Tangent Developable Orbit Space of an Octupole

TL;DR

This paper explores the geometric structure of the orbit space of an octupole tensor, revealing its relation to classical algebraic surfaces and moduli spaces of algebraic curves, with implications for understanding symmetries in mathematical physics.

Contribution

It identifies the orbit space of an octupole as a tangent developable surface of degree six and connects it to classical algebraic geometry and moduli spaces, providing new insights into its structure.

Findings

Orbit space is a 3D body with cusps and cuspidal edges.

Boundary is a tangent developable surface of degree six.

Links to moduli space of hyperelliptic curves of genus two.

Abstract

The orbit space of an octupole, a traceless symmetric third-rank tensor of $SO(3)$, is shown to be a three-dimensional body with three cusps and two cuspidal edges. It is demonstrated that for a unique choice of orbit space coordinates its boundary turns into a tangent developable surface of degree six which we identify with one in the classification by Chasles and Cayley. The close relation of the octupole's orbit space to the moduli space of a binary sextic form is described and relevant work by Clebsch and Bolza from the nineteenth century is recalled. Upon complexification, the octupole's orbit space yields the geometry of the moduli space of hyperelliptic curves of genus two. Its boundary is found to be a non-orientable tangent developable surface. As preamble, the orbit space of a set of three vectors is shown to be bounded by Cayley's nodal cubic surface and the boundary of the orbit space for a vector plus a quadrupole is found to correspond to Cremona's first class in his classification of quartic ruled surfaces.