Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Costantino, Francesco Harper, Matthew Robertson, Adam Young, Matthew B. |
| author_facet | Costantino, Francesco Harper, Matthew Robertson, Adam Young, Matthew B. |
| contents | We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 π\sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_12181 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$ Costantino, Francesco Harper, Matthew Robertson, Adam Young, Matthew B. Quantum Algebra Geometric Topology Representation Theory 81T45 (Primary), 20G42 (Secondary) We construct three-dimensional non-semisimple topological field theories from the unrolled quantum group of the Lie superalgebra $\mathfrak{osp}(1 \vert 2)$. More precisely, the quantum group depends on a root of unity $q=e^{\frac{2 π\sqrt{-1}}{r}}$, where $r$ is a positive integer greater than $2$, and the construction applies when $r$ is not congruent to $4$ modulo $8$. The algebraic result which underlies the construction is the existence of a relative modular structure on the non-finite, non-semisimple category of weight modules for the quantum group. We prove a Verlinde formula which allows for the computation of dimensions and Euler characteristics of topological field theory state spaces of unmarked surfaces. When $r$ is congruent to $\pm 1$ or $\pm 2$ modulo $8$, we relate the resulting $3$-manifold invariants with physicists' $\widehat{Z}$-invariants associated to $\mathfrak{osp}(1 \vert 2)$. Finally, we establish a relation between $\widehat{Z}$-invariants associated to $\mathfrak{sl}(2)$ and $\mathfrak{osp}(1 \vert 2)$ which was conjectured in the physics literature. |
| title | Non-semisimple topological field theory and $\widehat{Z}$-invariants from $\mathfrak{osp}(1 \vert 2)$ |
| topic | Quantum Algebra Geometric Topology Representation Theory 81T45 (Primary), 20G42 (Secondary) |
| url | https://arxiv.org/abs/2407.12181 |