Flat meromorphic connections of Frobenius manifolds with tt*-structure

Details Flat meromorphic connections of Frobenius manifolds with tt*-structure Flat meromorphic connections of Frobenius manifolds with tt*-structure

2011-05-08

arxiv.org/abs/1105.1542v1

Journal of Geometry and Physics 62 (2012), pp. 37-46

Mathematics

Differential Geometry

Scientific paper

10.1016/j.geomphys.2011.09.006

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 of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with 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 dim M = 2.

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