N=4 mechanics, WDVV equations and roots (note unusual date!)

by Olaf Lechtenfeld (Universitaet Hannover)

Monday 17 Mar 2008, 14:00 → 15:00 Europe/Zurich

TH Theory Conference Room (CERN)

I report on the recent work arXiv:0802.4386 of Galajinsky, Polovnikov and myself (sorry!) about the classification of N=4 superconformal n-particle quantum mechanics on the real line. The goal is to extend Calogero-type models to systems with su(1,1|2) invariance. The minimal ansatz for the Hamiltonian needs a classical potential U and a quantum potential F. The closure of the algebra says that F satisfies the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations of topological field theory and enters as a background in the pde for U. The solution for F is expressed in terms of deformations of finite Coxeter root systems, extending previously known WDVV solutions. In the A_n case it is given by orthocentric n-simplices, pointing towards a general classification by deformed Coxeter polytopes. We then try to turn on the classical potential U (which provides the central charge) in these F backgrounds, and find this impossible for more than three particles with irreducible interactions. The prime example is based on the G_2+A_1 root system.