Good afternoon.
Let $M$ be, say, a compact symplectic manifold. Both deformation quantization (as in Kontsevich) and quantum cohomology yield "deformations" (in the appropriate respective senses) of "classical" data — the Poisson algebra of functions $C^\infty(M)$ and the cohomology algebra (or rather, Frobenius algebra) $H^\ast(M; \mathbb{C})$ respectively.
Are these two things related somehow?
I am interested in both mathematical and physical answers.
I apologize if this question is naive. I feel like, with a proper understanding of the physics, the answer to this question is probably obviously "yes" or obviously "no". Unfortunately, I don't have a good understanding of the physics.
Edit: From the looks of the discussion below, deformation quantization is perhaps more directly related to the Fukaya category. I welcome any additional remarks on the Fukaya category.
Best Answer
Yep. The connection I have in mind comes from recent work of Gukov & Witten: "Branes and Quantization". If you've got a symplectic manifold $M$, you can sometimes make the Kontsevich deformation of $C^\infty(M)$ act on the vector space of $A$-model states which describe strings in complexification $Y$ of $M$ which begin on the canonical co-isotropic brane in $Y$ and end on $M$ (which you can think of as a Lagrangian brane in $Y$).