Szablikowski B. Classical r-matrix like approach to Frobenius manifolds and WDVV equations, talk at The Workshop on GDEq and Integrability, 11-15 October 2010, Hradec nad Moravici, Czech Republic (abstract)
Speaker: Błażej Szablikowski
Title: Classical -matrix like approach to Frobenius manifolds and WDVV equations
The theory of Frobenius manifolds was formulated by B. Dubrovin as a coordinate-free formulation of the WDVV associativity equations on a so-called prepotential function, appearing in the context of 2-dimensional topological field theories. A standard approach in the construction of Frobenius manifolds relies on the so-called Landau-Ginzburg model.
It is well known that the theory of Frobenius manifolds is closely related to the theory of bi-Hamiltonian hierarchies of hydrodynamic type. These systems together with their Hamiltonian structures are efficiently constructed by means of the so-called classical -matrix formalism.
Our goal is to exploit the classical -matrix formalism in order to build principal hierarchies of Frobenius manifolds in such a way that their Hamiltonians could be used to derive prepotential functions giving rise to associativity equations. We will present a method to construct Frobenius algebras, using the concept of double algebras, and show its relation to the linear Lie-Poisson brackets of the classical -matrix formalism applied to Poisson algebras. Besides, we derive a simple recursion relation for the generating Hamiltonians of the corresponding principal hierarchies. We illustrate this in the context of the integrable hierarchies associated with dispersionless KdV and dispersionless Toda as well as show the connectionwith with the -function in the Whitham theory proposed by Krichever and Landau-Ginzburg models.
Slides: Szablikowski B. Classical -matrix like approach to Frobenius manifolds and WDVV equations (presentation at The Workshop on Geometry of Differential Equations and Integrability, 11-15 October 2010, Hradec nad Moravici, Czech Republic).pdf