Elliptic constructions of hyperkaehler metrics III: Gravitons and Poncelet polygons

Abstract

In the generalized Legendre approach, the equation describing an asymptotically locally Euclidean space of type $D_n$ is found to admit an algebraic formulation in terms of the group law on a Weierstrass cubic. This curve has the structure of a Cayley cubic for a pencil generated by two transversal plane conics, that is, it takes the form $Y^2=\det (\mathcal{A}+X\mathcal{B})$, where $\mathcal{A}$ and $\mathcal{B}$ are the defining $3\times 3$ matrices of the conics. In this light, the equation can be interpreted as the closure condition for an elliptic billiard trajectory tangent to the conic $\mathcal{B}$ and bouncing into various conics of the pencil determined by the positions of the monopoles. Poncelet's porism guarantees then that once a trajectory closes to a star polygon, any trajectory will close, regardless of the starting point and after the same number of steps.

26 pages, 7 figures

References in corpus (1)

• Elliptic constructions of hyperkaehler metrics II: The quantum mechanics of a Swann bundle

Cited by in corpus (2)

• Elliptic constructions of hyperkaehler metrics I: The Atiyah-Hitchin manifold
• Elliptic constructions of hyperkaehler metrics II: The quantum mechanics of a Swann bundle