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It is one of the most important problems in mirror symmetry to obtainfunctorially Frobenius manifolds from smooth compact Calabi-Yau$A_\infty$-categories. This paper gives an approach to this problem based onthe theory of primitive forms. Under an assumption on the formality of acertain homotopy algebra, a formal primitive form for a smooth compactCalabi-Yau dg algebra can be constructed, which enable us to have a formalFrobenius manifold.

From Calabi-Yau dg Categories to Frobenius manifolds via Primitive Forms