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

Abstract: 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.