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

Abstract

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 $Vir\otimes \overline{Vir}$ 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 $r\in {\mathbb{N}}^{\ast }$) are truly zero, so these fields come with simple $Vir\otimes \overline{Vir}$ ("Kac") modules. Meanwhile, fields with weights $(h_{r,s},h_{r,-s})$ and $(h_{r,-s},h_{r,s})$ (with $r,s\in {\mathbb{N}}^{\ast }$) come in indecomposable but not fully reducible representations mixing four simple $Vir\otimes \overline{Vir}$ 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.

arXiv comment37 pages, 3 figures, 15 tables. v2: Minor edits to beginning of Section 7.1 and Acknowledgments v3: 38 pages. Addition of Appendix C, in which we correct some of our reasoning about the lattice evidence for the structure of the continuum modules. This correction does not change the results. Minor other edits

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