论文标题
图表上粒子的编织组的几何表现
Geometric presentations of braid groups for particles on a graph
论文作者
论文摘要
我们研究了被约束在图上移动的粒子的编织组的几何表现，即由节点和边缘组成的网络。我们提出的一组发电机由图形连接和某些圆形移动的粒子对交换组成，其中一个粒子围绕图形的简单循环传播。我们指出，如此定义的发电机通常无法满足2D物理学已知的编织关系。我们完成了恒星图的发电机之间关系的完整描述，我们得出了某些准编织关系。我们还描述了图编织组如何依赖图的（图理论）连接性。这是根据图形组的商的商来完成的，其中将一颗粒子的移动置于身份。特别是，我们表明，以$ 3 $连接的平面图，这些商的平面图重建了众所周知的平面编织组。对于$ 2 $连接的图，这种方法会导致Yang-Baxter方程的概括。我们的结果与网络上的非亚洲人的研究特别相关，显示了图形上非亚伯量子统计数据的新可能性。
We study geometric presentations of braid groups for particles that are constrained to move on a graph, i.e. a network consisting of nodes and edges. Our proposed set of generators consists of exchanges of pairs of particles on junctions of the graph and of certain circular moves where one particle travels around a simple cycle of the graph. We point out that so defined generators often do not satisfy the braiding relation known from 2D physics. We accomplish a full description of relations between the generators for star graphs where we derive certain quasi-braiding relations. We also describe how graph braid groups depend on the (graph-theoretic) connectivity of the graph. This is done in terms of quotients of graph braid groups where one-particle moves are put to identity. In particular, we show that for $3$-connected planar graphs such a quotient reconstructs the well-known planar braid group. For $2$-connected graphs this approach leads to generalisations of the Yang-Baxter equation. Our results are of particular relevance for the study of non-abelian anyons on networks showing new possibilities for non-abelian quantum statistics on graphs.
