There is a "folk theorem" (alternatively, a fun and easy exercise) which asserts that a 2D TQFT is the same as a commutative Frobenius algebra. Now, to every compact oriented manifold $X$ we can associate a natural Frobenius algebra, namely the cohomology ring $H^\ast(X)$ with the Poincare duality pairing. Thus to every compact oriented manifold $X$ we can associate a 2D TQFT.


Is this a coincidence? Is there any reason we might have expected this TQFT to pop up?


When $X$ is a compact *symplectic* manifold, perhaps the appearance of the Frobenius algebra can be explained by the fact that the quantum cohomology of $X$, which comes from the A-twisted sigma-model with target $X$, becomes the ordinary cohomology of $X$ upon passing to the "large volume limit".


But for a general compact oriented $X$? I don't see how we might interpret the appearance of the Frobenius algebra in some quantum-field-theoretic way. Maybe there is an explanation via Morse homology?



Cohomology rings and 2D TQFTs
These 2D TQFTs do not come from extended theories (unless X is discrete). I interpret this as saying that these theories are non-local (in the 2D bordism) and so you will have trouble interpreting them in a traditional QFT framework. You will have to do something funny and non-local, like squashing your circles to points and surfaces to graphs, as in the Cohen work mentioned by Tim. 


