Fermionization of fusion category symmetries in 1+1 dimensions
Abstract 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 Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion categ...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2023-10-01
|
| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP10(2023)101 |
| _version_ | 1797341453244956672 |
|---|---|
| author | Kansei Inamura |
| author_facet | Kansei Inamura |
| author_sort | Kansei Inamura |
| collection | DOAJ |
| description | Abstract 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 Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep( H $$ \mathcal{H} $$ u ), which is the supercategory of super representations of a weak Hopf superalgebra H $$ \mathcal{H} $$ u . The weak Hopf superalgebra H $$ \mathcal{H} $$ u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H $$ \mathcal{H} $$ u by explicitly constructing fermionic TQFTs with SRep( H $$ \mathcal{H} $$ 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. |
| first_indexed | 2024-03-08T10:18:17Z |
| format | Article |
| id | doaj.art-f7ce964a7c7842c6ab180193a85eb123 |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-08T10:18:17Z |
| publishDate | 2023-10-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-f7ce964a7c7842c6ab180193a85eb1232024-01-28T12:10:49ZengSpringerOpenJournal of High Energy Physics1029-84792023-10-0120231016310.1007/JHEP10(2023)101Fermionization of fusion category symmetries in 1+1 dimensionsKansei Inamura0Institute for Solid State Physics, University of TokyoAbstract 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 Rep(H) of a semisimple weak Hopf algebra H, the fermionized TQFT has a superfusion category symmetry SRep( H $$ \mathcal{H} $$ u ), which is the supercategory of super representations of a weak Hopf superalgebra H $$ \mathcal{H} $$ u . The weak Hopf superalgebra H $$ \mathcal{H} $$ u depends not only on H but also on a choice of a non-anomalous ℤ 2 subgroup of Rep(H) that is used for the fermionization. We derive a general formula for H $$ \mathcal{H} $$ u by explicitly constructing fermionic TQFTs with SRep( H $$ \mathcal{H} $$ 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.https://doi.org/10.1007/JHEP10(2023)101Discrete SymmetriesGlobal SymmetriesLattice Quantum Field TheoryTopological Field Theories |
| spellingShingle | Kansei Inamura Fermionization of fusion category symmetries in 1+1 dimensions Journal of High Energy Physics Discrete Symmetries Global Symmetries Lattice Quantum Field Theory Topological Field Theories |
| title | Fermionization of fusion category symmetries in 1+1 dimensions |
| title_full | Fermionization of fusion category symmetries in 1+1 dimensions |
| title_fullStr | Fermionization of fusion category symmetries in 1+1 dimensions |
| title_full_unstemmed | Fermionization of fusion category symmetries in 1+1 dimensions |
| title_short | Fermionization of fusion category symmetries in 1+1 dimensions |
| title_sort | fermionization of fusion category symmetries in 1 1 dimensions |
| topic | Discrete Symmetries Global Symmetries Lattice Quantum Field Theory Topological Field Theories |
| url | https://doi.org/10.1007/JHEP10(2023)101 |
| work_keys_str_mv | AT kanseiinamura fermionizationoffusioncategorysymmetriesin11dimensions |