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

TL;DR

This paper explores the relationship between Lagrangian Floer pairing in symplectic geometry and Kapustin-Li pairing in Landau-Ginzburg models, revealing a conformal equivalence involving a volume ratio and introducing a new B-invariant.

Contribution

It establishes a conformal equivalence between Floer and Kapustin-Li pairings, introduces a B-invariant in Floer cohomology, and proves a generalized Cardy identity under certain conditions.

Findings

Conformal equivalence with a specific volume ratio factor.

Introduction of B-invariant valued in the Jacobian ring.

Explicit computation of the conformal factor for an elliptic curve quotient.

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.