三体问题中的十二类单极子
Monopoles of Twelve Types in 3-Body Problems
摘要：
我们考虑十二种不同的方式建模三维问题的$\geq 2$。这些可以被看作是经典和量子背景独立的模型。我们表明，一种不同的类型的？实现在每个的关系空间：一种类型的减少配置空间。8例发生在2- $d$中，4例发生在3- $d$中;这些分别反映了$S_3 \times C_2$和$S_3$的非等效亚组作用计数。$S_3$作用于粒子标记;额外的$C_2$对应于是否识别镜像的纯2- $d$选项。非等价实现是由于一组子群、轨道空间和分层特征。我们的2- $d$磁单极子包括4个已知的磁单极子：狄拉克磁单极子在关系空间中的实现，而不是其更习惯的空间设置，岩井磁单极子的2- $d$版本，以及具有和不具有镜像识别的不可区分的粒子磁单极子。4个新的在2粒子标签开关下或在偶数排列下是不可区分的，在每种情况下具有可选的镜像识别。我们的4个3- $d$单极子是两个已知的：实际的Iwai单极子和它已经宣布的不可分割粒子对应物，以及两个新的：两粒子标签开关和偶排列情况。对所有4例3- $d$病例进行分层。三个偶排列的例子是轨道褶皱，两个有边界，3- $d$例子的边界构成一个单独的层，给出一个分层的轨道褶皱。我们记录了12种情况下的基本形状空间和关系空间，以及每种情况下的Hopf数学，全局截面与拓扑量子化二分法，Dirac弦定位，以及Chern积分与Gauss-Bonnet定理的拓扑贡献形式的一致性。
英文摘要：
We consider twelve different ways of modelling the 3-body problem in dimension $\geq 2$. These can be viewed as models of classical and quantum background independence. We show that a different type of monopole is realized in each's relational space: a type of reduced configuration space. 8 cases occur in 2-$d$, and 4 distinct ones in 3-$d$; these reflect counts of non-equivalent subgroup actions of $S_3 \times C_2$ and $S_3$ respectively. The $S_3$ acts on particle labels; the extra $C_2$ corresponds to the purely 2-$d$ option of whether or not to identify mirror images. The non-equivalent realization is due to a suite of subgroup, orbit space and stratification features. Our 2-$d$ monopoles include 4 known ones: a realization of Dirac's monopole in relational space rather than its more habitual setting of space, the 2-$d$ version of Iwai's monopole, and indistinguishable particle monopoles with and without mirror image identification. The 4 new ones are indistinguishable under a 2-particle label switch or under even permutations, in each case with optional mirror image identification. Our 4 3-$d$ monopoles are 2 known ones: the actual Iwai monopole and its already-announced indistinguishable-particles counterpart, and 2 new ones: the two-particle label switch and even permutation cases. All 4 3-$d$ cases are stratified. The three even-permutation cases are orbifolds, two with boundary, the 3-$d$ case's boundary constituting a separate stratum, giving a stratified orbifold. We document each of the 12 cases' underlying shape space and relational space, and each monopole's Hopf mathematics, global-section versus topological quantization dichotomy, Dirac string positioning, and Chern integral concordance with topological contributions form of Gauss--Bonnet Theorem.
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