In a previous article, a construction of the smooth Deligne-Beilinson cohomology groups $H^p_D(M)$ on a closed $3$-manifold $M$ represented by a Heegaard splitting $X_L \cup_f X_R$ was presented. Then, a determination of the partition functions of the $U(1)$ Chern-Simons and BF Quantum Field theories was deduced from this construction. In this second and concluding article we stay in the context of a Heegaard spitting of $M$ to define Deligne-Beilinson $1$-currents whose equivalent classes form the elements of $H^1_D(M)^\star$, the Pontryagin dual of $H^1_D(M)$. Finally, we use singular fields to first recover the partition functions of the $U(1)$ Chern-Simons and BF quantum field theories, and next to determine the link invariants defined by these theories. The difference between the use of smooth and singular fields is also discussed.

中文翻译：

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3D 拓扑模型和 Heegaard 分裂。二、庞特里亚金对偶性和可观察性

在前一篇文章中，提出了在封闭的 $3$-流形 $M$ 上构造光滑的 Deligne-Beilinson 上同调群 $H^p_D(M)$，由 Heegaard 分裂 $X_L \cup_f X_R$ 表示。然后，从这个构造推导出 $U(1)$Chern-Simons 和 BF 量子场理论的配分函数的确定。在第二篇也是最后一篇文章中，我们停留在 Heegaard 吐出 $M$ 的上下文中来定义 Deligne-Beilinson $1$-currents，其等效类形成 $H^1_D(M)^\star$ 的元素，庞特里亚金对$H^1_D(M)$。最后，我们使用奇异场首先恢复$U(1)$Chern-Simons 和BF 量子场论的配分函数，然后确定这些理论定义的链接不变量。还讨论了使用平滑场和奇异场之间的区别。