Homological Mirror Symmetry for Fano Orbifolds: Lagrangian Floer Theory and the Derived Fukaya Category

We establish a rigorous framework for Homological Mirror Symmetry (HMS) in the context of Fano orbifolds, extending the classical correspondence between the symplectic geometry of the A-model and the complex geometry of the B-model. Specifically, we construct the derived Fukaya category F(X) for a symplectic Fano orbifold X by incorporating twisted sectors into the Lagrangian Floer theory via Chen-Ruan cohomology classes. We prove that the A-structure on F(X) is unobstructed under suitable bulk deformations arising from the orbifold sectors. Conversely, we analyze the mirror Landau-Ginzburg model (Y, W), where Y is a non-compact Kähler manifold and W is a holomorphic superpotential. We demonstrate an equivalence of triangulated categories DF(X) .5-.5.5-.5.5-.5.5-.5MF(Y, W), where MF(Y, W) denotes the category of matrix factorizations associated with the singularities of W. This correspondence is validated through explicit computations of the quantum cohomology ring QH*orb(X) and the Jacobian ring of W, confirming the predictions of the generalized Dubrovin conjecture for orbifold targets.