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Three-manifold invariant from functional integration
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We give a precise definition and produce a path-integral computation of the normalized partition function of the abelian U(1) Chern-Simons field theory defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson formalism, we sum over the inequivalent U(1) principal bundles over the manifold and, for each bundle, we integrate over the gauge orbits of the associated connection 1- forms. The result of the functional integration is compared with the abelian U(1) Reshetikhin-Turaev surgery invariant.
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Reshetikhin-Turaev construction and $\mathrm{U}(1)^n$ Chern-Simons partition function
U(1)^n Chern-Simons partition functions coincide, up to normalization, with Reshetikhin-Turaev invariants built from a twisted (non-modular) category.
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