Local Moduli of Semisimple Frobenius Coalescent Structures

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.

105 pages

References in corpus (2)

• Helix Structures in Quantum Cohomology of Fano Varieties
• The Singularity of Kontsevich's Solution for $Q{H}^{\ast }(C{P}^2)$

Cited by in corpus (17)

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• On residual categories for Grassmannians
• Helix Structures in Quantum Cohomology of Fano Varieties
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• Isomonodromic Laplace Transform with Coalescing Eigenvalues and Confluence of Fuchsian Singularities
• Higher genera Catalan numbers and Hirota equations for extended nonlinear Schroedinger hierarchy
• Isomonodromic deformations along a stratum of the coalescence locus
• Notes on Non-Generic Isomonodromy Deformations
• Asymptotic solutions for linear ODEs with not-necessarily meromorphic coefficients: a Levinson type theorem on complex domains, and applications
• Taking limits in topological recursion
• Riemann-Hilbert-Birkhoff inverse problem for semisimple flat $F$-manifolds, and convergence of oriented associativity potentials
• Quantum differential equations and helices
• Frobenius manifolds and quantum groups
• Somonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold
• Integrable deformations and degenerations of some irregular singularities
• Cyclic stratum of Frobenius manifolds, Borel-Laplace $(\mathbf{\alpha },\mathbf{\beta })$-multitransforms, and integral representations of solutions of Quantum Differential Equations
• Borel $(\alpha ,\beta )$-multitransforms and Quantum Leray-Hirsch: integral representations of solutions of quantum differential equations for ${\mathbb{P}}^1$-bundles