Local Moduli of Semisimple Frobenius Coalescent Structures

TL;DR

This paper extends the analytic theory of Frobenius manifolds to semisimple points with coalescing eigenvalues, clarifies monodromy data ambiguities, and provides explicit computations in singularity and quantum cohomology contexts.

Contribution

It introduces a detailed analysis of monodromy data at coalescing points and connects these data to derived category mutations, advancing the understanding of Frobenius manifolds.

Findings

Computed monodromy data at Maxwell Stratum of A3-Frobenius manifold

Analyzed monodromy data in small quantum cohomology of G(2,4)

Linked monodromy data to characteristic classes of mutations in derived categories

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.