The Atiyah–Sutcliffe conjectures on configurations of points: linear independence of the direction polynomials and the determinant volume inequalities for general n

Atiyah and Sutcliffe's paper 'The geometry of point particles' (2002) associated to every configuration of $n$ distinct points in $\mathbb{R}^3$ a normalized determinant $D$ — the coefficient determinant of the $n$ degree-$(n-1)$ 'direction polynomials' $p_i$ divided by the product of their pairwise two-by-two determinants — and posed three conjectures: the polynomials are always linearly independent (Conjecture 2.1, equivalent to $D\neq0$); the volume $V=|D|$ satisfies $V\geq1$ (Conjecture 4.1); and the inductive inequality $V^{\,n-2}\geq\prod_i V_i$ over one-point deletions (Conjecture 4.2). These statements are the missing ingredient for Atiyah's explicit $\Sigma_n$-equivariant flag map realizing the Berry–Robbins approach to the geometry of spin-statistics, and they underlie the multi-particle energy $E=-\log V$ whose minimizers reproduce fullerene-, Skyrmion- and Coulomb-type polyhedra. All three conjectures are theorems for $n\leq4$ (elementary for $n\leq3$; Eastwood–Norbury 2001 and Bou Khuzam–Johnson 2014 for $n=4$), and the Minkowski-space variant has been disproved for every $n\geq3$ (Liu 2026), but in Euclidean space they remain open for every $n\geq5$. The problem is to prove or refute Conjectures 2.1, 4.1 and 4.2 for general configurations of $n$ points in $\mathbb{R}^3$.

Background

Identical quantum particles in three-dimensional space are described by configuration spaces rather than by individual coordinates: for $n$ particles one considers $\mathcal{C}_n(\mathbb{R}^3)$, the space of ordered $n$-tuples of pairwise distinct points, on which the symmetric group $\Sigma_n$ acts by permuting the labels. In a 1997 study of the spin–statistics connection (the theorem linking particle-exchange symmetry to spin), Berry and Robbins asked for a natural identification between particle exchange and the rotation of spin states, formalized as a continuous $\Sigma_n$-equivariant map from $\mathcal{C}_n(\mathbb{R}^3)$ to the flag manifold $U(n)/U(1)^n$, the space of $n$ mutually orthogonal complex lines in $\mathbb{C}^n$ (orthonormal bases up to phase), with $\Sigma_n$ acting on the target through its $n$-dimensional permutation representation. Atiyah constructed an explicit candidate for such a map, relaxing unitarity to a map into $GL(n,\mathbb{C})/(\mathbb{C}^*)^n$, as follows (Atiyah 2001; Atiyah–Sutcliffe 2002). Given distinct points $x_1,\dots,x_n\in\mathbb{R}^3$, let $v_{ij}=(x_j-x_i)/|x_j-x_i|$ be the unit vector pointing from $x_i$ toward $x_j$, and let $t_{ij}\in\mathbb{CP}^1$ be its image under the stereographic identification of the unit sphere $S^2$ with the Riemann sphere (the complex plane completed by a point at infinity); in Euclidean space $t_{ji}$ is exactly the antipode of $t_{ij}$. Choosing for each $t_{ij}$ a nonzero linear form $u_{ij}$ on $\mathbb{C}^2$ with root $t_{ij}$ (a binary form, i.e. a homogeneous linear polynomial in two variables), one forms the degree-$(n-1)$ binary forms

$$p_i \;=\; \prod_{j\neq i} u_{ij} \;\in\; S^{n-1}(\mathbb{C}^2)\cong\mathbb{C}^n,$$

where $S^{n-1}(\mathbb{C}^2)$ is the $(n-1)$-st symmetric power of $\mathbb{C}^2$, identified with the irreducible $n$-dimensional (spin-$(n-1)/2$) representation of $SL(2,\mathbb{C})$; in an inhomogeneous coordinate $t$, the polynomial $p_i$ simply has as its roots the directions from $x_i$ to the other points. The candidate flag map is obtained by normalizing the $n$ vectors $p_i$, so it is well defined precisely when they are linearly independent:

Conjecture 2.1 (Atiyah–Sutcliffe). For every configuration $(x_1,\dots,x_n)\in\mathcal{C}_n(\mathbb{R}^3)$ the polynomials $p_1,\dots,p_n$ are linearly independent over $\mathbb{C}$.

Independence is equivalent to the non-vanishing of a normalized determinant. Since the $p_i$ are defined only up to the individual scalings of the $u_{ij}$, Atiyah and Sutcliffe introduced the scale-invariant ratio

$$D(x_1,\dots,x_n) \;=\; \frac{\det(p_1,\dots,p_n)}{\prod_{i<j}\det(u_{ij},u_{ji})},$$

where $\det(p_1,\dots,p_n)$ is the determinant of the $n\times n$ coefficient matrix of the $p_i$ in the monomial basis and $\det(u_{ij},u_{ji})$ is the $2\times2$ determinant of a pair of linear forms; the ratio is unchanged by rescaling each $u_{ij}$, so $D$ is a well-defined complex function on configuration space, with $D\neq0$ equivalent to Conjecture 2.1. It is invariant under translations and dilations of $\mathbb{R}^3$ and covariant under its full rotation group, with reflections acting by complex conjugation (so $D$ is real for planar configurations). Writing $V=|D|$ (the volume) and letting $V_i$ denote the value of $V$ on the subconfiguration obtained by deleting $x_i$, the two stronger conjectures of the paper are:

Conjecture 4.1. $V\geq1$ for all configurations.

