I've been trying to understand the Berry phase (abelian/non-abelian) as the holonomy of some "natural connection". I almost have all the pieces together, but there are a few parts that are a bit fuzzy to me.
Setup. For simplicity, let me only consider finite-dimensions. Indeed, consider a smooth mapping $H$ from $\mathbb{S}^1$ into the space of Hamiltonians (hermitian operators), parametrized by $H(s)$ so that $H(0)=H(1)$. Let me assume that along this closed loop, $||H(s)-H(0)||$ remains sufficiently small for all $s$ so that a smooth projection operator $P(s)$ onto the (possibly degenerate) ground state eigenspace of $H(s)$ is well-defined. The Berry phase accumulated is (more or less) a unitary transform $W(s)$ mapping between $P(0)$ and $P(s)$ (i.e., $P(s)W(s)=W(s)P(0)$) obtained from the following differential equation [1] $$ W'(s)=[P'(s), P(s)]W(s) $$
Problem. Given the equation, one can obviously check that it satisfies all the necessary properties and gives the Berry phase. However, conversely, there should be something "natural" to this differential equation so that one can easily see that "yes, this is indeed the equation that we want". One way I attempted to formulate this was to think about holonomy (as done in [2], though I don't completely understand how to connect his paper with what I'm formulating here). Here is my attempt. Let the total space $\mathcal{P}$ be the space of projection operators and let the group of unitary operators $\mathcal{U}$ act on $\mathcal{P}$ via unitary action, i.e., $(P,U)\mapsto P\cdot U \equiv U^* PU$ is a right action. It's then clear that $P(s)$ should denote the horizontal lift of some "base-space path $\gamma \in \mathcal{P}/\mathcal{U}$" so that the holonomy of the base-space path is given by the unitary $W(s)$, i.e.,$P(s)\cdot W(s) = P(0)$ or more specifically, $W(s)^*P(s)W(s)=P(0)$. All horizontal lifts are found by the connection, which I think should be of the form, $$ W^{-1} [P,dP] W +W^{-1} dW $$ (Indeed, setting the formula to $=0$ is exactly the differential equation, and also has the "right form" of a connection).
Question. If my attempt in formulation is correct, what is so natural about the connection $W^{-1} [P,dP] W +W^{-1} dW $? Please be advised my knowledge in holonomy/differential geometry is relatively primitive.
[1] Kato, T. (1950). On the adiabatic theorem of quantum mechanics. Journal of the Physical Society of Japan, 5(6), 435-439.
[2] Simon, B. (1983). Holonomy, the quantum adiabatic theorem, and Berry's phase. Physical Review Letters, 51(24), 2167.
