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

### Submission summary

 Authors (as Contributors): Sylvain Ribault
Submission information
Date accepted: 2023-02-14
Date submitted: 2023-01-19 09:10
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Ontological classification
Specialties:
• High-Energy Physics - Theory
• Mathematical Physics
Approach: Theoretical

### Abstract

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. 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 Jones--Temperley--Lieb 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 four-point functions of the corresponding CFTs.

###### Current status:
Accepted in target Journal

Editorial decision: For Journal SciPost Selections: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

### 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 Jones-Temperley-Lieb 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 Temperley-Lieb 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.

### Submission & Refereeing History

Resubmission 2208.14298v2 on 19 January 2023
Submission 2208.14298v1 on 13 October 2022

## Reports on this Submission

### Report

After these revisions I recommend the paper to be published

• validity: -
• significance: -
• originality: -
• clarity: -
• formatting: -
• grammar: -