Title: Towards a complexity-theoretic dichotomy for TQFT invariants Show Chinese title

Title: 关于拓扑量子场论不变量的复杂性理论二分法

Abstract: We show that for any fixed $(2+1)$-dimensional TQFT over $\mathbb{C}$ of either Turaev-Viro-Barrett-Westbury or Reshetikhin-Turaev type, the problem of (exactly) computing its invariants on closed 3-manifolds is either solvable in polynomial time, or else it is $\#\mathsf{P}$-hard to (exactly) contract certain tensors that are built from the TQFT's fusion category. Our proof is an application of a dichotomy result of Cai and Chen [J. ACM, 2017] concerning weighted constraint satisfaction problems over $\mathbb{C}$. We leave for future work the issue of reinterpreting the conditions of Cai and Chen that distinguish between the two cases (i.e. $\#\mathsf{P}$-hard tensor contractions vs. polynomial time invariants) in terms of fusion categories. We expect that with more effort, our reduction can be improved so that one gets a dichotomy directly for TQFTs' invariants of 3-manifolds rather than more general tensors built from the TQFT's fusion category. Show Chinese abstract

Abstract: 我们证明了对于任何固定的定义在$\mathbb{C}$上的维度为$(2+1)$的拓扑量子场论（TQFT），无论是 Turaev-Viro-Barrett-Westbury 类型还是 Reshetikhin-Turaev 类型，在闭三维流形上精确计算其不变量的问题要么可以在多项式时间内解决，要么计算某些由该 TQFT 的融合范畴构造的张量的精确收缩问题是$\#\mathsf{P}$- 难的。 我们的证明是基于 Cai 和 Chen [J. ACM, 2017] 关于$\mathbb{C}$上带权约束满足问题的二分性结果的一个应用。 我们将重新解释 Cai 和 Chen 提出的区分两种情况（即$\#\mathsf{P}$- 难张量收缩与多项式时间不变量）的条件表述为融合范畴的工作留作未来研究。 我们预计通过更多努力，我们的归约可以改进，从而直接得到三维流形上的 TQFT 不变量的二分性，而不是更一般的由 TQFT 的融合范畴构建的张量的二分性。