Title: MTC$[M_3, G]$: 3d Topological Order Labeled by Seifert Manifolds Show Chinese title

Title: MTC$[M_3, G]$：由Seifert流形标记的三维拓扑序

Abstract: We propose a correspondence between topological order in 2+1d and Seifert three-manifolds together with a choice of ADE gauge group $G$. Topological order in 2+1d is known to be characterized in terms of modular tensor categories (MTCs), and we thus propose a relation between MTCs and Seifert three-manifolds. The correspondence defines for every Seifert manifold and choice of $G$ a fusion category, which we conjecture to be modular whenever the Seifert manifold has trivial first homology group with coefficients in the center of $G$. The construction determines the spins of anyons and their S-matrix, and provides a constructive way to determine the R- and F-symbols from simple building blocks. We explore the possibility that this correspondence provides an alternative classification of MTCs, which is put to the test by realizing all MTCs (unitary or non-unitary) with rank $r\leq 5$ in terms of Seifert manifolds and a choice of Lie group $G$. Show Chinese abstract

Abstract: 我们提出了一种对应关系，将2+1维拓扑序与Seifert三维流形以及选择的ADE规范群$G$联系起来。已知2+1维拓扑序可以用模张量范畴（MTCs）来刻画，因此我们提出了MTCs与Seifert三维流形之间的联系。 这种对应关系为每个Seifert流形和$G$的选择定义了一个融合范畴，并且我们推测当Seifert流形在以$G$中心为系数时具有平凡的一阶同调群时，该融合范畴是模的。 构造确定了任意子的自旋及其S矩阵，并提供了一种从简单构建块确定R和F符号的构造方法。 我们探讨了这种对应关系是否可以提供MTCs的一种替代分类方法，并通过利用Seifert流形和Lie群$G$的选择来实现秩为$r\leq 5$的所有MTCs（单位化或非单位化）来检验这一假设。