On the conjectures of Atiyah and Sutcliffe

Abstract

Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of $n$ distinct points $x_1,..., x_n$ in $\mathbb R^3$ assigns a complex number $D(x_1,..., x_n)$. In a joint work, he and Sutcliffe stated three intriguing conjectures about this determinant. They provided compelling numerical evidence for the conjectures and an interesting physical interpretation of the determinant. The first conjecture asserts that the determinant never vanishes, the second states that its absolute value is at least one, and the third says that $|D(x_1,..., x_n)|^{n-2}\geq \prod_{i=1}^n |D(x_1,..., x_{i-1},x_{i+1},..., x_n)|$. Despite their simple formulation, these conjectures appear to be notoriously difficult. Let $D_n$ denote the Atiyah determinant evaluated at the vertices of a regular $n-$gon. We prove that $\lim_{n\to \infty} \frac{\ln D_n}{n^2}= \frac{7\zeta(3)}{2\pi^2}-\frac{\ln 2}{2}=0.07970479...$ and establish the second conjecture in this case. Furthermore, we prove the second conjecture for vertices of a convex quadrilateral and the third conjecture for vertices of an inscribed quadrilateral.