Does the flatness of a gauge field has anything to do with whether it's dynamical? One common way in studying Symmetry Protected Topological(SPT) phases with a global symmetry G is to promote G to a gauge symmetry and couple the system to a flat gauge field A for G. Then one can integrate out the matter fields and obtain an effective action which depends on A as well as background geometry. The gauge field A must be flat because coupling the system to such a field is unambiguous (this is essentially the minimal coupling prescription which tells us to replace ordinary derivatives with covariant derivatives) (c.f. section 2 of http://arxiv.org/abs/1404.6659). In these cases, the gauge field A is some background gauge field which is non-dynamical. 
I was a bit confused about the distinction between dynamical gauge fields and non-dynamical gauge fields, and in particular, if it has anything to do with whether the gauge field is flat. I know that for non-dynamical gauge fields, we do not add its kinetic term $F\wedge*F$ in the action, and we do not sum over $A$ configurations in the path integral, and for dynamical gauge fields we do the converse. However, if the gauge field $A$ is flat, which is always the case when the gauge group $G$ is finite, what is the distinction between dynamical and non-dynamical gauge fields. The curvature in this case always vanishes, and there is no kinetic term in the action for both cases. I know there is still the difference of whether we sum over $A$ configurations in the path integral. But physically what is the distinction. 
 A: Take 2D SPT as an example. When the gauge fields are not dynamical, physically it really means that we are just modifying the Hamiltonian to a certain gauge field configuration (i.e. by changing the coupling of some of the terms, for example). These "gauge fields" are just extrinsic parameters in the Hamiltonian. 
To detect the SPT, we need to introduce extrinsic symmetry fluxes into the Hamiltonian, or when it is on a torus, introducing twisted boundary conditions (still flat connections). We might want to use the braiding statistics of these fluxes to distinguish the SPT phases. However, one needs to be careful about the braiding statistics of the fluxes of non-dynamical nature, the reason of which being there is a branch cut attached (in the dynamical case, the "magnetic" term in the Hamiltonian renders the branch cut completely invisible in the ground state). Most notable example is the topological spin of a flux. Usually, one would define topological spin as the phase acquired after rotating the flux by $2\pi$. However, in the process the branch cut is dragged around, and to return to the initial configuration, one needs to pull the branch cut over the flux. Since the flux is just some extrinsic defect, it is possible that there are some gauge charges trapped near the flux, and additional phase can come in when the branch cut is pulled over. Therefore, the topological spin is only defined up to the braiding phase of charge and flux.
To summarize, in a dynamical gauge theory the branch cut is invisible, and a flux has a well-defined charge (thus a dyon). If the gauge fields are non-dynamical, then we do not have a way to measure absolutely how much charges a flux carries (we can tell the difference). I think this is the most important distinction between dynamical and non-dynamical gauge fields. Similar ideas also work for 3D SPT phases.
