The fully packed loop model as a non-rational $W_3$ conformal field theory

The fully packed loop (FPL) model is a statistical model related to the integrable $U_q(\hat{\mathfrak{sl}}_3)$ vertex model. In this paper we study the continuum limit of the FPL. With the appropriate weight of non-contractible loops, we give evidence of an extended $W_3$ symmetry in the continuum. The partition function on the torus is calculated exactly, yielding new modular invariants of $W_3$ characters. The full CFT spectrum is obtained, and is found to be in excellent agreement with exact diagonalisation.