标题： 四维2-陈-西蒙斯理论的组合量子化II：$D^4$中高维带状结构的量子不变量 显示英文标题

标题： Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$

摘要： 这是本系列的第一篇论文（arXiv:2501.06486）的延续，在其中建立了以紧致结构李二群 $\mathbb{G}$ 为背景的4d 2-陈-西蒙斯理论的组合量子化框架。 本文中，我们继续我们的研究，并刻画了加法模*函子 $\omega:\mathfrak{C}_q(\mathbb{G}^{\Gamma^2})\rightarrow\mathsf{Hilb}$，它们作为线性*泛函（即态）在 $C^*$-代数上的范畴化。 这些使得我们能够在离散的2维简单多面体 $P$（将三维流形划分为部分）上构造非阿贝尔威尔逊面相关量 $\widehat{\mathfrak{C}}_q(\mathbb{G}^{P})$。 通过证明其在三维手柄体变换下的稳定等价性，这些威尔逊面态扩展到了装饰化的三维标记流形边界在4维圆盘 $D^4$ 中。 这为从给定的紧致李2-群 $\mathbb{G}$ 的数据中提供了有框架的定向2-带的不变量 $D^4$。 我们发现这些2-陈-西蒙斯型2-带不变量由双分级 $\mathbb{Z}$-模给出，类似于Manolescu-Walker-Wedrich的意大利面结型模。 显示英文摘要

摘要： 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 $\omega:\mathfrak{C}_q(\mathbb{G}^{\Gamma^2})\rightarrow\mathsf{Hilb}$, 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.