# BF theory without perturbation theory (spin network appoach)

The BF model given by the action

$$S = \int \mathrm{tr}(B \wedge F)$$

with a 2-form field $B$ and fibre bundle curvature tensor $F$ will have the following partition function after integration over $B$:

$$Z = \int d[A] \, \delta(F) = \int d[A] \, \delta (dA+A\wedge A).$$

Clearly $A=0$ is a (trivial) solution, but a more general solution is $A = g^{-1}dg$ for a group $g$. This group action must not necessarily be a gauge transformation; it can be a group determined by the fibre bundle topology.

Question: Let $H^k$ be the $k$-th cohomology class for Lie algebras (base manifold of the bundle is simply connected and isomorphic to Minkowski spacetime). Will the solution for $A$ be the sum over all possible cohomology classes, i.e. is

$$Z = \sum_{k} \sum_{g \in H^k}1$$

correct or for observables $O(A)$ is

$$\langle O(A)\rangle = \frac{1}{Z}\sum_{k} \sum_{g \in H^k}O(g^{-1}dg)?$$

• Your use of terminology is not entirely clear to me: $A=0$ is a solution to what (the classical equations of motion, perhaps?)? $A= g^{-1}\mathrm{d}g$ is not for a "group" $g$, but for a group-valued function $g : M \to G$, where $M$ is the spacetime and $G$ the group. What does "group determined by the fibre bundle topology" mean? What has the connectedness of the base to do with the $k$-th Lie algebra cohomology, and why do you think the $k$-th Lie algebra cohomology enters here at all? Is your first sum really a sum over ones? How is the $g$ a supposed to be a cohomology class? – ACuriousMind Jan 6 '17 at 14:34
• I try to use a Group $g$ that has a topological degree $\neq 1$ which is the "topological interesting" object I want to consider. – kryomaxim Jan 6 '17 at 14:36
• I would write a detailed answer, but unfortunately, the moderator of this site decided to close the question (probably based on personal hatred towards something he doesn't understand). – Prof. Legolasov Jan 7 '17 at 6:18
• @SolenodonParadoxus 1. Please don't make guesses about other users motives - I did close this question because it is unclear to me, but I also asked a host of specific questions above which were intended to show what needs to be improved in order to make this question clear. Do you claim that the terminology in this question is used correctly, and that you can tell what it is asking without guessing? Then I will admit my error and reopen it. 2. You have enough rep to cast a reopen vote on your own if you think a closure is wrong. – ACuriousMind Jan 7 '17 at 13:45
• @ACuriousMind I didn't mean to offend you. I wanted to emphasize how unhappy I am with the questions which show interest and attempts to research the subject getting closed. Don't you think that the answer which explains in details what is wrong with the original question is a good one? – Prof. Legolasov Jan 7 '17 at 15:27