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 (12)

• Gromov-Witten classes, quantum cohomology, and enumerative geometry
• The Verlinde Algebra And The Cohomology Of The Grassmannian
• The structure of 2D semi-simple field theories
• Geometry and analytic theory of Frobenius manifolds
• (Semi)simple exercises in quantum cohomology
• Frobenius Manifolds: Natural submanifolds and induced bi-Hamiltonian structures
• Constraints For Topological Strings In $D\ge 1$
• Topological Mirrors and Quantum Rings
• Frobenius submanifolds
• Helix Structures in Quantum Cohomology of Fano Varieties
• Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers
• The Singularity of Kontsevich's Solution for $Q{H}^{\ast }(C{P}^2)$

Cited by in corpus (19)

• On the origins of Riemann-Hilbert problems in mathematics
• On residual categories for Grassmannians
• Helix Structures in Quantum Cohomology of Fano Varieties
• Coalescence Phenomenon of Quantum Cohomology of Grassmannians and the Distribution of Prime Numbers
• Degenerate Riemann-Hilbert-Birkhoff problems, semisimplicity, and convergence of WDVV-potentials
• 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
• A construction of Frobenius manifolds from stability conditions
• 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