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arXiv · 2608.11416

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

Guido Franchetti·Derek Harland
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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 $\tfrac{1}{2}$ 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.

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Guido Franchetti, Derek Harland. 2026-08-11. The $L^2$ metric for hyperbolic 2-monopoles. https://arxiv.org/abs/2608.11416

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