energy conservation in a topological quantum field theory Suppose I have a BF-theory with action $S_{BF} = \int_M tr (B \wedge F)$ and the manifold $M$, curvature tensor $F$ and 2-form field $B$. Now there is also coupling to ordinary matter with action $S_m[A]$ and a gauge connection $A$. The final action has the form: 
$S = S_{BF}+S_m[A]+S_{FP}$.
Here, the extra term $S_{FP}$ is the Faddeev-Popov term for nonabelian case. If I compute the partition function $Z = \int d[A] \int d[B] e^{iS}$ I will get the partition function generated by the matter action with additional topological terms. These topological terms depend on the structure of spacetime manifold $M$. Thus, I will have an inhomogenity in spacetime, right?
Spacetime is not homogeneous if it features "holes" at some regions, while at some regions it has no "holes" and therefore Noether's theorem implies that energy is not conserved if the particles are within the region of nontrivial topology. Is my argumentation right?
 A: The question whether energy is conserved or not depends also on the time interval in which the system is considered. It is obvious to make this time interval sufficiently large to avoid "energy excesses" due to Heisenberg's uncertainty principle. $M$ can be decomposed into $n$ components
$M = \cup_{i=1}^N M_i$,
where $M_i$ is the manifold component within the $i$-th (sufficiently large) time interval. The topologies ($H^k$ is $k$-th cohomology class) can differ in general, i.e. $H^k(M_i) \neq H^k(M_j),i \neq j$. Therefore, a well-known particle scattering process gives results that depend explicitely on $M_i$. For a $M_i$ with trivial topology (meaning that $H^k = 0$ for all $k \neq 0$), ordinary scattering amplitudes can be obtained, but otherwise, additional topological corrections must be taken into account while computing partition functions dependent on the manifold components $Z(M_i)$.
Energy is not conserved when regarding finite time intervals in this case, but for time intervals tending to infinity, energy and momentum is clearly conserved.
