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

TL;DR

This paper extends the study of 3D topological models by exploring Pontryagin duality and observables within the framework of Heegaard splittings, providing new insights into partition functions and link invariants in quantum field theories.

Contribution

It introduces the use of Deligne-Beilinson 1-currents and singular fields to analyze $U(1)$ Chern-Simons and BF theories on 3-manifolds with Heegaard splittings, advancing the understanding of their partition functions and link invariants.

Findings

Defined Deligne-Beilinson 1-currents and their duals on 3-manifolds.

Reconstructed partition functions using singular fields.

Analyzed differences between smooth and singular field approaches.

Abstract

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.