Is topological string theory a topological field theory? Topological string theory comes from a topological twist (in the suitable spacetimes) of the sigma model defining it. Does the topological twist make the sigma model a topological field theory?
 A: The answer is essentially yes. Topological string theory is a topological field theory of the Witten-type. This is evident when you study the Witten's construction as appeared in the classical references Topological Sigma Models and Mirror Manifolds and Topological Field theory or as is reviewed in the excellent Mini-Course in Topological Strings. 
A subtelty should be recalled. Topological string theory satisfy Witten's axioms (BRST-exact stress tensor and graviton vertex operators, topological observables and metric-independent correlation functions) in the weakly coupled limit (large target space volume) but the holy grail of the theory is to find a definition for the topological string in the compact (target space) case. In that regime things become different because at finite volume Newton's constant becomes finite and the graviton vertex operator is no longer BRST-exact. Interesting developments (related to 6d SCFTS) have been discovered recently: Divulgative , SCFTs, Holography, and Topological Strings.
