Fermionization of fusion category symmetries in 1+1 dimensions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910022387105792 |
|---|---|
| author | Inamura, Kansei |
| author_facet | Inamura, Kansei |
| contents | We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}(\mathcal{H}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra $\mathcal{H}^u$. The weak Hopf superalgebra $\mathcal{H}^u$ depends not only on $H$ but also on a choice of a non-anomalous $\mathbb{Z}_2$ subgroup of $\mathrm{Rep}(H)$ that is used for the fermionization. We derive a general formula for $\mathcal{H}^u$ by explicitly constructing fermionic TQFTs with $\mathrm{SRep}(\mathcal{H}^u)$ symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on $F$-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_13159 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Fermionization of fusion category symmetries in 1+1 dimensions Inamura, Kansei Strongly Correlated Electrons High Energy Physics - Theory We discuss the fermionization of fusion category symmetries in two-dimensional topological quantum field theories (TQFTs). When the symmetry of a bosonic TQFT is described by the representation category $\mathrm{Rep}(H)$ of a semisimple weak Hopf algebra $H$, the fermionized TQFT has a superfusion category symmetry $\mathrm{SRep}(\mathcal{H}^u)$, which is the supercategory of super representations of a weak Hopf superalgebra $\mathcal{H}^u$. The weak Hopf superalgebra $\mathcal{H}^u$ depends not only on $H$ but also on a choice of a non-anomalous $\mathbb{Z}_2$ subgroup of $\mathrm{Rep}(H)$ that is used for the fermionization. We derive a general formula for $\mathcal{H}^u$ by explicitly constructing fermionic TQFTs with $\mathrm{SRep}(\mathcal{H}^u)$ symmetry. We also construct lattice Hamiltonians of fermionic gapped phases when the symmetry is non-anomalous. As concrete examples, we compute the fermionization of finite group symmetries, the symmetries of finite gauge theories, and duality symmetries. We find that the fermionization of duality symmetries depends crucially on $F$-symbols of the original fusion categories. The computation of the above concrete examples suggests that our fermionization formula of fusion category symmetries can also be applied to non-topological QFTs. |
| title | Fermionization of fusion category symmetries in 1+1 dimensions |
| topic | Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2206.13159 |