Moduli Spaces of Lumps on Real Projective Space

TL;DR

This paper studies the geometric structure and symmetries of charge three lumps on real projective space, providing explicit formulas for their moduli space metrics and exploring implications for lump decay.

Contribution

It characterizes the moduli space of charge three lumps as a 7-dimensional cohomogeneity one manifold with explicit metric formulas and symmetry analysis.

Findings

The moduli space is a 7-dimensional manifold with cohomogeneity one.

Explicit formulas for the metric and geometric quantities are derived.

Discussions on lump decay and symmetry implications are included.

Abstract

Harmonic maps that minimise the Dirichlet energy in their homotopy classes are known as lumps. Lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement. This has consequences for the behaviour of lumps and their symmetries. An interesting feature is that the moduli space of charge three lumps is a $7$-dimensional manifold of cohomogeneity one which can be described as a one-parameter family of symmetry orbits of $D_2$ symmetric maps. In this paper, we discuss the charge three moduli spaces of lumps from two perspectives: discrete symmetries of lumps and the Riemann-Hurwitz formula. We then calculate the metric and find explicit formulas for various geometric quantities. We also discuss the implications for lump decay.