标题： 退化的共形块对于c=2的$W_3$代数和Specht多项式 显示英文标题

标题： Degenerate conformal blocks for the $W_3$ algebra at c=2 and Specht polynomials

摘要： 我们研究一个由二维共形场论（CFTs）产生的具有$W_3$对称代数的同质系统，该系统包含$d+8$个线性偏微分方程（PDEs），涉及$d$个变量。在CFT背景下，$d$个PDE为三阶方程，对应于零态方程，而其余8个PDE（其中五个为二阶方程，三个为一阶方程）对应于$W_3$全局Ward恒等式。在中心荷为$c=2$的情况下，我们构造了所有解空间的一个子空间，该子空间的增长速度不超过幂律。我们将这个子空间称为$W_3$共形块空间，并提供了一个基于与三列列严格、矩形杨图相关的Specht多项式的基。该空间的维数是一个Kotska数，它与CFT的预测一致，因此我们猜想该空间涵盖了所有具有幂律界限的解空间。 此外，我们证明了$W_3$共形块的空间是某个从$\mathfrak{sl}_3$网络定义的图代数的不可约表示，我们称之为库珀伯格代数。 最后，我们提出了一个精确的猜想，将$W_3$共形块在$c=2$处与肯尼思和施最近研究的三重二甲模型中概率的标度极限联系起来。 我们验证了该猜想在$d=6$以内的显式例子。 对于更一般的中心荷，我们预期$W_3$共形块与基于$\mathfrak{sl}_3$网络的格点模型中概率的标度极限有关。 显示英文摘要

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 Kotska number and it 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 formulate a precise conjecture relating the $W_3$ conformal blocks at $c=2$ to scaling limits of probabilities in the triple dimer model recently studied by Kenyon and Shi. We verify the conjecture for explicit examples up to $d=6$. 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.