Frobenius structures, Coxeter discriminants, and supersymmetric mechanics

[摘要] 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 $G_2$.

[发布机构] University:University of Glasgow;Department:School of Mathematics and Statistics

[关键词] Frobenius structures, Coxeter discriminants, supersymmetric mechanics.