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Anyon and Loop Braiding Statistics in Field Theories with a Topological $Θ-$term

We demonstrate that the anyon statistics and three-loop statistics of various 2d and 3d topological phases can be derived using semiclassical nonlinear Sigma model field theories with a topological $Θ$-term. In our formalism, the braiding statistics has a natural geometric meaning: The braiding process of anyons or loops leads to a nontrivial field configuration in the space-time, which will contribute a braiding phase factor due to the $Θ$-term.