Lets briefly explain the whole of picture:
A-model and B-model is study of twisted conformal field theory
If you want to study the invariants of complex manifolds , then you will face with two notion due to Edward Witten philosophy
A) Closed string B-model which is study of Hodge structure or periods of holomorphic differential forms
B-model on Calabi-Yau three-fold can be described by a Kodaira-Spencer gauge theory(due to Bershadsky-Cecotti-Ooguri-Vafa). sometimes called BCOV theory.
B) Open string B-model is study of Category of coherent sheaves on complex manifold(for example complex submanifolds equiped with holomorphic vector bundles)
If you want to study the invariants of symplectic manifolds, then we are facing with two object
A) Closed string A-model , which is study of Gromov-Witten invariants about counting holomorphic curves
B) Open string A-model which is Fukaya category (object is lagrangian submanifolds and morphisms is Floer chain )which corresponding to the Calabi–Yau category
From homological mirror symmetry point of view (due to M. Kontsevich conjecture), the A-model is dual to the B-model. $$D^\pi\mathcal F(M)\cong D^b Coh(M^\vee)$$ which called Meta-Mirror symmetry
Moreover, in the language of moduli spaces if $M$ and $M^\vee$ are two mirror to each other then $$\mathcal M_{sym}(M)\cong \mathcal M_{cpx}(M^\vee)$$ where $\mathcal M_{sym}(M)$ and $\mathcal M_{cpx}(M)$ are moduli spaces of symplectic structures and moduli spaces of complex structures respectively. This gives an equivalence of A-module and B-module for families
For your question see the Kontsevich paper
M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, around page 19 in The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997
and this thesis
