Configurations of points and the symplectic Berry-Robbins problem

Abstract

We present a new problem on configurations of points, which is a new version of a similar problem by Atiyah and Sutcliffe, except it is related to the Lie group Sp(n), instead of the Lie group U(n). Denote by h a Cartan algebra of Sp(n), and Δ the union of the zero sets of the roots of Sp(n) tensored with R3, each being a map from h⊗R3→R3. We wish to construct a map (h⊗R3)∖Δ→Sp(n)/Tn which is equivariant under the action of the Weyl group Wn of Sp(n) (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of Sp(n), and Tn is the diagonal n-torus. The existence of such a map was proved by Atiyah and Bielawski in a more general context. We present an explicit smooth candidate for such an equivariant map, which would be a genuine map provided a certain linear independence conjecture holds. We prove the linear independence conjecture for n=2.