Moduli spaces of d-connections and difference Painlevé equations

We show that difference Painlevé equations can be interpreted as isomorphisms of moduli spaces of difference connections (d-connections) on P1 with given singularity structure. In particular, we derive a difference equation that lifts to an isomorphism between A2(1)*-surfaces in Sakai's classification (see [29]); it degenerates to both difference Painlevé V and classical (differential) Painlevé VI equations. This difference equation has been known before under the name of asymmetric discrete Painlevé IV equation