Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model

• Main Authors: Cho, Cheol-hyun, Lee, Sangwook, Shin, Hyung-Seok
• Format: Journal Article
• Language: English
• Published: 25.10.2018
• Subjects: Mathematics - Mathematical Physics; Mathematics - Symplectic Geometry; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.1810.11172

Comm. Math. Phys. (2020), 375(1), 345-390 We find a relation between Lagrangian Floer pairing of a symplectic manifold and Kapustin-Li pairing of the mirror Landau-Ginzburg model under localized mirror functor. They are conformally equivalent with an interesting conformal factor $(vol^{Floer}/vol)^2$, which can be described as a ratio of Lagrangian Floer volume class and classical volume class. For this purpose, we introduce $B$-invariant of Lagrangian Floer cohomology with values in Jacobian ring of the mirror potential function. And we prove what we call a multi-crescent Cardy identity under certain conditions, which is a generalized form of Cardy identity. As an application, we discuss the case of general toric manifold, and the relation to the work of Fukaya-Oh-Ohta-Ono and their $Z$-invariant. Also, we compute the conformal factor $(vol^{Floer}/vol)^2$ for the elliptic curve quotient $\mathbb{P}^1_{3,3,3}$, which is expected to be related to the choice of a primitive form.