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From a class of Calabi-Yau dg algebras to Frobenius manifolds via primitive forms. (English) Zbl 1452.14041

Hori, Kentaro (ed.) et al., Primitive forms and related subjects – Kavli IPMU 2014. Proceedings of the international conference, University of Tokyo, Tokyo, Japan, February 10–14, 2014. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 83, 389-415 (2019).
Summary: 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.
For the entire collection see [Zbl 1446.53004].

MSC:

14J33 Mirror symmetry (algebro-geometric aspects)
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
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