Flat meromorphic connections of Frobenius manifolds with tt*-structure

The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back bundle π*T_M ^(1,0) by the projection π:C{double-struck}×M→M carries two natural holomorphic structures equipped with two flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dimM=2.