We present basic constructions and properties in arithmetic Chern–Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set $S$ of finite primes of a number field $k$, we construct arithmetic analogues of the Chern–Simons $1$-cocycle, the prequantization bundle for a surface and the Chern–Simons functional for a $3$-manifold. We then construct arithmetic analogues for $k$ and $S$ of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf–Witten partition function in $(2+1)$-dimensional Chern–Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally, we show decomposition and gluing formulas for arithmetic Chern–Simons invariants and arithmetic Dijkgraaf–Witten partition functions.

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关于算术 Dijkgraaf-Witten 理论

我们沿着拓扑量子场论的路线，提出了具有有限规范群的算术 Chern-Simons 理论的基本结构和性质。对于数域 $k$ 的有限素数的有限集 $S$，我们构造 Chern-Simons $1$-cocycle 的算术类似物，表面的预量化束和 $3$-流形的 Chern-Simons 泛函. 然后，我们为量子希尔伯特空间（共形块空间）的 $k$ 和 $S$ 以及 $(2+1)$ 维 Chern-Simons TQFT 中的 Dijkgraaf–Witten 配分函数构造算术模拟。我们展示了这些算术类似物的一些基本和函数特性。最后，我们展示了算术 Chern-Simons 不变量和算术 Dijkgraaf-Witten 配分函数的分解和粘合公式。