Connection matrices and Berry phases on the homogeneous spaces $\mathcal{D}^J_1$ and $\mathcal{X}^J_1$ attached to Jacobi groups

We analyze the relation between connection matrices and Berry phases on the Siegel-Jacobi disk $\mathcal{D}^J_1$ and Siegel-Jacobi upper half-plane $\mathcal{X}^J_1$ and we explicitly compute the connection matrices and the Berry connections on these homogeneous spaces in different coordinates.