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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.08608 |
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| _version_ | 1866914984819163136 |
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| author | Carqueville, Nils Meir, Ehud Szegedy, Lorant |
| author_facet | Carqueville, Nils Meir, Ehud Szegedy, Lorant |
| contents | For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2 . In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Sigma. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Sigma, we construct a symmetric monoidal category C and a C-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the associated algebras are necessarily non-semisimple. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_08608 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Invariants of r-spin TQFTs and non-semisimplicity Carqueville, Nils Meir, Ehud Szegedy, Lorant Quantum Algebra Mathematical Physics 57K16, 18M05 For a positive integer r, an r-spin topological quantum field theory is a 2-dimensional TQFT with tangential structure given by the r-fold cover of SO_2 . In particular, such a TQFT assigns a scalar invariant to every closed r-spin surface Sigma. Given a sequence of scalars indexed by the set of diffeomorphism classes of all such Sigma, we construct a symmetric monoidal category C and a C-valued r-spin TQFT which reproduces the given sequence. We also determine when such a sequence arises from a TQFT valued in an abelian category with finite-dimensional Hom spaces. In particular, we construct TQFTs with values in super vector spaces that can distinguish all diffeomorphism classes of r-spin surfaces, and we show that the associated algebras are necessarily non-semisimple. |
| title | Invariants of r-spin TQFTs and non-semisimplicity |
| topic | Quantum Algebra Mathematical Physics 57K16, 18M05 |
| url | https://arxiv.org/abs/2306.08608 |