Smoothness/properness of the Fukaya category and comparison with quantum cohomology

Establish that the Fukaya category F(X) is smooth and proper for general compact symplectic manifolds X in the A-model monotone setting, and prove that the Hochschild cohomology HH^*(F(X)) agrees with the quantum cohomology QH^*(X).

Background

The categorical form of ‘quantization commutes with reduction’ would have broad consequences if F(X) possessed standard finiteness properties and if its Hochschild cohomology coincided with the quantum cohomology of X.

The authors note that these properties are not currently known in general, limiting immediate Gromov–Witten consequences derivable from the conjectured categorical equivalence.