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

Source abstract

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 $E^{(1)}_7$ and $E^{(1)}_8$, respectively. Furthermore, we realize the extended affine Weyl group symmetries of the space of initial conditions in terms of connections.