WDVV Equations and Frobenius Manifolds

Abstract A well-known construction of B. Dubrovin and K. Saito endows the parameter space of a universal unfolding of a simple singularity with a Frobenius manifold structure. In our paper, we present a generalization of this construction for the singularities of types $A$ and $D$ that gives a solution of the open WDVV equations. For the $A$-singularity, the resulting solution describes the intersection numbers on the moduli space of $r$-spin disks, introduced recently in a work of the 2nd author, E. Clader and R. Tessler. In the 2nd part of the paper, we describe the space of homogeneous polynomial solutions of the open WDVV equations associated to the Frobenius manifolds of finite irreducible Coxeter groups.

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Classicalr-matrix like approach to Frobenius manifolds, WDVV equations and flat metrics

Journal of Physics A Mathematical and Theoretical ◽

10.1088/1751-8113/48/31/315203 ◽

2015 ◽

Vol 48 (31) ◽

pp. 315203 ◽

Cited By ~ 3

Author(s):

Błażej M Szablikowski

Keyword(s):

Wdvv Equations ◽

Frobenius Manifolds

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Second-order integrable Lagrangians and WDVV equations

Letters in Mathematical Physics ◽

10.1007/s11005-021-01403-3 ◽

2021 ◽

Vol 111 (2) ◽

Author(s):

E. V. Ferapontov ◽

M. V. Pavlov ◽

Lingling Xue

Keyword(s):

Lagrangian Density ◽

Theta Functions ◽

Second Order ◽

Elementary Functions ◽

Lagrange Equations ◽

Wdvv Equations ◽

Integrability Conditions ◽

Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

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