Appendix 1. Moduli spaces of meromorphic connections, quiver varieties, and integrable deformations

Abstract

This is a note in which we first review symmetries of moduli spaces of stable meromorphic connections on trivial vector bundles over the Riemann sphere, and next discuss symmetries of their integrable deformations as an application. We shall give an interpretation of the list of 4-dimensional Painlevé type equations given by Kawakami-Nakamura-Sakai from our classification of 4-dimensional moduli spaces of connections. In the study of the symmetries, a realization of the moduli spaces as quiver varieties is given and plays an essential role.