Level lines of the Gaussian free field and $c=1$ degenerate conformal blocks
We consider Gaussian free field (GFF) on simply connected domains with piecewise constant Dirichlet boundary data. We show that the crossing probabilities for its level lines are determined by conformal blocks of primary fields in a conformal field theory (CFT) with central charge $c = 1$ which are degenerate at each insertion. Alternatively, the crossing probabilities are ratios of explicit partition functions of fused multiple $\mathrm{SLE}_4$ curves, which can be written in terms of fused Specht polynomials introduced recently by Lafay, Peltola, & Roussillon in a representation-theoretic context. We also prove that for the metric graph GFF introduced by Lupu, with appropriate boundary conditions, the crossing probabilities for its level sets converge in the scaling limit to our formulas. In particular, the geometry of the level-set percolation for both the continuum GFF and the metric graph GFF has a CFT description in terms of the aforementioned $c=1$ conformal blocks, which are linearly independent and solve the Belavin-Polyakov-Zamolodchikov (BPZ) PDEs of arbitrary orders. Interestingly, not all combinatorial boundary conditions are amenable for the GFF models -- while the CFT contains conformal blocks of primary fields labeled by generalized Dyck paths (i.e., semi-standard Young tableaux), the ones appearing in the above models satisfy specific monotonicity constraints.