Givental graphs and inversion symmetry

TL;DR

This paper explores the inversion symmetry of Frobenius manifolds through Givental graphs, revealing its connection to the Givental group action and implications for Schlesinger transformations and hierarchy Hamiltonians.

Contribution

It provides a novel interpretation of inversion symmetry using Givental graphs, linking it to the Givental group action and its effects on related transformations and hierarchies.

Findings

Interpretation of inversion symmetry via Givental graphs

Connection between inversion symmetry and Givental group action

Implications for Schlesinger transformations and hierarchy Hamiltonians

Abstract

Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarchy.