The $L^2$ metric for hyperbolic 2-monopoles

Abstract

It has been recently shown that the notoriously divergent $L^2$ metric on the moduli space of hyperbolic monopoles can be made finite by the introduction of a modified gauge fixing condition \cite{franchetti:2024}. In this paper we compute this modified $L^2$ metric for the mass $\frac{1}{2}$ 2-monopoles. The resulting metric is actually a complex non-degenerate bilinear form, which restricts to a Riemannian metric on the 4-dimensional subspace of inversion symmetric 2-monopoles. Remarkably, we have been able to compute the $L^2$ form explicitly in terms of elementary functions and elliptic integrals. Its asymptotic form matches what was expected on the basis of previous results in the literature.

42 pages, 1 figure