Abstract
Recent years have seen an upsurge of interest in dynamical realizations of the superconformal group SU(1,1|2) in mechanics. Remarking that SU(1,1|2) is a particular member of a chain of supergroups SU(1,1|n) parametrized by an integer n, here we begin a systematic study of SU(1,1|n) multi-particle mechanics. A representation of the superconformal algebra su(1,1|n) is constructed on the phase space spanned by m copies of the (1,2n,2n-1) supermultiplet. We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation. All solutions to the latter in terms of root systems reveal decoupled models only. An extension of the dynamical content of the (1,2n,2n-1) supermultiplet by angular variables in a way similar to the SU(1,1|2) case is problematic.
Highlights
In this work, we initiate a systematic study of SU(1, 1|n) many-body mechanics
A representation of the superconformal algebra su(1, 1|n) is constructed on the phase space spanned by m copies of the (1, 2n, 2n−1) supermultiplet
We show that the dynamics is governed by two prepotentials V and F, and the Witten-Dijkgraaf-Verlinde-Verlinde equation for F shows up as a consequence of a more general fourth-order equation
Summary
The leftmost equation in the first line in (3.3) is a variant of the WDVV equation. With regard to the SU(1, 1|2) mechanics it has been extensively studied in [4, 6, 7, 27, 28]. Each solution of the WDVV equation satisfying (3.6) will qualify to describe some SU(1, 1|n) superconformal mechanics. The known WDVV solutions are based on socalled ∨-systems [29], which are certain deformations of Coxeter root systems. The projection antisymmetric in i and l ensures the WDVV equation; it is assumed to be fulfilled for our root systems. The symmetric projection gives further algebraic conditions: the vanishing of the double residues of the poles (α · x)−1(β · x)−1 for any pair (α, β) yields (α · β) (αiαjβkβl + βiβjαkαl) = 0. For any pair of distinct roots (α, β). This admits only the direct sum of mutually orthogonal one-dimensional (i.e. rank-one) systems. We have arrived at a no-go theorem for interacting SU(1, 1|n) mechanics based on the on-shell supermultiplet of type (1, 2n, 2n−1)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
