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 [1]. However, the exact nature of the corresponding Vir ⊗ $$ \overline{\mathrm{Vir}} $$ 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” [2], 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 [3] on the analytical side. We find that null-descendants of diagonal fields having weights ( h r, 1 , h r, 1 ) (with r ∈ ℕ * ) are truly zero, so these fields come with simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ Vir ¯ (“Kac”) modules. Meanwhile, fields with weights ( h r,s , h r,−s ) and ( h r,−s , h r,s ) (with r, s ∈ ℕ * ) come in indecomposable but not fully reducible representations mixing four simple Vir ⊗ $$ \overline{\mathrm{Vir}} $$ 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 $$ {\overline{L}}_0 $$ 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.