Symmetry theta angles and topological Witten effects

We introduce a large class of $\theta$ angles in quantum field theory that we call symmetry $\theta$ angles. Unlike conventional $\theta$ angles whose definition depends on a choice of a path integral, symmetry $\theta$ angles are intrinsic parameters of a quantum field theory that depend only on its symmetries. A frequent consequence of symmetry $\theta$ angles is a phenomenon we call the topological Witten effect, which is a generalization of the standard Witten effect. Topological Witten effects modify which charged operators are attached to topological operators as a function of $\theta$. Physically, topological Witten effects induce generalized Aharonov-Bohm effects. We show that these new $\theta$ angles and Witten effects can appear in many familiar field theories.