具有相同 $S$、$T$ 和 $W$ 的非等价模张量范畴

中文摘要 AI 辅助

模数据 $(S,T)$ 连同 Whitehead 链环矩阵 $W$ 是否在带饰等价意义下确定一个模张量范畴，这仍然是一个未解决的问题。我们通过构造两个带饰不等价的 Dijkgraaf--Witten 模范畴来给出否定回答，这两个范畴针对素数 $p\ge5$ 的群 $(\mathbb Z/p\mathbb Z)^3$，具有相同的 $(S,T,W)$，并且更一般地，在简单对象的一个双射下，所有至多两个分支的带框定向链环的 Reshetikhin--Turaev 不变量都相同。该构造在固定一个二次上同调类的同时变化一个交错 $3$-上循环，利用对秩至多二的子群的限制来匹配链环不变量，并利用二次类和交错类一起来阻碍带饰等价。因此，对模张量范畴在带饰等价意义下的完全分类需要超越至多两个分支的彩色带框链环不变量的不变量。我们还解析地评估了一个闭定向三维流形上的未着色配分函数，该函数区分了该族中所有 $(p-1)/2$ 个等价类。

英文摘要

It remains an open question whether the modular data $(S,T)$ together with the Whitehead-link matrix $W$ determine a modular tensor category up to ribbon equivalence. We answer negatively by constructing two braided-inequivalent Dijkgraaf--Witten modular categories for $(\mathbb Z/p\mathbb Z)^3$ for prime $p\ge5$ with identical $(S,T,W)$ and, more generally, identical Reshetikhin--Turaev invariants of all framed oriented links with at most two components under a single bijection of simple objects. The construction varies an alternating $3$-cocycle while fixing a quadratic cohomology class, using restriction to subgroups of rank at most two to match link invariants and the quadratic and alternating classes together to obstruct braided equivalence. A complete classification of modular tensor categories up to ribbon equivalence therefore requires invariants beyond those of colored framed links with at most two components. We also analytically evaluate an uncolored partition function on a closed oriented three-manifold that separates all $(p-1)/2$ equivalence classes in the family.

发表机构

• School of Physics, Peking University（北京大学物理学院）
• School of Physics, East China Normal University（华东师范大学物理学院）

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