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I'm trying to understand non abelian berry phase, but I'm having trouble with it. I'm reading the article "Appearance of Gauge Structure in Simple Dynamical Systems" of Wilczek and Zee (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.52.2111).

After they establish the relevant equations for the berry phase, a particular system is studied for illustration of the non abelian phase. The hamiltonian of the system is described as $H=R(t)H_0 R(t)^{-1}$ where $H_0$ is a $n+1$ dimensional matrix with entries $(H_0)_{ij}=0$ unless $i=j=n+1$. and $R(t)=R(\theta(t))$ is the rotation matrix defined as:

$$R=exp(i\theta_n T_{n,n+1})...exp(i\theta_2 T_{2,n+1})exp(i\theta_1 T_{1,n+1}) $$

It is not exactly mentioned in the article what are the $T_{n,n+1},...,T_{1,n+1}$, what are these? are they matrices? generators of the rotations on the $(n,n+1)$ plane for example?. As I understand it, $R$ is an element of $SO(n+1)$. There is a commentary about the embedding of $SO(n)$ in $SO(n+1)$ which depends on time which is not entirely clear for me either. Could someone explain the meaning of this comment?.

Finally, the calculation of the berry gauge potential and the gauge field is confusing me. Int he article it is stated that the berry gauge potential is

$$A_\mu = \pi R^{-1}\frac{\partial R}{\partial \theta^{\mu}} \pi$$

where $\pi$ is the projection on the first $n$ degenerate states. In the article it is mentioned that this expression for the gauge potential comes from equation (11) but it is not entireley clear how it is derived from this equation, since equation (11) concerns quantum states, and this definition of the gauge potential concerns only operators. Could someone clear this derivation?. The field strength is

$$-F_{\mu\nu} = \pi R^{-1}\frac{\partial R}{\partial \theta^{\mu}}(1-\pi) R^{-1}\frac{\partial R}{\partial \theta^{\mu}}-(\mu \leftrightarrow\nu)$$.

In this definition there seems to be a typo in the second $\theta$ derivative which should be $\theta^{\nu}$, but yet I dont understand how is this field strength derived from the gauge potential. Could someone help by saying how it is derived? I would greatly appreciate help for understanding all this questions which would help me in understanding the article.

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