3D Topological Models and Heegaard Splitting II: Pontryagin duality and Observables

International audience; In Paper I [F. Thuillier, “3D topological models and Heegaard splitting I: Partition function,” J. Math. Phys. 60, 32 (2019)], a construction of the smooth Deligne–Beilinson cohomology groups HDp(M) on a closed 3-manifold M represented by a Heegaard splitting XL ∪fXR was presented. Then, the partition functions of the U(1) Chern–Simons and BF Quantum field theories were determined 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 HD1(M)⋆, the Pontryagin dual of HD1(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.