BF theory secretly has another name - Zn gauge theory in the deconfined limit. The parameter n is what appears in front of the BF action. Zn gauge theory can be defined on any manifold you want - just introduce a lattice approximation of the manifold and compute the lattice gauge theory partition function. By taking the extreme deconfined limit of the lattice gauge theory one can verify that this construction is independent of the way you approximated the manifold.
Basic examples
In the deconfined limit the Zn flux through every plaquette of the lattice gauge theory is zero. (This is like the constraint imposed by integrating out B.) We have a residual freedom to specify the holonomy around all non-contractible loops. Hence Zn(M)=Zn(S4)nb1(M) where Zn(S4) is a normalization constant. Requiring that Zn(S3×S1)=1 gives Zn(S4)=1/n. This condition, Zn(S3×S1)=1, is the statement that the theory has one unique ground state on S3. In general, Zn(Σ3×S1) is tr(e−βH), but since H=0 we are simply counting ground states.
As a quick check, Zn(S2×S1×S1)=n, which is the number of ground states on S2×S1, and Zn(T3×S1)=n3, which is the number of ground states on T3=(S1)3.
Further relations
Another check on the value of Zn(S4): this is the renormalized topological entanglement entropy of a ball in the 3+1d topological phase described by deconfined Zn gauge theory. More precisely, the topological entanglement entropy of a ball is −lnZn(S4) which gives −logn in agreement with explicit wavefunction calculations.
We can also consider defects. The BF action is n2π∫B∧dA. Pointlike particles (spacetime worldlines) that minimally couple to A carry Zn charge of 1. Similarly, string excitations (spacetime worldsheets) that minimally couple to B act like flux tubes carry Zn flux of 2π/n. Now when a minimal particle encircles a minimal flux, we get a phase of 2π/n (AB phase), but this also follows from the BF action. Without getting into two many details, the term in the action like B12∂tA3 fixes the commutator of A3 and B12 to be [A3(x),B12(y)]=2πinδ3(x−y) (flat space). The Wilson-line like operators given by WA=ei∫dx3A3 and WB=ei∫dx1dx2B12 thus satisfy WAWB=WBWAe2πi/n which is an expression of the braiding flux above that arises since the particle worldline pierced the flux string worldsheet.
Comments on the comments
Conservative thoughts
If I understand you correctly, what you want to do is sort of argue directly from the continuum path integral and show how the asymmetry you mentioned arises. I haven't done this calculation, so I can't be of direct help on this point right now. I'll ponder it though.
That being said, it's not at all clear to me that treating the action as ∫AdB leads one to just counting 2-cycles. Of course, I agree that 2-cycles are how you get non-trivial configurations of B, but in a conservative spirit, it's not at all clear to me, after gauge fixing and adding ghosts and whatever else you need to do, that the path integral simply counts these.
The way that I know the path integral counts 1-cycles is that I have an alternative formulation, the lattice gauge theory, that is easy to define and unambiguously does the counting. However, I don't know how it works for the other formulation. I suppose a naive guess would be to look at lattice 2-gauge theory for B, but this doesn't seem to work.
Ground states
One thing I can address is the issue of ground states. In fact, I favor this approach because one doesn't have to worry about so many path integral subtleties. All you really needs is the commutator.
Take the example you raise, S2×S1. There are two non-trivial operators in this case, a WB for the S2 and a WA for the S1. Furthermore, these two operators don't commute, which is why there are only n ground states. If we define |0⟩ by WA|0⟩=|0⟩, then the n states {|0⟩,WB|0⟩,...,Wn−1B|0⟩} span the ground state space and are distinguished by their WA eigenvalue. Importantly, you can also label states by WB eigenvalue, but you still only get n states. Poincare duality assures you that this counting of states will always work out no matter how you do it.
Furthermore, the operator WB has a beautiful interpretation in terms of tunneling flux into the non-contractible loop measured by WA. It's easier to visualize the analogous process in 3d, but it still works here.
You can also see the difference between the theory in 4d and 4d. Since both B and A are 1-forms you have different possibilities. The analogue of S2×S1 might be S1×S1 and this space does have n2 states. However, that is because both A and B can wrap both cycles.
The remarkable thing is that the Zn gauge theory formulation always gets it right. You simply count the number of one-cycles and that's it.
Getting at other spaces
The state counting approach gets you everything of the form Zn(Σ3×S1), but even spaces like S2×S2 can be accessed. You can either do the direct euclidean gauge theory computation, or you can regard Zn(S2×S2)=|Z(S2×D2)|2 i.e. the inner product of a state on S2×S1=∂(S2×D2) generated by imaginary time evolution.
This post imported from StackExchange Physics at 2014-04-05 03:00 (UCT), posted by SE-user Physics Monkey