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The Atiyah-Sutcliffe determinant

This post is a continuation of my series of posts, which includes previous posts such as “the spin-statistics theorem and the Berry-Robbins problem” and “the Atiyah problem on configurations”. I will make use of notation introduced there, particularly in the latter post.

I will explain in this post, how to define the Atiyah-Sutcliffe normalized determinant function, which is a smooth complex-valued function on $C_n(\mathbb{R}^3)$.

Given a configuration $\mathbf{x} = (\mathbf{x}_1, \ldots, \mathbf{x}_n) \in C_n(\mathbb{R}^3)$ of $n$ distinct points in $\mathbb{R}^3$, I have explained in my previous post how to associate to $\mathbf{x}$ complex polynomials $p_i(\zeta)$, for $i = 1, \ldots, n$ of degree (at most) $n-1$ using the Hopf map. I will paraphrase this construction (due to Atiyah and Sutcliffe) here.

Given any $i,j \in \{1,\ldots,n\}$, with $i \neq j$, we define $\zeta_{ij}$ to be the complex number (possibly $\infty$) corresponding to the normalized direction from $\mathbf{x}_i$ to $\mathbf{x}_j$, after identifying the unit sphere $S^2$ with the Riemann sphere $\hat{S}$ (see my previous post). Let $p_{ij}(\zeta)$ be the polynomial of degree (at most) $1$ having $\zeta_{ij}$ as its unique root. Note that $p_{ij}(\zeta)$ is only defined up to scaling by a nonzero complex number. For instance, if $\zeta_{ij}$ is “finite”, one may for instance take $p_{ij}(\zeta) = \zeta – \zeta_{ij}$, while if $\zeta_{ij} = \infty$, one may take for example $p_{ij}(\zeta) = 1$. This last convention may seem odd at first, but it makes sense when one thinks of the Riemann sphere as $P^1(\mathbb{C})$ and uses a pair of homogeneous coordinates to describe the location of a point on it.

Then one may define, for $1 \leq i \leq n$:

$$p_i(\zeta) = \prod_{j \neq i} p_{ij}(\zeta).$$

Denote by $\det(p_1, \ldots, p_n)$ the determinant of the complex $n$ by $n$ matrix containing the coefficients of $p_j$ (ordered say by increasing powers of $\zeta$) as its $j$-th column. Note that, due to the scaling ambiguity of each polynomial $p_{ij}(\zeta)$, which therefore results in a scaling ambiguity of each of the $p_j(\zeta)$, the expression $\det(p_1,\ldots, p_n)$ is ill-defined. We may however divide it by an expression with exactly the same kind of scaling ambiguity and then the resulting quotient will be well defined. More precisely, Atiyah and Sutcliffe defined

$$D(\mathbf{x}) = \frac{ \det(p_1, \ldots, p_n) } { \prod_{1 \leq i < j \leq n} \det(p_{ij}, p_{ji}) }.$$

The Atiyah-Sutcliffe conjecture $1$ states that for any $\mathbf{x} \in C_n(\mathbb{R}^3)$, the $n$ polynomials $p_i(\zeta)$, for $i = 1, \ldots, n$, are linearly independent over $\mathbb{C}$. This conjecture is also equivalent to $D$ being non-vanishing on $C_n(\mathbb{R}^3)$.

The Atiyah-Sutcliffe conjecture $2$ states that for any $\mathbf{x} \in C_n(\mathbb{R}^3)$, $| D(\mathbf{x} | \geq 1$. It is clear that the Atiyah-Sutcliffe conjecture $2$ is stronger than conjecture $1$.

As of the time of writing, both conjectures are open for a general $n > 4$. The case $n = 3$ was proved by Atiyah and Sutcliffe. Conjecture 1 for $n = 4$ was proved by Eastwood and Norbury, using symbolic manipulations in Maple, shortly after the conjectures appeared. They also came close to proving conjecture 2. More precisely, they showed that for any $\mathbf{x} \in C_4(\mathbb{R}^3)$, $|D(\mathbf{x})| \geq \frac{15}{16}$, so their lower bound is just a little below $1$.

Many years later, a proof using linear programming was done by Bou Khuzam and Johnson, published in SIGMA. They even proved a stronger conjecture, called the Atiyah-Sutcliffe conjecture 3 (for $n = 4$), which is even stronger than conjecture 2. Roughly at the same time and independently, alternative proofs of conjectures 2 and 3 for $n = 4$ were also made and presented by D. Svrtan. There are currently no other general results for $n > 4$. There are some special cases (meaning for some special classes of configurations) for which some of the conjectures are known (see the works of Svrtan and Urbiha for instance).

I mention also that Sir Michael Atiyah used to say that he would offer a bottle of liquor to whoever solves the conjectures (a bottle of Vodka if they are Russian, a bottle of Arak if they are Lebanese and so on). I had the great honor to meet with him a few times and discuss with him this problem and other mathematical topics. I will probably write at some point about that in a later post.

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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.