Spaces of states of the two-dimensional $O(n)$ and Potts models
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 t...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SciPost
2023-05-01
|
| Series: | SciPost Physics |
| Online Access: | https://scipost.org/SciPostPhys.14.5.092 |
| _version_ | 1797834072927502336 |
|---|---|
| author | Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur |
| author_facet | Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur |
| author_sort | Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-04-09T14:33:53Z |
| format | Article |
| id | doaj.art-970811605224408a977cfa6d57d76bff |
| institution | Directory Open Access Journal |
| issn | 2542-4653 |
| language | English |
| last_indexed | 2024-04-09T14:33:53Z |
| publishDate | 2023-05-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics |
| spelling | doaj.art-970811605224408a977cfa6d57d76bff2023-05-03T14:15:32ZengSciPostSciPost Physics2542-46532023-05-0114509210.21468/SciPostPhys.14.5.092Spaces of states of the two-dimensional $O(n)$ and Potts modelsJesper Lykke Jacobsen, Sylvain Ribault, Hubert SaleurWe 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.https://scipost.org/SciPostPhys.14.5.092 |
| spellingShingle | Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur Spaces of states of the two-dimensional $O(n)$ and Potts models SciPost Physics |
| title | Spaces of states of the two-dimensional $O(n)$ and Potts models |
| title_full | Spaces of states of the two-dimensional $O(n)$ and Potts models |
| title_fullStr | Spaces of states of the two-dimensional $O(n)$ and Potts models |
| title_full_unstemmed | Spaces of states of the two-dimensional $O(n)$ and Potts models |
| title_short | Spaces of states of the two-dimensional $O(n)$ and Potts models |
| title_sort | spaces of states of the two dimensional o n and potts models |
| url | https://scipost.org/SciPostPhys.14.5.092 |
| work_keys_str_mv | AT jesperlykkejacobsensylvainribaulthubertsaleur spacesofstatesofthetwodimensionalonandpottsmodels |