A note on the relationship between rational and trigonometric solutions of the WDVV equations

The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class-the so-called Legendre transformations-were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.

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“…This field -called a twisted Legendre field -was first studied in [11]. However the full geometric properties were not fully described there.…”

Section: Twisted Legendre Transformation and Almost-dualitymentioning

confidence: 99%

Generalized Legendre transformations and symmetries of the WDVV equations

Strachan¹,

Stedman

2017

J. Phys. A: Math. Theor.

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“…In a literal sense, this Laurent polynomial is used for the Landau-Ginzburg (or "mirror") description of the topological sigma model of CP 1 [17,18]. The associated Frobenius structure is also a significant example of Dubrovin's duality [13,29].…”

Section: Examples: Two-variable Reductionsmentioning

confidence: 99%

Old and New Reductions of Dispersionless Toda Hierarchy

Takasaki¹

2012

SIGMA

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Abstract. This paper is focused on geometric aspects of two particular types of finitevariable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the Löwner equations. Consistency of these Löwner equations yields a radial version of the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Flat coordinates of the underlying Egorov metrics are presented.

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“…More recently, Dubrovin introduced a dual formulation of Frobenius manifolds [17]. Although it would be interesting to study symmetries between dual Frobenius manifolds [18], we do not cover that part here. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 97%

The WDVV symmetries in two-primary models

Chen¹,

Lee

2009

Theor Math Phys

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From the bi-Hamiltonian standpoint, we investigate symmetries of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV ) equations proposed by Dubrovin. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-Hamiltonian systems of hydrodynamic type. In particular, we show that the moduli space of two-primary models under symmetries of the WDVV equations can be parameterized by the polytropic exponent h. We discuss the transformation properties of the free energy at the genus-one level.

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A note on the relationship between rational and trigonometric solutions of the WDVV equations

Cited by 17 publications

References 5 publications

Generalized Legendre transformations and symmetries of the WDVV equations

Generalized Legendre transformations and symmetries of the WDVV equations

Old and New Reductions of Dispersionless Toda Hierarchy

The WDVV symmetries in two-primary models

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