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

Abstract

In a previous article, a construction of the smooth Deligne-Beilinson cohomology groups $H_D^p(M)$ 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 $H_D^1(M{)}^{\star }$, the Pontryagin dual of $H_D^1(M)$. 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.

References in corpus (1)

• Geometric aspects of interpolating gauge fixing in Chern-Simons Theory