TQFT's as effective theories of the groundstate subspace I often hear: "The degenerate groundstate subspace of a QFT is often a TQFT". 
I'm trying to work out an example of this for, say, superconductors: In the context of condensed matter physics, the spacetime of a 2D superconductor is
$$\Sigma\times \mathbb R$$
Where $\Sigma$ is some compact, oriented 2-manifold. Now let's consider the Ginzburg-Landau model, which is a good effective QFT for a superconductor:
$$S_\text{GL}(\phi,A)\equiv \int_{\Sigma\times \mathbb R} (|\nabla^{2A}\phi|^2+|F_A|^2)\,d^3x$$
Now this model has classical solution space $H^1(\Sigma;\mathbb Z_2)$, based off of recognizing that, classically, $\nabla^{2A}$ is a flat spin connection and thus we are just counting spin structures on $\Sigma$. Moreover, this action is supposed to represent, at $T=0$, the groundstate energy of a superconductor.
Anyways, my questions are, (1), does this mean that the classical solutions of the GL model correspond to quantum mechanical groundstates?  (2) how is the GL model a topological quantum field theory (because I see a $U(1)$ Yang-Mills term and a Bochner Laplacian)?
If the GL model is a TQFT, then if we write 
$$Z_\text{GL}(\Sigma\times [0,t])=\int_{C^\infty(\cdots)} D\phi\,DA\, \exp(iS_\text{GL}/\hbar),$$
This is supposed to represent a linear map $U(t):H^1(\Sigma;\mathbb Z_2)\to H^1(\Sigma;\mathbb Z_2)$. However, since we're on a cylinder, the time-evolution should be trivial, no? 
Also: Where do we get all the nice quasiparticle braiding? if we add quasiparticles to the action, they certainly do not affect the topology of $\Sigma$ (they're not massive enough). But then, because of diffeomorphism-invariance, quasiparticle processes do not affect the partition function - so we can't switch between groundstates - and so topological quantum computing is impossible!
 A: One can analyze this in a limit where the LG potential is very strong, ie. we study
$$\int |\nabla_A \phi|^2 + |dA|^2 + g^2(|\phi|^2 - a^2)^2$$
with $g^2 >> 1$. We separate $\phi$ into amplitude and phase parts separately, and the amplitude fluctuations of $\phi$ are very massive, centered around their vev $|\phi|=a$. Meanwhile, the phase fluctuations are quick. The effective Lagrangian for the phase $\theta$ is
$$a^2|d\theta - 2A|^2 + |dA|^2.$$
Now we study the limit $a^2 >>1$, for which we obtain the constraint
$$2A = d\theta.$$
We see that this forces $dA = 0$, which knocks out the Maxwell term, and also forces $A$ to have holonomy $\int A = 0$ or $\pi$ around all cycles. So we see that the TQFT we get is the (untwisted) $\mathbb{Z}_2$ gauge theory. For more calculations like these, I recommend https://arxiv.org/abs/1307.4793 and https://arxiv.org/abs/1308.2926 .
To actually "see" the nontrivial braiding takes a bit of work. In this case the theory has a fermionic quasiparticle, which is a bound state between the unit charge
$$\exp i \int_\gamma A$$
and the $\pi$-flux, which is a disorder operator along a worldline $\gamma'$ which forces $\int_S dA = \pi$ through every surface $S$ intersecting $\gamma'$, so you can think of it as a narrow flux tube around $\gamma'$ containing $\pi$ units of magnetic flux. It's easy to see that this braids the Wilson loop above with a minus sign, and therefore that their bound state is a fermion (they are both bosons on their own, which can also be checked).
