Geometriae Dedicata

, Volume 158, Issue 1, pp 329–342

On the conjectures of Atiyah and Sutcliffe

Original Paper


Motivated by certain questions in physics, Atiyah defined a determinant function which to any set of n distinct points x1, . . . , xn in \({\mathbb R^3}\) assigns a complex number D(x1, . . . , xn). 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 \({|D(x_1,\ldots, x_n)|^{n-2} \geq \prod_{i=1}^n |D(x_1,\ldots, x_{i-1},x_{i+1},\ldots, x_n)|}\). Despite their simple formulation, these conjectures appear to be notoriously difficult. Let Dn denote the Atiyah determinant evaluated at the vertices of a regular n-gon. We prove that \({\lim_{n\to \infty}\frac{\ln D_n}{n^2}=\frac{7\zeta(3)}{2\pi^2}-\frac{\ln 2}{2}=0.07970479\ldots}\) 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.


Atiyah–Sutcliffe conjecture Atiyah determinant Configuration space 

Mathematics Subject Classification (2000)

51M04 51M16 70G10 


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  1. 1.
    Atiyah, M.F.: The geometry of classical particles. In: Surveys in Differential Geometry, vol VII, pp. 1–15, Internatinal Press, Somerville (2000)Google Scholar
  2. 2.
    Atiyah M.F.: Configuration of points. Philos. Trans. R. Soc. Lond. A 359, 1375–1387 (2001)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Atiyah M.F., Sutcliffe P.M.: The geometry of point particles. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 458, 1089–1115 (2002)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Berry M.V., Robbins J.M.: Indistinguishability for quantum particles: spin, statistics and geometric phase. Proc. R. Soc. London Ser. A 453, 1771–1790 (1997)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bradley, D.M.: Representations of Catalan’s constant. unpublished note (1998) available at
  6. 6.
    Crelle, A.L.: Einige Bemerkungen über die dreiseitige Pyramide. Sammlung mathematischer Aufsätze u. Bemerkungen 1, 105–132 (1821) (available at
  7. 7.
    Doković D.Z.: Verification of Atiyah’s conjecture for some nonplanar configurations with dihedral symmetry. Publ. Inst. Math. (Beograd) (N.S.) 72, 23–28 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Eastwood M., Norbury P.: A proof of Atiyah’s conjecture on configurations of four points in Euclidean three-space. Geom. Topol. 5, 885–893 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Finch S.R.: Mathematical Constants. Cambridge University Press, Cambridge (2003)MATHGoogle Scholar
  10. 10.
    Miller E.A., Srivastava H.M.: A simple reducible case of double hypergeometric series involving Catalan’s constant and Riemann’s ζ-function. Int. J. Math. Educ. Sci. Technol. 21, 375–377 (1990)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of MathematicsBinghamton UniversityBinghamtonUSA
  2. 2.Department of MathematicsSUNY BrockportBrockportUSA

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