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Abstract

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 A3-Frobenius manifold, as well as at the small quantum



cohomology of the Grassmannian G(2,4). 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.