发表机构
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明定义域大小为3的复值对称约束函数Holant*问题的复杂性二分定理，给出可判定判据，利用群作用与旋量表示，并引入框架和壳统一证明#P-难性。
AI 中文摘要
我们证明了在定义域大小为3的复值对称约束函数族$\mathcal{F}$上，$\mathrm{Holant}^*$问题的复杂性二分定理。我们给出了一个可判定的可解性判据，并证明若$\mathcal{F}$满足该判据，则$\mathrm{Holant}^*(\mathcal{F})$可在多项式时间内求解，否则为#P-难。这是首个针对高维定义域上复值约束函数集的Holant二分定理。我们表明，复值约束函数具有实值约束函数中未观察到的丰富结构。这种结构仅在非标准双线性形式的二分Holant设置中分析时才显现，并为二分定理的证明提供了主干。我们利用群作用以及$\mathrm{SL}(2,\mathbb{C})$在$\mathrm{SO}(3,\mathbb{C})$和广义正交群中的旋量表示来促进这一分析。我们还引入了$\textit{frames}$（框架）和$\textit{shells}$（壳）。框架线性化群作用，并与壳结合使用时为证明#P-难性提供了统一框架。此外，我们通过$\textit{annihilators}$（零化子）刻画低维约束函数，这使得在定义域大小为3时分析$\textit{essentially Boolean domain}$（本质布尔域）函数成为可能。
英文摘要
We prove a complexity dichotomy theorem for $\mathrm{Holant}^*$ problems over complex-valued symmetric constraint functions $\mathcal{F}$ on domain size $3$. We give a decidable tractability criterion and prove that if $\mathcal{F}$ satisfies the criterion, then $\mathrm{Holant}^*(\mathcal{F})$ is solvable in polynomial time, and otherwise it is #P-hard. This is the first Holant dichotomy for a set of complex-valued constraint functions on higher domains. We show that complex-valued constraint functions have a rich structure not observed in real-valued constraint functions. This structure is only revealed when we analyze them in a bipartite Holant setting with a non-standard bilinear form and provides the backbone to the proof of the dichotomy. We use group actions and the spin representation of $\mathrm{SL}(2, \mathbb{C})$ in $\mathrm{SO}(3, \mathbb{C})$ and the generalized orthogonal group to facilitate this analysis. We also introduce $\textit{frames}$ and $\textit{shells}$. Frames linearize the group action and provide a unified framework for proving #P-hardness when used in conjunction with shells. Furthermore, we characterize the lower dimensional constraint functions by $\textit{annihilators}$, which makes it possible to analyze $\textit{essentially Boolean domain}$ functions in domain size $3$.
