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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2506.05785 |
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| _version_ | 1866917046916218880 |
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| author | Chen, Hank |
| author_facet | Chen, Hank |
| contents | This is a continuation of the first paper (arXiv:2501.06486) of this series, where the framework for the combinatorial quantization of the 4d 2-Chern-Simons theory with an underlying compact structure Lie 2-group $\mathbb{G}$ was laid out. In this paper, we continue our quest and characterize additive module *-functors $ω:\mathfrak{C}_q(\mathbb{G}^{Γ^2})\rightarrow\mathsf{Hilb}$, which serve as a categorification of linear *-functionals (ie. a state) on a $C^*$-algebra. These allow us to construct non-Abelian Wilson surface correlations $\widehat{\mathfrak{C}}_q(\mathbb{G}^{P})$ on the discrete 2d simple polyhedra $P$ partitioning 3-manifolds. By proving its stable equivalence under 3d handlebody moves, these Wilson surface states extend to decorated 3-dimensional marked bordisms in a 4-disc $D^4$. This provides invariants of framed oriented 2-ribbonsin $D^4$ from the data of the given compact Lie 2-group $\mathbb{G}$. We find that these 2-Chern-Simons-type 2-ribbon invariants are given by bigraded $\mathbb{Z}$-modules, similar to the lasagna skein modules of Manolescu-Walker-Wedrich. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_05785 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$ Chen, Hank Mathematical Physics High Energy Physics - Theory Quantum Algebra 20G42, 57K45 This is a continuation of the first paper (arXiv:2501.06486) of this series, where the framework for the combinatorial quantization of the 4d 2-Chern-Simons theory with an underlying compact structure Lie 2-group $\mathbb{G}$ was laid out. In this paper, we continue our quest and characterize additive module *-functors $ω:\mathfrak{C}_q(\mathbb{G}^{Γ^2})\rightarrow\mathsf{Hilb}$, which serve as a categorification of linear *-functionals (ie. a state) on a $C^*$-algebra. These allow us to construct non-Abelian Wilson surface correlations $\widehat{\mathfrak{C}}_q(\mathbb{G}^{P})$ on the discrete 2d simple polyhedra $P$ partitioning 3-manifolds. By proving its stable equivalence under 3d handlebody moves, these Wilson surface states extend to decorated 3-dimensional marked bordisms in a 4-disc $D^4$. This provides invariants of framed oriented 2-ribbonsin $D^4$ from the data of the given compact Lie 2-group $\mathbb{G}$. We find that these 2-Chern-Simons-type 2-ribbon invariants are given by bigraded $\mathbb{Z}$-modules, similar to the lasagna skein modules of Manolescu-Walker-Wedrich. |
| title | Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$ |
| topic | Mathematical Physics High Energy Physics - Theory Quantum Algebra 20G42, 57K45 |
| url | https://arxiv.org/abs/2506.05785 |