Local Moduli of Semisimple Frobenius Coalescent Structures

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2020
• Main Authors: Cotti, Giordano, Dubrovin, Boris, Guzzetti, Davide
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2020
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210710
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Local Moduli of Semisimple Frobenius Coalescent Structures. Giordano Cotti, Boris Dubrovin and Davide Guzzetti. SIGMA 16 (2020), 040, 105 pages

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 ₂(ℂ⁴). In the latter case, we analyse in detail 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.