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

Understand

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.