Topological defects in general and Chern-Simons in particular I'm trying to gain intuition on some physical concepts that I cannot yet fully understand, and I think many of you can help me.
Is it correct to think of of a topological defect as the addition ad hoc of some term to an action that couples appropiately with the physical fields of the theory? For example, if I have a gauge connection, this can couple to a curve, and I can integrate the gauge connection along the curve.
Is it correct to say that adding this term to the (hypothetical) action "creates" (clearly in a  a topological defect in the theory? Can there be dynamically created defects that werent ad hoc inserted? In the first case, I'm adding a defect classically, so this is present before any quantization, But I also see that one considers the expectation value of the term, I dont understand what's the meaning of this, one considers how quantum fluctuations alter the classical value?
What happens if I do the same procedure whitout adding the defect classically?
Maybe my questions are a bit vague, I apologize but I'm really struggling to understand this things, that are not so mathematically precise I think are more of an intuitional nature. I thank you in advance.
 A: Your questions are a bit vague so if my answer isn't satisfying please specify what exactly you want to know. I am also still struggling to get my head around the concept and I am far from an expert but here is what I know.
So the Chern-Simons term,
\begin{equation} 
-\theta_{ab} F^a_{\mu \nu} \tilde{F}^{a \mu \nu}.
\end{equation}
is present because it can be. This might seem ad-hoc but we are trying to construct the most general gauge-invariant Lagrangian. Having said this, there is also a method of introducing such a term through anomalies. The term is actually equal to a divergence, the Chern-Simons current,
\begin{equation} 
F^{a}_{ \mu \nu} \tilde{F}^{a \mu \nu} = \partial_\mu K^\mu, \quad K^\mu = 2 \epsilon^{\mu \nu \rho \sigma} \left( A_\nu^a \partial_\rho A_\sigma^a + \frac{g}{3} f^{abc} A^a_\nu A^b_\rho A^c_\sigma \right).
\end{equation}
This implies that the term has no influence on perturbation theory, but can create non-perturbative effects. This is why there is no such term in QED.
One way of interpreting this term and where it becomes important is in the study of the ground state. The ground state of a gauge theory is pure gauge, i.e.
\begin{equation} 
A^\mu = U^{-1} \partial U.
\end{equation}
This is obvious because $A^\mu = 0$ is definitely a ground state and we can still perform gauge transformations. Now consider $U \in SU(2)$ and work in the temporal gauge $A_0 = 0$. Then we can identify $SU(2)$ with the 3-sphere $\mathcal{S}_3$ and we can also identify $A^i \in \mathbb{R}^3$ with the 3-sphere for example through a stereographic projection. Thus a gauge transformation is a mapping $\mathcal{S}_3 \longrightarrow \mathcal{S}_3$. These kinds of mappings are classified into homotopy classes distinguished by the winding number $\nu$. One can show that the difference between two winding numbers is
\begin{equation} 
\nu_2 - \nu_1 = - \frac{g^2}{32 \pi^2} \int_{t_1 \le x^0 \le t_2} d^4x\; F^{a}_{\mu \nu} \tilde{F}^{a \mu \nu}.
\end{equation}
So here the Chern-Simons current becomes important again. Transitions between different winding numbers must therefore proceed through non-vanishing $F^{a\mu \nu}$. The vacuum states can then tunnel through these potentials. The true vacuum of a theory is then a superposition of states with different winding numbers. If one takes this into account, then the Chern-Simons term has to be included naturally in the Lagrangian,
\begin{equation} 
\mathcal{L}_\text{eff} = \mathcal{L} - \theta \frac{g^2}{32 \pi^2} F^{a \mu \nu} \tilde{F}^a_{\mu \nu}.
\end{equation}
Furthermore, the energy of the vacuum state is dependent on the mixing angle $\theta$.
By the way, this does not occur classically because it relies on the tunneling effect which is only a feature of quantum mechanics.
