Remarks on the complex geometry of the 3-monopole

Berry's connection, Kähler geometry and the Nahm construction of monopoles

We study supersymmetric deformations of N = 4 quantum mechanics with a Kahler target space admitting a holomorphic isometry. We show that the twisted mass deformation generalises to a deformation constructed from matrix-valued functions of the moment map, which obey the Nahm equations. We also explain how N = 4 supersymmetry implies that the Berry connection on the vacuum bundle for this theory satisfies the BPS monopole equations. In the case where the target space is a Riemann sphere, our analysis reduces to the standard Nahm construction of monopoles. This generalises an earlier result by S…

Geometry of Periodic Monopoles

BPS monopoles on $\mathbb{R}^2\times S^1$ correspond, via the generalized Nahm transform, to certain solutions of the Hitchin equations on the cylinder $\mathbb{R}\times S^1$. The moduli space M of two monopoles with their centre-of-mass fixed is a 4-dimensional manifold with a natural hyperkähler metric, and its geodesics correspond to slow-motion monopole scattering. The purpose of this paper is to study the geometry of M in terms of the Nahm/Hitchin data, i.e. in terms of structures on $\mathbb{R}\times S^1$. In particular, we identify the moduli, derive the asymptotic metric on M, and disc…

磁单极子的几何与准经典量子化

Geometry and quasi-classical quantization of magnetic monopoles

本文梳理了磁荷概念的物理与数学起源，构造了黎曼流形上磁单极子的磁拉普拉斯算子，并给出了其本征截面的准经典近似结果。

作为统一Reissner-Nordstrom几何中对偶体的磁单极场与库仑场

The Monopole and the Coulomb field as duals within the unifying Reissner-Nordstrom geometry

该研究证明磁单极场与库仑场在Reissner-Nordstrom几何中互为对偶，二者都是构造局域协变对角化应力-能量张量的标架所必需的，同属该几何的固有组成部分。

The asymptotic geometry of $\rm{G}_2$-monopoles

This article investigates the asymptotics of $\rm{G}_2$-monopoles. First, we prove that when the underlying $\rm{G}_2$-manifold is nonparabolic (i.e. admits a positive Green's function), finite intermediate energy monopoles with bounded curvature have finite mass. The second main result restricts to the case when the underlying $\rm{G}_2$-manifold is asymptotically conical. In this situation, we deduce sharp decay estimates and that the connection converges, along the end, to a pseudo-Hermitian--Yang--Mills connection over the asymptotic cone. Finally, our last result exhibits a Fredholm setup…

Magnetic monopoles, squashed 3-spheres and gravitational instantons from exotic R^4

We show that, in some limit, gravitational instantons correspond to exotic smooth geometry on R^4. The geometry of this exotic R^4 represents magnetic monopoles of Polyakov-'t Hooft type and the BPS condition allows for the generation of the charges from the gravitational sources of exotic R^4. Higgs field is, as usual, present in the monopole configurations. We indicate some possible scenarios in cosmology, particle physics and condensed matter where pure SU(2) Yang-Mills theory on the exotic R^4 aquires non-zero mass for its gauge boson when the smoothness is changed to the standard R^4 and,…