AI 中文总结
该研究证明 zesting 构造关联的 Reshetikhin-Turaev 不变量计算复杂性不变，开发了计算相关链环不变量的多项式时间算法，将其识别为 rack 上同调不变量，为拓扑量子场论等的复杂性层次结构提供了依据。
AI 中文摘要
我们证明，当简单染色链环的 Reshetikhin-Turaev 不变量对应的基础 ribbon 融合范畴通过 zesting 构造关联时，其计算复杂性保持不变。Zesting 会修改一个 A-graded ribbon 融合范畴 $\boldsymbol{\textit{C}}$，添加额外代数数据 $\boldsymbol{\textit{\u03b6}}$，生成新范畴 $\boldsymbol{\textit{C}}^\boldsymbol{\textit{\u03b6}}$，已知该新范畴的链环不变量与原范畴的差异仅为依赖于 $\boldsymbol{\textit{\u03b6}}$ 的 A-染色链环不变量 $\boldsymbol{\textit{J}}_\boldsymbol{\textit{\u03b6}}$。基于这一理解及此前关于 zesting 下量子辫群表示的研究，我们的结果表明 zesting 如何助力将 (2+1)D 拓扑量子场论和拓扑相组织成复杂性理论层次结构。为证明核心结果，我们开发了类似 Reshetikhin-Turaev 构造的局部形式体系，用于计算缠结不变量 $\boldsymbol{\textit{J}}_\boldsymbol{\textit{\u03b6}}(T)$，这导出了计算链环不变量 $\boldsymbol{\textit{J}}_\boldsymbol{\textit{\u03b6}}(L)$ 的多项式时间算法。该构造的一个附带成果是（符号除外）将链环不变量 $\boldsymbol{\textit{J}}_\boldsymbol{\textit{\u03b6}}(L)$ 识别为 rack 上同调不变量，这可能具有独立研究价值。我们的形式体系还可扩展以定义带 A-结构的闭 3-流形的不变量，且我们从 A-模融合范畴构建的同伦量子场论也得到了类似的复杂性结果。
英文摘要
We show that the computational complexity of Reshetikhin-Turaev invariants of simply colored links is preserved when their underlying ribbon fusion categories are related by the zesting construction. Zesting modifies an $A$-graded ribbon fusion category $\mathcal{C}$ with additional algebraic data $ζ$ to produce a new category $\mathcal{C}^ζ$ whose link invariants are known to differ from those of $\mathcal{C}$ by an invariant of $A$-colored links $\mathcal{J}_ζ$ depending only on $ζ$. Building on this understanding and on earlier work on quantum braid group representations under zesting, our result suggests how zesting contributes to the organization of (2+1)D topological quantum field theories and topological phases into complexity-theoretic hierarchies. To prove our main result we develop a local formalism analogous to the Reshetikhin-Turaev construction to compute \emph{tangle} invariants $\mathcal{J}_ζ(T)$, which leads to a polynomial time algorithm to compute invariants of links $\mathcal{J}_ζ(L)$. A byproduct of our construction is an identification (up to a sign) of the link invariants $\mathcal{J}_ζ(L)$ as rack cocycle invariants, which may be of independent interest. Our formalism also extends to define invariants of closed $3$-manifolds with $A$-structure and we obtain similar complexity results for homotopy quantum field theories built from $A$-modular fusion categories.
Comments44 pages, many figures
