A Remark on Deformations of Hurwitz Frobenius Manifolds

Buryak, Alexandr; Shadrin, Sergey

doi:10.1007/s11005-010-0410-8

Cited by 4 publications

(14 citation statements)

References 13 publications

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“…Proof. Any Bergman kernel satisfies the Cauchy property (4). If the poles of dx dominate the poles of dy then dy/dx has poles only at the zeros P i of dx.…”

Section: Theorem 4 Produces a Mapmentioning

confidence: 99%

“…[15], Shramchenko [24] defined deformations of Dubrovin's Frobenius manifold structures on H g,µ . See also Buryak-Shadrin [4]. Recall that once we are given (Σ, x, {A i , B i } i=1,...,g ) and D, we define a Bergman kernel and use that to define primary differentials φ α for α ∈ D. Instead of the Bergman kernel B(p, p ′ ) Shramchenko considered arbitrary Bergman kernels ω [κ] 0,2 (p, p ′ ) on Σ which is a symmetric bidifferential on Σ × Σ, with a double pole on the diagonal of zero residue, double residue equal to 1, and no further singularities.…”

Section: Vector Fields Cycles and Meromorphic Differentials Let Us mentioning

confidence: 99%

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Primary invariants of Hurwitz Frobenius manifolds

Dunin-Barkowski¹,

Norbury²,

Orantin³

et al. 2018

Proceedings of Symposia in Pure Mathematics

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Hurwitz spaces parameterizing covers of the Riemann sphere can be equipped with a Frobenius structure. In this review, we recall the construction of such Hurwitz Frobenius manifolds as well as the correspondence between semisimple Frobenius manifolds and the topological recursion formalism. We then apply this correspondence to Hurwitz Frobenius manifolds by explaining that the corresponding primary invariants can be obtained as periods of multidifferentials globally defined on a compact Riemann surface by topological recursion. Finally, we use this construction to reply to the following question in a large class of cases: given a compact Riemann surface, what does the topological recursion compute? CONTENTS 2010 Mathematics Subject Classification. Primary 53D45; Secondary 32G15, 81T45.

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“…Proof. Any Bergman kernel satisfies the Cauchy property (4). If the poles of dx dominate the poles of dy then dy/dx has poles only at the zeros P i of dx.…”

Section: Theorem 4 Produces a Mapmentioning

confidence: 99%

Section: Vector Fields Cycles and Meromorphic Differentials Let Us mentioning

confidence: 99%

Primary invariants of Hurwitz Frobenius manifolds

Dunin-Barkowski¹,

Norbury²,

Orantin³

et al. 2018

Proceedings of Symposia in Pure Mathematics

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“…That is, the off-diagonal elements of the matrix n.d.P are equal to the corresponding elements of P, and the diagonal of n.d.P is zero. The notation n.d. is taken from [8]. It is easy to check that [δP, δQ] = 0, δ[δP, Q] = 0 (6) for any square matrices P, Q of the same size.…”

Section: Conventions and Notationmentioning

confidence: 99%

“…. , s. Symmetries of a given system (7) form a Lie algebra, where the Lie bracket of U and U ′ is defined as follows (8) are sometimes called local symmetries of (7). (In contrast to nonlocal symmetries, which are discussed below.)…”

Section: Preliminaries On Symmetries and Recursion Operatorsmentioning

confidence: 99%

“…Using the infinitesimal version of the Givental-van de Leur twisted loop group action on the space of semisimple Frobenius manifolds [14,17,18,27,28,29], in the framework of multi-KP hierarchies and Sato's semi-infinite Grassmannian, Buryak and Shadrin [8] presented an action of the Lie algebra T by nonlocal symmetries of the Darboux-Egoroff system (3). (In [8] these symmetries are called infinitesimal deformations of solutions of the Darboux-Egoroff system. )…”

Section: Introductionmentioning

confidence: 99%

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On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

Igonin

Marvan

2014

Journal of Geometry and Physics

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The Darboux-Egoroff system of PDEs with any number n ≥ 3 of independent variables plays an essential role in the problems of describing n-dimensional flat diagonal metrics of Egoroff type and Frobenius manifolds. We construct a recursion operator and its inverse for symmetries of the Darboux-Egoroff system and describe some symmetries generated by these operators.The constructed recursion operators are not pseudodifferential, but are Bäcklund autotransformations for the linearized system whose solutions correspond to symmetries of the Darboux-Egoroff system. For some other PDEs, recursion operators of similar types were considered previously by Papachristou, Guthrie, Marvan, Pobořil, and Sergyeyev.In the structure of the obtained third and fifth order symmetries of the Darboux-Egoroff system, one finds the third and fifth order flows of an (n − 1)-component vector modified KdV hierarchy.The constructed recursion operators generate also an infinite number of nonlocal symmetries. In particular, we obtain a simple construction of nonlocal symmetries that were studied by Buryak and Shadrin in the context of the infinitesimal version of the Givental-van de Leur twisted loop group action on the space of semisimple Frobenius manifolds.We obtain these results by means of rather general methods, using only the zero-curvature representation of the considered PDEs.

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Givental Graphs and Inversion Symmetry

2013

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Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarchy.2010 Mathematics Subject Classification. 53D45, 37K10.

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A Remark on Deformations of Hurwitz Frobenius Manifolds

Cited by 4 publications

References 13 publications

Primary invariants of Hurwitz Frobenius manifolds

Primary invariants of Hurwitz Frobenius manifolds

On construction of symmetries and recursion operators from zero-curvature representations and the Darboux–Egoroff system

Givental Graphs and Inversion Symmetry

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