双曲2单极子的$L^2$度量
The $L^2$ metric for hyperbolic 2-monopoles
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中文总结 AI 辅助
本文计算了质量为$\ frac{1}{2}$的2单极子的修正$L^2$度量，该度量为复非退化双线性型，在反演对称2单极子子空间上为黎曼度量，且可通过初等函数与椭圆积分显式算出，其渐近形式与文献预期一致。
中文摘要 AI 辅助
近期研究表明，通过引入修正规范固定条件可使双曲单极子模空间上原本臭名昭著的发散$L^2$度量变为有限。本文针对质量为$\ frac{1}{2}$的2单极子计算该修正$L^2$度量，所得度量实为复非退化双线性型，其在4维反演对称2单极子子空间上限制为黎曼度量。值得注意的是，我们已能以初等函数和椭圆积分显式计算该$L^2$形式，其渐近形式与文献中先前结果的预期相符。
英文摘要
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.
