c=2时W₃代数的简并共形块与三重 dimer 模型中的连接概率
Degenerate conformal blocks for the $W_3$ algebra at c=2 and connection probabilities in the triple dimer model
AI总结：
研究了W₃对称二维共形场论产生的d+8个线性偏微分方程系统，构造了c=2时增长不超幂律的W₃共形块子空间，证明其为Kuperberg代数的不可约表示，并建立了与三重dimer模型连接概率标度极限的关系。
摘要：
我们研究了由具有W₃对称代数的二维共形场论（CFTs）产生的d个变量中的d+8个线性偏微分方程（PDEs）的齐次系统。在CFT背景下，d个PDEs是三阶的，对应于零态方程，而其余8个PDEs（5个是二阶的，3个是一阶的）对应于W₃全局Ward恒等式。在中心荷c=2的情况下，我们构造了所有解空间的一个子空间，其增长不超过幂律。我们将这个子空间称为W₃共形块空间，并提供了一个用与三列列严格矩形Young表相关的Specht多项式表示的基。该空间的维数是一个Kostka数，与CFT预测一致，因此我们推测它穷尽了所有具有幂律界的解空间。此外，我们证明W₃共形块空间是由我们称为Kuperberg代数的𝔰𝔩₃网定义的某个图代数的不可约表示。最后，我们证明了一个公式，将我们构造的c=2时的W₃共形块与Kenyon和Shi的三重dimer模型中连接概率的标度极限联系起来。对于更一般的中心荷，我们期望W₃共形块与基于𝔰𝔩₃网的格模型中概率的标度极限有关。
英文摘要：
We study a homogeneous system of $d+8$ linear partial differential equations (PDEs) in $d$ variables arising from two-dimensional Conformal Field Theories (CFTs) with a $W_3$-symmetry algebra. In the CFT context, $d$ PDEs are third-order and correspond to the null-state equations, whereas the remaining 8 PDEs (five being second-order and three being first-order) correspond to the $W_3$ global Ward identities. In the case of central charge $c=2$, we construct a subspace of the space of all solutions which grow no faster than a power law. We call this subspace the space of $W_3$ conformal blocks, and we provide a basis expressed in terms of Specht polynomials associated with column-strict, rectangular Young tableaux with three columns. The dimension of this space is a Kostka number which coincides with CFT predictions, hence we conjecture that it exhausts the space of all solutions having a power law bound. Moreover, we prove that the space of $W_3$ conformal blocks is an irreducible representation of a certain diagram algebra defined from $\mathfrak{sl}_3$ webs that we call Kuperberg algebra. Finally, we prove a formula relating the $W_3$ conformal blocks at $c=2$ we constructed to Kenyon and Shi's scaling limits of connection probabilities in the triple dimer model. For more general central charges, we expect that $W_3$ conformal blocks are related to scaling limits of probabilities in lattice models based on $\mathfrak{sl}_3$ webs.
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