It is well known that the theta term
$\int d^4x\frac{\theta}{4\pi}Tr[F\wedge F]=\int d^4x\frac{\theta}{4\pi}\epsilon_{\mu\nu\sigma\lambda}Tr[F^{\mu\nu}F^{\sigma\lambda}]$
is a topological term, since the integral is the instanton number on a 4d manifold (mathematically, the integral is the 2nd Chern character).
Besides, it is definitely metric independent. As far as I know, metric independence is at least a necessary condition for a theory to be a topological quantum field theory (TQFT). For example, ChernSimons and BF are metric independent.
So, now, the questions come:

Is metric independence a sufficient condition for a theory to be a TQFT?

Does the theta term above give rise to a TQFT?
My guess is that both answers are negative, but I am not sure. I checked out some reference and found the mathematical definition of a TQFT, it has to satisfy the AtiyahSegal axioms (see [1] or [2]). But I do not know how to prove whether or not the theta term gives a TQFT, because category is too abstract to me. Could someone give help?
Thanks in advance.
[1] http://en.wikipedia.org/wiki/Topological_quantum_field_theory
[2] http://webusers.imjprg.fr/~christian.blanchet/Articles/EMP_axiomatic_TQFT.pdf
This post imported from StackExchange Physics at 20150402 13:06 (UTC), posted by SEuser Blue