The U(1) Chern–Simons theory can be extended to a topological 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 U(1)n Chern–Simons theory. This partition function is clearly a topological invariant of the closed oriented three-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 U(1)n Chern–Simons theories is demonstrated.
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