(t)=\exp(-iH(t)\cdot\tau), \quad H(t)=\frac{1}{2}(\hat{n}\times \delta\hat{n}) $$ And $\tau$ is the vector of Pauli matrices so that $\psi \mapsto U^{\delta t}\psi$ corresponds to an infinitesimal rotation which maps $\hat{n} \mapsto \hat{n}+\delta\hat{n}$ when projected onto $\mathbb{S}^2$.
Question. If you work through all the algebra, the above formulation indeed seems to works. However, I'm confused on how one comes up with this and how this can be formulated rigorously. It seems to be related to how the holonomy is related to the path-ordered integral of the connection, but I'm a little fuzzy on this concept and I was hoping someone could clarify and possibly provided references.
EDIT. Just as a review, the Hopf fibration from the unit sphere in $\mathbb{C}^2$ onto $\mathbb{S}^2$ can be described by the map given as follows: let $\psi\in \mathbb{C}^2$ with $|\psi|=1$, then the projection operator $P_\psi = |\psi\rangle \langle \psi|$ (physics notation) onto the span of $\psi$ is given by $$ P_\psi=\frac{1}{2}(\tau_0+\hat{n}\cdot \tau) $$ (Where $\tau_0$ is the $2\times 2$ identity operator) for some unique $\hat{n}\in \mathbb{S}^2$. Hence, $\psi\mapsto \hat{n}$ is the corresponding fibration.