We prove a complete complexity classification theorem for the planar eight-vertex model. For every parameter setting in ${\mathbb C}$ for the eight-vertex model, the partition function is either (1) computable in P-time for every graph, or (2) \#P-hard for general graphs but computable in P-time for planar graphs, or (3) \#P-hard even for planar graphs. The classification has an explicit criterion. In (2), we discover new P-time computable eight-vertex models on planar graphs beyond Kasteleyn's algorithm for counting planar perfect matchings. They are obtained by a combinatorial transformation to the planar {\sc Even Coloring} problem followed by a holographic transformation to the tractable cases in the planar six-vertex model. In the process, we also encounter non-local connections between the planar eight vertex model and the bipartite Ising model, conformal lattice interpolation and Möbius transformation from complex analysis. The proof also makes use of cyclotomic fields.

翻译：我们证明了平面八顶点模型的完全复杂性分类定理。对于八顶点模型在${\mathbb C}$中的每个参数设定，配分函数要么(1)对任意图均可在多项式时间内计算，要么(2)对一般图是#P难问题但对平面图可在多项式时间内计算，要么(3)即使对平面图也是#P难问题。该分类具有明确的判定准则。在情形(2)中，我们发现了超越Kasteleyn平面完美匹配计数算法的、可在多项式时间内计算的平面八顶点模型新类别。这些模型通过组合变换转化为平面偶着色问题，再经全息变换转化为平面六顶点模型中的可处理情形。在此过程中，我们还发现了平面八顶点模型与二分伊辛模型之间的非局域关联、共形格点插值以及复分析中的莫比乌斯变换。证明过程还利用了分圆域理论。