cohomological field theory

TQFT: intersection theory, equivariant cohomology
Cohomological field theory is the Witten type TQFT. Common examples are BRST quantization and CFT.

1. Q nilpotent: $Q^2=0$;
2. physical states are Q-closed ((anti-)commutator);
3. energy-momentum tensor is Q-exact;
4. vacuum is Q-closed (annihilate).

Then, in the meaning of correlators, Q-exact is s trivial insertion and correlators are independent of the metric (suppose only the action explicitly contains the metric). If we set the action be Q-exact, we can similarly prove that correlators are independent of $\hslash$.
Non-local observables can be constructed from a local observable $\mathcal{O}$: $[Q,G_{\mu }]={\partial }_{\mu }$ (local translation is Q-exact so the theory is topological), $G=G_{\mu }d{x}^{\mu }$, $e^{G}Q{e}^{-G}=Q+d\to (Q+d)(e^G\mathcal{O})=0\to Q\int e^G\mathcal{O}=0$.

In 2d, correlators can be factorized w.r.t. genus.