Topics on the theory of Frobenius manifolds

• Autor(a) principal: Castro, Thales Novelli
• Data de Publicação: 2024
• Tipo de documento: Dissertação
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/55/55135/tde-22052024-142742/
• topic: Differential geometry; Estruturas de Poisson; Frobenius manifolds; Geometria diferencial; Hamiltonian; Hamiltoniano; Poisson structures; Sistemas de tipo hidrodinâmico.; Systems of hydrodyamic type; Variedades de Frobenius

This work aims to present a connection between Frobenius manifolds, a concept of differential geometry which shows up in topological field theory, and systems of differential equations of hydrodynamic type. Formulated by Dubrovin in the 1990s, Frobenius manifolds aim to give a geometric interpretation to the so-called associativity equations, or WDVV equations, a nonlinear system whose solution is a quasi-homogeneous function describing structure constants of an associative algebra. Hydrodynamic-type systems arise, as the name suggests, in studies on fluid mechanics, especially gas dynamics. From the geometric approach, the relation between these two entities is given by means of a Hamiltonian representation for these equations, arising from a specific type of Poisson structure. Specifically, the work presents an overview of the main geometric aspects of the theory, leading to a theorem according to which the loop-space of a Frobenius manifold carries a so-called bi-Hamiltonian structure of hydrodynamic type.