Pairings in Mirror Symmetry Between a Symplectic Manifold and a Landau–Ginzburg B-Model

  • Cheol-Hyun ChoEmail author
  • Sangwook Lee
  • Hyung-Seok Shin


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 gives a modular form.



We would like to thank Kaoru Ono, Atsushi Takahashi, Lino Amorim, Hansol Hong and Siu-Cheong Lau for helpful conversations. We would like to thank Dohyeong Kim at Seoul National University for the proof of mirror identity in “Appendix B”. The work of C.H. Cho is supported by Samsung Science and Technology Foundation under Project Number SSTF-BA1402-05.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical Sciences, Research Institute in MathematicsSeoul National UniversityGwanak-gu, SeoulSouth Korea
  2. 2.Korea Institute for Advanced StudySeoulKorea

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