3-dimensional F-manifolds

Basalaev, Alexey; Hertling, Claus

doi:10.1007/s11005-021-01432-y

Cited by 3 publications

(1 citation statement)

References 20 publications

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“…As we will see, the classification of germs of 2-dimensional F -manifolds will play a crucial role. Recently significant progress has been made in the classification of germs of 3-dimensional F -manifolds; but as seen in [3] this classification is much more vast and complicated than the two-dimensional case.…”

Section: F Reyesmentioning

confidence: 99%

Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

Reyes

2023

SIGMA

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We study the family of ordinary differential equations associated to a Dubrovin-Frobenius manifold along its caustic. Upon just loosing an idempotent at the caustic and under a non-degeneracy condition, we write down a normal form for this family and prove that the corresponding fundamental matrix solutions are strongly isomonodromic. It is shown that the exponent of formal monodromy is related to the multiplication structure of the Dubrovin-Frobenius manifold along its caustic.

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Section: F Reyesmentioning

confidence: 99%

Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

Reyes

2023

SIGMA

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Regular F-manifolds with eventual identities

Perletti,

Strachan

2024

J. Phys. A: Math. Theor.

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Given an F-manifold one may construct a dual multiplication (generalizing the idea of an almost-dual Frobenius manifold introduced by Dubrovin) using a so-called eventual identity, the definition of which ensure that the dual object is also an F-manifold. In this paper we solve the equations for an eventual identity for a regular (so non-semi-simple) F-manifold and construct a dual coordinate system in which dual multiplication is preserved. As an application, families of Nijenhuis operators are constructed.

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Nijenhuis operators with a unity and F$F$‐manifolds

Antonov,

Konyaev

2024

Journal of London Math Soc

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The core object of this paper is a pair , where is a Nijenhuis operator and is a vector field satisfying a specific Lie derivative condition, that is, . Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for ‐regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and ‐manifolds. Specifically, we prove that the class of regular ‐manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding ‐manifolds around singularities.

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3-dimensional F-manifolds

Abstract: F-manifolds are complex manifolds with a multiplication with unit on the holomorphic tangent bundle with a certain integrability condition. Here, the local classification of 3-dimensional F-manifolds with or without Euler fields is pursued.

Cited by 3 publications

References 20 publications

Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

Isomonodromic Deformations Along the Caustic of a Dubrovin-Frobenius Manifold

Regular F-manifolds with eventual identities

Nijenhuis operators with a unity and F$F$‐manifolds

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