From a class of Calabi–Yau dg algebras to Frobenius manifolds via primitive forms

Abstract

It is one of the most important problems in mirror symmetry to obtain functorially Frobenius manifolds from smooth compact Calabi–Yau $A_\infty$-categories. This paper gives an approach to this problem based on the theory of primitive forms and, in particular, reduces it to a formality conjecture of certain homotopy algebra. Under this formality conjecture, a formal primitive form for a non-negatively graded connected smooth compact Calabi–Yau dg algebra can be constructed, which enable us to have a formal Frobenius manifold.