单连通图上粒子的同调群
Homology groups for particles on one-connected graphs
摘要：
我们提出了一个数学框架来描述单连通图上粒子的位形空间的拓扑。特别地，我们计算了不同类型的单连通图的整数上的同调群。我们的方法是基于一些基本的组合性质的配置空间，Mayer-Vietoris序列的不同部分的配置空间和一些有限的使用离散Morse理论。作为结果之一，我们给出了树图上不可区分粒子的同调群秩的封闭公式。我们还详细讨论了可区分粒子和不可区分粒子的位形空间的第二同调群。我们的动机是寻找新的量子统计。
英文摘要：
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces and some limited use of discrete Morse theory. As one of the results, we derive a closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
