The Atiyah Problem on Configurations

# The Atiyah Problem on Configurations

This post is a continuation of my previous post “the spin-statistics theorem and the Berry-Robbins problem”. The reader is referred to that post to understand the origins of the Atiyah problem on configurations and the Atiyah-Sutcliffe conjectures.

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^3$. Given $\mathbf{x} = (\mathbf{x}_1,\ldots,\mathbf{x}_n) \in C_n(\mathbb{R}^3)$, so that each $\mathbf{x}_i \in \mathbb{R}^3$, for $i = 1, \ldots, n$ and the $\mathbf{x}_i$ are distinct, we will associate to the configuration $\mathbf{x}$ $n$ complex polynomials $p_i(\zeta)$, for $i = 1, \ldots, n$ of degree (at most) $n-1$, each of them only defined up to multiplication by a nonzero complex factor.

Let $i \in \{1,\ldots,n\}$. For each $j \neq i$, $1 \leq j \leq n$, we form the normalized vector

$$v_{ij} = \frac{\mathbf{x}_j-\mathbf{x}_i}{\lVert \mathbf{x}_j-\mathbf{x}_i \rVert} \in S^2.$$

Using stereographic projection, there is a natural complex structure on $S^2$, which turns it into the Riemann sphere $\hat{S}$ and a natural complex coordinate $\zeta$ defined on $S^2 \setminus \{N\}$, where $N = (0,0,1)^T$ is the “North” pole, so to speak.

So $v_{ij}$ corresponds to a complex number $\zeta_{ij}$ if $v_{ij} \neq N$ and to the point at infinity $\infty$ on the Riemann sphere $\hat{S}$ if $v_{ij} = N$.

Let $p_i(\zeta)$ be the complex polynomial of degree (at most) $n-1$, having the $\zeta_{ij}$, for $1 \leq j \leq n$ and $j \neq i$, as its set of roots, taking multiplicity into account. Note that each polynomial $p_i(\zeta)$ is only defined up to a nonzero complex factor.

Atiyah conjectured that given any $\mathbf{x} \in C_n(\mathbb{R}^3)$, the corresponding polynomials $p_i(\zeta)$, for $1 \leq i \leq n$, are always linearly independent over $\mathbb{C}$. This is also known as the Atiyah-Sutcliffe conjecture $1$. If it is true, then it would imply that the map

$$\mathbf{x} \mapsto (p_1(\zeta), \ldots, p_n(\zeta))$$

gives, after orthogonalization, a smooth solution of the Berry-Robbins problem.

This is a problem I have been working on for a very long time. It is still open for general $n$, though it was proved to be true by Atiyah for $n = 3$ and by Eastwood and Norbury for $n = 4$.

There is more to the story. Indeed, Atiyah and Sutcliffe have defined a very interesting normalized determinant function $D$

$$D: C_n(\mathbb{R}^3) \to \mathbb{C}$$

whose non-vanishing is equivalent to the linear independence conjecture above (the Atiyah-Sutcliffe conjecture 1). However, Atiyah and Sutcliffe made a stronger conjecture, known as the Atiyah-Sutcliffe conjecture 2, which states that

$$|D(\mathbf{x})| \geq 1 \text{ for any \mathbf{x} \in C_n(\mathbb{R}^3).}$$

However, I should tell the reader first how to define $D$. This will be the subject of a future post.