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The action of the Virasoro algebra in the two-dimensional Potts and loop models at generic $Q$

The spectrum of conformal weights for the CFT describing the two-dimensional critical $Q$-state Potts model (or its close cousin, the dense loop model) has been known for more than 30 years. However, the exact nature of the corresponding $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ representations has remained unknown up to now. Here, we solve the problem for generic values of $Q$. This is achieved by a mixture of different techniques: a careful study of "Koo--Saleur generators" [arXiv:hep-th/9312156], combined with measurements of four-point amplitudes, on the numerical side, and OPEs and the four-point amplitudes recently determined using the "interchiral conformal bootstrap" in [arXiv:2005.07258] on the analytical side. We find that null-descendants of diagonal fields having weights $(h_{r,1},h_{r,1})$ (with $r\in \mathbb{N}^*$) are truly zero, so these fields come with simple $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ ("Kac") modules. Meanwhile, fields with weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ (with $r,s\in\mathbb{N}^*$) come in indecomposable but not fully reducible representations mixing four simple $\hbox{Vir}\otimes\overline{\hbox{Vir}}$ modules with a familiar "diamond" shape. The "top" and "bottom" fields in these diamonds have weights $(h_{r,-s},h_{r,-s})$, and form a two-dimensional Jordan cell for $L_0$ and $\bar{L}_0$. This establishes, among other things, that the Potts-model CFT is logarithmic for $Q$ generic. Unlike the case of non-generic (root of unity) values of $Q$, these indecomposable structures are not present in finite size, but we can nevertheless show from the numerical study of the lattice model how the rank-two Jordan cells build up in the infinite-size limit.