Flat coordinates of algebraic Frobenius manifolds in small dimensions

Wright, Johan (2024) Flat coordinates of algebraic Frobenius manifolds in small dimensions. PhD thesis, University of Glasgow.

Abstract

The orbit space of the reflection representation of a finite irreducible Coxeter groupW gives a polynomial Frobenius manifold. The intersection form g on these Frobenius manifolds is given by the W-invariant inner product on the reflection representation, while the metric η is a Lie derivative of g. Both metrics g and η are flat and flat coordinates of the metric η expressed via the flat coordinates of g are Saito polynomials, which are a distinguished set of basic invariants of the Coxeter group.

Algebraic Frobenius manifolds are typically related to quasi-Coxeter conjugacy classes in finite irreducible Coxeter groups. This class of Frobenius manifolds comes naturally after the polynomial case but it is understood less well. In this thesis we are interested in the relations between the flat coordinates of the flat metrics η and g on the algebraic Frobenius manifolds.

We find explicit relations between flat coordinates of the metric η and flat coordinates of the intersection form g for most known examples of algebraic Frobenius manifolds up to dimension 4. In all the cases, flat coordinates of the metric η appear to be algebraic functions on the orbit space of a Coxeter group.

The dual prepotentials for the polynomial Frobenius manifolds are easy to write down explicitly in terms of root systems of Coxeter groups. We find dual prepotentials for a particular family of two-dimensional algebraic Frobenius manifolds. Special functions are needed to give the answer already in this case.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Additional Information: Supported by funding from the Engineering and Physical Sciences Research Council (EPSRC).
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics
• Funder's Name: Engineering and Physical Sciences Research Council (EPSRC)
• Supervisor's Name: Feigin, Professor Misha, Valeri, Dr. Daniele and Korff, Professor Christian
• Date of Award: 2024
• Depositing User: Theses Team
• Unique ID: glathesis:2024-84525
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 02 Sep 2024 08:34
• Last Modified: 02 Sep 2024 08:35
• Thesis DOI: 10.5525/gla.thesis.84525
• URI: https://theses.gla.ac.uk/id/eprint/84525