nLab Atiyah-Sutcliffe conjecture

Idea

In the course of providing a geometric proof of the spin-statistics theorem, Berry & Robbins 1997 asked, at each natural number $n \in \mathbb{N}$, for a continuous and $Sym(n)$-equivariant function

(1)$$\underset{ {}^{\{1,\cdots, n\}} }{Conf}\big(\mathbb{R}^3\big) \xrightarrow{\phantom{AAA}} \mathrm{U}(n)/\big(\mathrm{U}(1)\big)^n$$

• from the configuration space of $n$ points (ordered and unlabeled) in Euclidean space $\mathbb{R}^3$

• to the coset space of the unitary group $\mathrm{U}(n)$ by its maximal torus, hence the complete flag manifold of flags in $\mathbb{C}^n$,

both equipped with the evident group action by the symmetric group $Sym(n)$.

For the first non-empty case $n = 2$ this readily reduces to asking for a continuous map of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is equivariant with respect to passage to antipodal points. This is immediately seen to be given by the radial projection. But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.

That a continuous and $Sym(n)$-equivariant Berry-Robbins map (1) indeed exists for all $n$ was proven in Atiyah 2000.

In this article, Atiyah turned attention to the stronger question asking for a function (1) which is smooth and $Sym(n) \times$$SO(3)$-equivariant and provided an elegant proof strategy for this stronger statement, which however hinges on some conjectural positivity properties of a certain determinant (discussed in more detail and with first numerical evidence in Atiyah 2001), interpreted as the electrostatic energy of $n$-particles in $\mathbb{R}^3$.

Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in Atiyah & Sutcliffe 2002, together with a refined formulation of the conjecture, whence it came to be known as the Atiyah-Sutcliffe conjecture.

The Atiyah-Sutcliffe conjecture has been proven for $n = 3$ in Atiyah 2000/01 and for $n = 4$ by Eastwood & Norbury 01.

References

The origin of the question in investigation of the spin-statistics theorem:

• Michael V. Berry, Jonathan M. Robbins, Indistinguishability for quantum particles: spin, statistics and the geometric phase, Proceedings of the Royal Society A 453 1963 (1997) 1771-1790 (doi:10.1098/rspa.1997.0096)

First form and first checks of the conjecture:

• Michael F. Atiyah, The geometry of classical particles, in Surveys in Differential Geometry, Surv. Differ. Geom. 7, Int. Press 2000, 1–15 (doi:10.4310/SDG.2002.v7.n1.a1)

• Michael F. Atiyah, Configurations of points, R. Soc. Lond. Philos. Trans. Ser. A, Math. Phys. Eng. Sci. 359 (2001) 1375–1387 (doi:10.1098/rsta.2001.0840)

Generalization of the codomain to flag manifolds of other compact Lie groups:

• Michael F. Atiyah, Roger Bielawski, Nahm’s equations, configuration spaces and flag manifolds, Bull. Braz. Math. Soc. (N.S.) 33 (2002), 157–176 (math.RT/0110112, doi:10.1007/s005740200007)

  (using Nahm's equation)

Full formulation of the Atiyah-Sutcliffe conjecture:

• Michael F. Atiyah, Paul M. Sutcliffe, The geometry of point particles, Proc. Roy. Soc. London Ser. A 458 (2002), 1089–1115 (hep-th/0105179, doi:10.1098/rspa.2001.0913)

Proof for $n = 4$:

• Michael Eastwood, Paul Norbury, A proof of Atiyah’s conjecture of four points in Euclidean three-space, Geom. Topol. 5 (2001) 885–893 (arXiv:math/0109161, doi:10.2140/gt.2001.5.88510.2140/gt.2001.5.885)

Further discussion:

• Dragutin Svrtan, Igor Urbiha, Atiyah-Sutcliffe conjectures for almost collinear configurations and some new conjectures for symmetric functions, (math.AG/0406386)

• Dragutin Svrtan, Igor Urbiha, Verification and strengthening of the Atiyah–Sutcliffe conjectures for several types of configurations (math.MG/0609174)

• Marcin Mazur, Bogdan V. Petrenko, On the conjectures of Atiyah and Sutcliffe, Geom Dedicata 158 (2012) 329–342 (doi:10.1007/s10711-011-9636-6, arxiv:1102.4662)

• Joseph Malkoun, Root Systems and the Atiyah-Sutcliffe Problem, Journal of Mathematical Physics 60, 101702 (2019) (arXiv:1903.00325)

• Joseph Malkoun, The Atiyah-Sutcliffe determinant (arXiv:1903.05957)