Moduli spaces of logarithmic connections associated with difference Painlevé equations of type $E_n^{(1)}$

In this paper, we provide an explicit correspondence between moduli spaces of parabolic logarithmic connections on the projective line with spectral type $(2^2,1^4,1^4)$ and $(3^2,2^3,1^6)$, and the spaces of initial conditions for difference Painlevé equations with symmetry of type $E_7^{(1)}$ and $E_8^{(1)}$, respectively. Furthermore, we realize the extended affine Weyl group symmetries of the space of initial conditions in terms of connections.