What does it mean to say we have a QFT? When can we say that the particular action/ Lagrangian we write is an QFT? Does it have to be perturbatively renormalisable? Not allow for negative norm states? The gauge theories we write in 4D, Witten says they are incomplete.

one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time.

What does he mean by the above statement?
 A: QFT (quantum field theory) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. In particle physics it is used to construct physical models of subatomic particles. The equations of motion are obtained by means of an action principle, which is built via a Lagrangian density or simply Lagrangian.  
QFT treats particles as excited states (quanta) of the underlying fields. Interactions between particles are described by interaction terms in the Lagrangian, involving their corresponding fields, as a perturbation expansion.  
A QFT can be renormalizable, that is requiring a finite number of counterterms to cancel all the infinities, or non-renormalizable, i.e. requiring an infinite number of counterterms. However, what is expected in either occurrences is that the theory is predictive, at least in some energy regimes.  
The Lagrangian should exibit Lorentz-invariance, but can show global or local symmetries as well. A local symmetry is a gauge symmetry and it fixes redundant degrees of freedom the theory may have.  
A negative norm state is unphysical. It does not describe a real particle.
A: The key words in Witten's statement is 'mathematically complete'.  He's referring to the lack of a mathematically rigorous construction of an interacting 4d QFT. (Non-interacting 4d QFTs -- free scalar, free Dirac, free Maxwell theory, etc - are mathematically complete. They're just not that exciting.) Saying that you've mathematically constructed a QFT means that you can exhibit all of its observables as operators on a Hilbert space at a level of detail that would satisfy mathematicians.
