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arXiv · 1810.11172

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

Cheol-hyun Cho·Sangwook Lee·Hyung-Seok Shin
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Abstract

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.

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Cheol-hyun Cho, Sangwook Lee, Hyung-Seok Shin. 2018-10-26. Pairings in mirror symmetry between a symplectic manifold and a Landau-Ginzburg $B$-model. https://doi.org/10.1007/s00220-019-03611-4

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