Anyon and loop braiding statistics in field theories with a topologicalΘterm

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.