Frobenius structures, Coxeter discriminants, and supersymmetric mechanics

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

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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: 25 Feb 2020 10:10
URI: http://theses.gla.ac.uk/id/eprint/79019

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