Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

Riley, Andrew; Strachan, Ian A. B.

doi:10.1007/s11005-006-0096-0

Cited by 8 publications

(15 citation statements)

References 17 publications

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“…We expect to find a prototype in differential geometry of Hurwitz spaces [31,32] and associated Whitham-type hierarchies [13,27], because they are generalizations of the rational reductions that we considered as a prototype of general multi-variable reductions for the genus zero case. Presumably, we should start with the genus one case, for which an explicit description of Hurwitz spaces are available in the literature [31,33,34,35] along with a candidate of Löwner-type equations [36].…”

Section: Discussionmentioning

confidence: 99%

“…We shall present these conditions later on when we consider the hodograph method. As in the case of one-variable reduction, the dual equations (53) imply that the z-functions are functionally related with each other by functions f α (z) of one variable z as (34) shows. Though Guil et al [9] further assumed the special form (35), the following consideration is not limited to that case.…”

Section: Multi-variable Reductionsmentioning

confidence: 97%

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Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

Takasaki¹,

Takebe²

2008

J. Phys. A: Math. Theor.

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This paper reconsiders finite variable reductions of the universal Whitham hierarchy of genus zero in the perspective of dispersionless Hirota equations. In the case of one-variable reduction, dispersionless Hirota equations turn out to be a powerful tool for understanding the mechanism of reduction. All relevant equations describing the reduction (Löwner-type equations and diagonal hydrodynamic equations) can be thereby derived and justified in a unified manner. The case of multi-variable reductions is not so straightforward. Nevertheless, the reduction procedure can be formulated in a general form, and justified with the aid of dispersionless Hirota equations. As an application, previous results of Guil, Mañas and Martínez Alonso are reconfirmed in this formulation.

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Section: Discussionmentioning

confidence: 99%

Section: Multi-variable Reductionsmentioning

confidence: 97%

Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

Takasaki¹,

Takebe²

2008

J. Phys. A: Math. Theor.

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“…This formula was found by the author during the researches that led to[27] and it has also appeared recently, with proof, in the work of Calaque, Enriques and Etingof[6]. However it is a classical formula; in terms of Weierstrass functions it is just the well-known Frobenius-Stickelberger equation[34] ζ(a)+ ζ(b) + ζ(c) 2 = ℘ (a) + ℘ (b) + ℘ (c) (a + b + c = 0)re-written in terms of ϑ -functions, an observation due to Prof. H.W.…”

mentioning

confidence: 77%

Weyl groups and elliptic solutions of the WDVV equations

Strachan

2010

Advances in Mathematics

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A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-VerlindeVerlinde (or WDVV) equations. This ansatz is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a ∨-system.Rational and trigonometric limits are studied together with examples of elliptic ∨-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding 'almost dual' structure is shown, in the A N and B N cases and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations.Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius-Stickelberger relations.

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“…The set of vectors U is constructed via a Landau-Ginzburg/Hurwitz space construction [18]. These satisfy the conditions…”

Section: Modular Almost-dual Solutionsmentioning

confidence: 99%

Polynomial modular Frobenius manifolds

Morrison

Strachan

2012

Physica D: Nonlinear Phenomena

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Abstract. The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class -so-called modular Frobenius manifolds -lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.

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Duality for Jacobi Group Orbit Spaces and Elliptic Solutions of the WDVV Equations

Cited by 8 publications

References 17 publications

Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

Weyl groups and elliptic solutions of the WDVV equations

Polynomial modular Frobenius manifolds

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