Local Moduli of Semisimple Frobenius Coalescent Structures

We extend the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues of the operator of multiplication by the Euler vector field. We clarify which freedoms, ambiguities and mutual constraints are allowed in the definition of monodromy data, in view of their importance for conjectural relationships between Frobenius manifolds and derived categories. Detailed examples and applications are taken from singularity and quantum cohomology theories. We explicitly compute the monodromy data at points of the Maxwell Stratum of the $A_3$-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian $\mathbb G_2\big(\mathbb C^4\big)$. In the latter case, we analyse in details the action of the braid group on the monodromy data. This proves that these data can be expressed in terms of characteristic classes of mutations of Kapranov's exceptional $5$-block collection, as conjectured by one of the authors.