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 $\mathrm{Sp}(n)$, instead of the Lie group $U(n)$. Denote by $\mathfrak{h}$ a Cartan algebra of $\mathrm{Sp}(n)$, and $\Delta$ the union of the zero sets of the roots of $\mathrm{Sp}(n)$ tensored with ${\mathbb{R}}^3$, each being a map from $\mathfrak{h}\otimes {\mathbb{R}}^3\to {\mathbb{R}}^3$. We wish to construct a map $(\mathfrak{h}\otimes {\mathbb{R}}^3)\backslash \Delta \to \mathrm{Sp}(n)/T^n$ which is equivariant under the action of the Weyl group $W_n$ of $\mathrm{Sp}(n)$ (the symplectic Berry-Robbins problem). Here, the target space is the flag manifold of $\mathrm{Sp}(n)$, and $T^n$ 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 $n=2$.