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 SIGMA, 2020, Volume 16, 040, 105 pp. (Mi sigma1577)

Local Moduli of Semisimple Frobenius Coalescent Structures

Giordano Cottia, Boris Dubrovinb, Davide Guzzettib

a Max-Planck-Institut für Mathematik, Vivatsgasse 7 - 53111 Bonn, Germany
b SISSA, Via Bonomea 265 - 34136 Trieste, Italy

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 $A_3$-Frobenius manifold, as well as at the small quantum cohomology of the Grassmannian $\mathbb G_2(\mathbb C^4)$. 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.

Keywords: Frobenius manifolds, isomonodromic deformations, singularity theory, quantum cohomology, derived categories.

 Funding Agency Grant Number EU Framework Programme for Research and Innovation 778010 IPaDEGAN The third author is a member of the European Union's H2020 research and innovation programme under the Marie Skłlodowska-Curie grant No. 778010 IPaDEGAN.

DOI: https://doi.org/10.3842/SIGMA.2020.040

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Full text: https://www.imath.kiev.ua/~sigma/2020/040/
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Bibliographic databases:

ArXiv: 1712.08575
MSC: 34M56, 53D45, 18E30
Received: June 18, 2019; in final form April 13, 2020; Published online May 7, 2020
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Citation: Giordano Cotti, Boris Dubrovin, Davide Guzzetti, “Local Moduli of Semisimple Frobenius Coalescent Structures”, SIGMA, 16 (2020), 040, 105 pp.

Citation in format AMSBIB
\Bibitem{CotDubGuz20}
\by Giordano~Cotti, Boris~Dubrovin, Davide~Guzzetti
\paper Local Moduli of Semisimple Frobenius Coalescent Structures
\jour SIGMA
\yr 2020
\vol 16
\papernumber 040
\totalpages 105
\mathnet{http://mi.mathnet.ru/sigma1577}
\crossref{https://doi.org/10.3842/SIGMA.2020.040}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85084845991}