(T)-structures over 2-dimensional F-manifolds: formal classification

Abstract

A $(TE)$-structure $\nabla$ over a complex manifold $M$ is a meromorphic connection defined on a holomorphic vector bundle over $\mathbb{C}\times M$, with poles of Poincar\'e rank one along $\{ 0 \} \times M.$ Under a mild additional condition (the so called unfolding condition), $\nabla$ induces a multiplication on $TM$ and a vector field on $M$ (the Euler field), which make $M$ into an $F$-manifold with Euler field. By taking the pull-backs of $\nabla$ under the inclusions $\{ z\} \times M \rightarrow \mathbb{C}\times M$ we obtain a family of flat connections on vector bundles over $M$, parameterized by $z\in \mathbb{C}^{*}$. The properties of such a family of connections give rise to the notion of $(T)$-structure. Therefore, any $(TE)$-structure underlies a $(T)$-structure but the converse is not true. The unfolding condition can be defined also for $(T)$-structures. A $(T)$-structure with the unfolding condition induces on its parameter space the structure of an $F$-manifold (without Euler field). After a brief review on the theory of $(T)$ and $(TE)$-structures, we determine normal forms for the equivalence classes, under formal isomorphisms, of $(T)$-structures which induce a given irreducible germ of $2$-dimensional $F$-manifolds.