Conjecture 4.2. $V^{\,n-2}\geq\prod_{i=1}^n V_i$ for all configurations.

Conjecture 4.2 implies 4.1 by induction from the settled small cases, and 4.1 implies 2.1. Collinear configurations satisfy $V=1$ (a theorem of the source paper, proved separately from the conjecture), and the numerical minimization below always found the minimum at collinear sets; the conjectured target of 4.1 is the inequality $V\geq1$ itself, not an equality characterization. For $n=3$ one has the explicit formula $V=\frac12[\cos^2(\theta_1/2)+\cos^2(\theta_2/2)+\cos^2(\theta_3/2)]$ in the triangle angles $\theta_i$, with global minimum $1$ at collinearity and maximum $9/8$ at the equilateral triangle. The paper verified all three conjectures numerically, by simulated annealing (a stochastic numerical minimization heuristic) over configuration space, for all $n\leq20$.

The determinant also supports a multi-particle energy $E=-\log V$ whose minimizing configurations, computed for $n\leq32$, are near-spherical polyhedra with $n$ vertices and $2(n-2)$ triangular faces arranged as twelve pentagons and $n-12$ hexagons — the same fullerene-type cages (carbon-atom polyhedral molecules) found for the Thomson problem of charges repelling on a sphere with Coulomb forces, and dual to configurations arising in the Skyrmion model of nuclei (a nonlinear field theory whose topological solitons cluster on polyhedral shells) — a universality the paper documents but does not explain. The paper also extends the construction to hyperbolic $3$-space (where $t_{ij}$ becomes the point at infinity of the oriented geodesic through $x_i$ and $x_j$) and to worldlines in Minkowski space (where the $t_{ij}$ are retarded celestial directions), leaving the corresponding conjectures and the physical interpretation of $D$ as further open issues.

The conjectures are known for $n\leq3$ (elementary) and for $n=4$: Eastwood and Norbury proved Conjecture 2.1 for every configuration of four points, and Bou Khuzam and Johnson later gave a computer-aided proof of Conjectures 4.1 and 4.2 for all four-point configurations; special families of any size (regular polygons, convex and inscribed quadrilaterals, dihedral-symmetric and almost-collinear configurations) were settled by Mazur–Petrenko, Djokovic, and Svrtan–Urbiha. All three conjectures remain open for every $n\geq5$, and this is the problem formulated here.

Problem Statement

The problem is to prove or refute the three Atiyah–Sutcliffe conjectures on configurations of $n$ distinct points in Euclidean three-space for general $n$ — the open range being $n\geq5$, since all cases $n\leq4$ are settled.

Let $x_1,\dots,x_n\in\mathbb{R}^3$ be distinct. For each ordered pair $i\neq j$, let $t_{ij}\in\mathbb{CP}^1$ be the stereographic image of the unit vector $(x_j-x_i)/|x_j-x_i|$, let $u_{ij}$ be a nonzero linear form on $\mathbb{C}^2$ with root $t_{ij}$, and let $p_i=\prod_{j\neq i}u_{ij}\in S^{n-1}(\mathbb{C}^2)\cong\mathbb{C}^n$. Define the normalized determinant and volume

$$D \;=\; \frac{\det(p_1,\dots,p_n)}{\prod_{i<j}\det(u_{ij},u_{ji})}, \qquad V \;=\; |D|,$$

and let $V_i$ be the value of $V$ on the subconfiguration with $x_i$ deleted. Determine the truth of the following three statements for every configuration and every $n\geq5$:

1. (Conjecture 2.1) The polynomials $p_1,\dots,p_n$ are linearly independent over $\mathbb{C}$; equivalently, $D\neq0$ for every configuration.
2. (Conjecture 4.1) $V\geq1$ for every configuration.
3. (Conjecture 4.2) $V^{\,n-2}\geq\prod_{i=1}^n V_i$ for every configuration.

Because Conjecture 4.2 implies 4.1 (by induction from the settled cases) and 4.1 implies 2.1, a proof of the inductive inequality 4.2 for all $n\geq5$ settles the entire problem; conversely, a single configuration of some $n\geq5$ points with $D=0$ refutes all three, and a configuration with $0<V<1$ refutes Conjectures 4.1 and 4.2. The three conjectures are posed as separate questions, and the common objective is the determination of the non-vanishing and the lower bounds of the Atiyah–Sutcliffe determinant on the configuration space of points in Euclidean three-space.

References

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