$\hat{Z}$-TQFT, Surgery Formulas, and New Algebras
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915500359942144 |
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| author | Guicardi, Pedro Jagadale, Mrunmay |
| author_facet | Guicardi, Pedro Jagadale, Mrunmay |
| contents | The $\hat{Z}$ invariants of three-manifolds introduced by Gukov-Pei-Putrov-Vafa have influenced many areas of mathematics and physics. However, their TQFT structure remains poorly understood. In this work, we develop a framework of decorated $\mathrm{Spin}$-TQFTs and construct one based on Atiyah-Segal-like axioms that computes the $\hat{Z}$ invariants. Central to our approach is a novel quantization of $SL(2,\mathbb{C})$ Chern-Simons theory and a $\mathbb{Q}$-extension of the algebra of observables on the torus, from which we obtain the torus state space of the $\hat{Z}$-TQFT. Using the torus state space and topological invariance, we uniquely determine the $\hat{Z}$ invariants for negative-definite plumbed manifolds. Within this TQFT framework, we establish gluing, rational surgery, partial surgery, satellite, and cabling formulas, as well as explicit closed-form expressions for Seifert manifolds and torus link complements. We also generalize these constructions to higher-rank gauge groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_14311 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\hat{Z}$-TQFT, Surgery Formulas, and New Algebras Guicardi, Pedro Jagadale, Mrunmay High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra The $\hat{Z}$ invariants of three-manifolds introduced by Gukov-Pei-Putrov-Vafa have influenced many areas of mathematics and physics. However, their TQFT structure remains poorly understood. In this work, we develop a framework of decorated $\mathrm{Spin}$-TQFTs and construct one based on Atiyah-Segal-like axioms that computes the $\hat{Z}$ invariants. Central to our approach is a novel quantization of $SL(2,\mathbb{C})$ Chern-Simons theory and a $\mathbb{Q}$-extension of the algebra of observables on the torus, from which we obtain the torus state space of the $\hat{Z}$-TQFT. Using the torus state space and topological invariance, we uniquely determine the $\hat{Z}$ invariants for negative-definite plumbed manifolds. Within this TQFT framework, we establish gluing, rational surgery, partial surgery, satellite, and cabling formulas, as well as explicit closed-form expressions for Seifert manifolds and torus link complements. We also generalize these constructions to higher-rank gauge groups. |
| title | $\hat{Z}$-TQFT, Surgery Formulas, and New Algebras |
| topic | High Energy Physics - Theory Mathematical Physics Geometric Topology Quantum Algebra |
| url | https://arxiv.org/abs/2509.14311 |