Spaces of states of the two-dimensional O(n) and Potts models

Understand

We determine the spaces of states of the two-dimensional $O(n)$ and $Q$-state Potts models with generic parameters $n,Q\in \mathbb{C}$ as representations of their known symmetry algebras.

• While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the $O(n)$ model, and the symmetric group $S_Q$ for the $Q$-state Potts model.
• We do this by two independent methods.
• First we compute the twisted torus partition functions of the models at criticality.