$\mathrm{U}(1)^{n}$ Chern-Simons theory: partition function, reciprocity formula and Chern-Simons duality

TL;DR

This paper extends $ ext{U}(1)$ Chern-Simons theory to a $ ext{U}(1)^n$ version using combined actions, computes its topological partition function, and explores a reciprocity formula and duality, linking it to Reshetikhin-Turaev invariants.

Contribution

It introduces a $ ext{U}(1)^n$ Chern-Simons theory framework, computes its partition function via Deligne-Beilinson cohomology, and establishes a duality and reciprocity formula for the invariant.

Findings

Partition function is a topological invariant of 3-manifolds.

Derived a reciprocity formula leading to a Reshetikhin-Turaev type invariant.

Demonstrated a duality between different $ ext{U}(1)^n$ Chern-Simons theories.

Abstract

The $\mathrm{U}(1)$ Chern-Simons theory can be extended to a topological $\mathrm{U}(1)^n$ theory by taking a combination of Chern-Simons and BF actions, the mixing being achieved with the help of a collection of integer coupling constants. Based on the Deligne-Beilinson cohomology, a partition function can then be computed for such a $\mathrm{U}(1)^n$ Chern-Simons theory. This partition function is clearly a topological invariant of the closed oriented $3$-manifold on which the theory is defined. Then, by applying a reciprocity formula a new expression of this invariant is obtained which should be a Reshetikhin-Turaev invariant. Finally, a duality between $\mathrm{U}(1)^n$ Chern-Simons theories is demonstrated.