Configurations of Points and the Symplectic Berry-Robbins Problem

TL;DR

This paper introduces a new configuration problem related to the symplectic group, proposing an explicit candidate map and proving a key conjecture for the case n=2, extending the Berry-Robbins problem to symplectic groups.

Contribution

It formulates a symplectic version of the Berry-Robbins problem, constructs an explicit candidate map, and proves the linear independence conjecture for n=2.

Findings

Proposed an explicit smooth candidate map for the symplectic Berry-Robbins problem.

Proved the linear independence conjecture for the case n=2.

Extended the Berry-Robbins problem framework to the symplectic Lie group.

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