Pairings in Mirror Symmetry Between a Symplectic Manifold and a Landau-Ginzburg B-Model
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(vo lFloer/v o l ) 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 (vo lFloer/v o l ) 2 for the elliptic curve quotient P3,3 ,3 1, which gives a modular form.
- Publication:
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Communications in Mathematical Physics
- Pub Date:
- November 2019
- DOI:
- 10.1007/s00220-019-03611-4
- arXiv:
- arXiv:1810.11172
- Bibcode:
- 2019CMaPh.375..345C
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematical Physics
- E-Print:
- 35 pages, 5 figures. Comments are welcome