拓扑多边形中的均匀生成树、SLE(8)配分函数及$c=-2$对数共形场论中的关联函数
Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
AI总结：
该论文通过Coulomb气体积分和超椭圆行列式给出拓扑多边形中均匀生成树Peano曲线的SLE(8)配分函数，并解释为中心荷-2对数共形场论的关联函数，揭示了其对数现象和融合规则。
摘要：
我们找到了在具有一般边界条件的拓扑多边形中，均匀生成树（UST）的Peano曲线缩放极限的显式SLE(8)配分函数。它们以Coulomb气体积分公式给出，这些公式也可以用超椭圆黎曼曲面的a-周期的行列式来表达。我们还确定了UST Peano曲线的穿越概率为这些配分函数的比值。这些配分函数被解释为中心荷$c=-2$的对数共形场论（log-CFT）中的关联函数。确实，从我们的结果可以清楚地看出，这个理论不是最小模型，并且表现出对数现象——极限函数具有我们显式计算的对数渐近行为。从显式公式中还可以推断出它们的一般融合规则。所发现的代数结构与已知的Virasoro交错模分类相匹配，因此在这个意义上，我们给出了描述UST模型的中心荷$-2$的log-CFT中关联函数的直接概率构造。
英文摘要：
We find explicit SLE(8) partition functions for the scaling limits of Peano curves in the uniform spanning tree (UST) in topological polygons with general boundary conditions. They are given in terms of Coulomb gas integral formulas, which can also be expressed in terms of determinants involving a-periods of a hyperelliptic Riemann surface. We also identify the crossing probabilities for the UST Peano curves as ratios of these partition functions. The partition functions are interpreted as correlation functions in a logarithmic conformal field theory (log-CFT) of central charge $c= -2$. Indeed, it is clear from our results that this theory is not a minimal model and exhibits logarithmic phenomena -- the limit functions have logarithmic asymptotic behavior, that we calculate explicitly. General fusion rules for them could also be inferred from the explicit formulas. The discovered algebraic structure matches the known Virasoro staggered module classification, so in this sense, we give a direct probabilistic construction for correlation functions in a log-CFT of central charge $-2$ describing the UST model.
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