Frobenius structures, Coxeter discriminants, and supersymmetric mechanics

Antoniou, Georgios (2020) Frobenius structures, Coxeter discriminants, and supersymmetric mechanics. PhD thesis, University of Glasgow.

Abstract

This thesis contains two directions both related to Frobenius manifolds.
In the first part we deal with the orbit space $M_W = V/W$ of a finite Coxeter group $W$ acting in its reflection representation $V$. The orbit space $M_W$ carries the structure of a Frobenius manifold and admits a pencil of flat metrics of which the Saito flat metric $η$, defined as the Lie derivative of the $W$-invariant form $g$ on $V$ is the key object. In the main result of the first part we find the determinant of Saito metric restricted on the Coxeter discriminant strata in $M_W$ . It is shown that this determinant in the flat coordinates of the form $g$ is proportional to a product of linear factors. We also find multiplicities of these factors in terms of Coxeter geometry of the stratum.
In the second part we study $N = 4$ supersymmetric extensions of quantum mechanical systems of Calogero–Moser type. We show that for any $∨$-system, in particular, for any Coxeter root system, the corresponding Hamiltonian can be extended to the supersymmetric Hamiltonian with $D(2,1;α)$ symmetry. We also obtain $N = 4$ supersymmetric extensions of Calogero–Moser–Sutherland systems. Thus, we construct supersymmetric Hamiltonians for the root systems $BC_N$, $F_4$ and $G_2$.

• Item Type: Thesis (PhD)
• Qualification Level: Doctoral
• Keywords: Frobenius structures, Coxeter discriminants, supersymmetric mechanics.
• Subjects: Q Science > QA Mathematics
• Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Supervisor's Name: Feigin, Dr. Misha
• Date of Award: 2020
• Depositing User: MR Georgios Antoniou
• Unique ID: glathesis:2020-79019
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 25 Feb 2020 10:07
• Last Modified: 31 Aug 2022 09:11
• Thesis DOI: 10.5525/gla.thesis.79019
• URI: https://theses.gla.ac.uk/id/eprint/79019