SciPost Submission Page
Spaces of states of the twodimensional O(n) and Potts models
by Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Jesper Lykke Jacobsen · Sylvain Ribault 
Submission information  

Preprint Link:  https://arxiv.org/abs/2208.14298v2 (pdf) 
Date accepted:  20230214 
Date submitted:  20230119 09:10 
Submitted by:  Ribault, Sylvain 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We determine the spaces of states of the twodimensional $O(n)$ 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. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the $O(n)$ model, we decompose representations of the Brauer algebra into representations of its unoriented JonesTemperleyLieb subalgebra. For the $Q$state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on fourpoint functions of the corresponding CFTs.
List of changes
We have tried to clarify the article as suggested by the first referee. Let us answer the specific points that were raised:
1. We have tried to clarify the counting of clusters with cross topology right after (3.29).
2. We have used $\delta_{N(C), 0}$ in (3.30) as suggested. The alternative formulation $0^{N(C)}$ is now mentioned before (3.31), in order to justify the use of (2.39).
3. The translation generator $u$ is defined on page 25 when we introduce the unoriented JonesTemperleyLieb algebra. In order to help the reader remember it when reaching (4.17), we have added the name 'translation generator', and a reference to (A.16).
4. Before (4.21), we have added the precision "modulo permutations of the upper sites", which was mistakenly omitted, although it appeared before (4.24).
5. 6. We have expanded the explanations between (4.22) and (4.23), in order to better describe the relevant TemperleyLieb representations and their properties.
7. We have corrected the typo.
We did not follow the suggestion to move several parts of Appendix A to the main text, because which parts should be moved is not clear to us, and may depend on the reader. We hope that Sections 4 and 5 are readable by people already familiar with diagram algebras. For the others, we have added an explicit recommendation to read Appendix A at the beginning of Section 4.1.
Published as SciPost Phys. 14, 092 (2023)