$N$-dimensional flat F-manifold $(M,\nabla,\circ)$ is called *semisimple* if $T_pM$ has a basis of idempotents $\pi_1,\dots,\pi_N$ satisfying $\pi_k \circ \pi_l = \delta_{k,l} \pi_k$. Moreover, locally around such a point one can choose coordinates $u^i$ such that $\frac{\d}{\d u^k}\circ\frac{\d}{\d u^l}=\delta_{k,l}\frac{\d}{\d u^k}$. These coordinates are called the [*canonical coordinates*]{}. In particular, this means that the semisimplicity is an open property. The flat F-manifold $(M,\nabla,\circ)$ is called semisimple, if a generic point of $M$ is semisimple.

Frobenius manifolds
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For a complex analytic manifold $M$ we denote by ${\mathcal{T}}_M$ the analytic tangent sheaf of $M$.

A flat F-manifold $(M,\nabla,\circ)$ is called a Frobenius manifold if the tangent spaces $T_pM$ are equipped with a symmetric non-degenerate bilinear form $\eta$ analytically depending on the point $p\in M$ such that $\nabla \eta = 0$ and for any $X,Y,Z \in \T_M$ the following condition is satisfied: $$\eta(X\circ Y, Z) = \eta(X,Y\circ Z).$$ The connection $\nabla$ is then the Levi-Civita connection associated to the form $\eta$. A Frobenius manifold will be denoted by the triple $(M,\eta,\circ)$. The form $\eta$ is traditionally called a metric.

Let $(M,\eta,\circ)$ be a Frobenius manifold and consider the flat coordinates $t^1,\ldots,t^N$ of the metric $\eta$ and the vector potential $(F^1,\ldots,F^N)$. Then locally there exists 