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 (vol F l oer /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 F l oer /vol ) 2 for the elliptic curve quotient P 1 3,3,3 , which is expected to be related to the choice of a primitive form.

Ungraded matrix factorizations as mirrors of non-orientable Lagrangians

“…At this point it is still conjectural, but we give the outline here since this should be useful in more general settings, for example for RP 2n . In [CLS20], the second author and his collaborators showed that the localized mirror functor is compatible with the open-closed maps. This relied on results from [FOOO16] which do not directly generalize to positive characteristic.…”

“…Although the map OC MF was defined in [PV12] for the category of Z/2-graded factorizations, in characteristic zero, its construction still goes through in our setting. The vertical arrow on the right is the composition of the Kodaira-Spencer map ks : QH • (CP 2n ) → Jac(W ) (see [FOOO16]) with a duality isomorphism I which is an identity in this case (see [CLS20]).…”

Kodaira-Spencer map, Lagrangian Floer theory and orbifold Jacobian algebras

“…The theory for exponentiated variables can be constructed analogously following the localized mirror construction of [10] of Lagrangian torus. We refer readers to Section 3.3 and Lemma 4.2 of [13] to see the summary and basic constructions for the case of exponentiated variables. For a gapped filtered A ∞ -algebra A = (V, {m k }), we assume that at least its cohomology is finite dimensional: If Λ 0 -module V itself is not finitely generated, we take its canonical model following [24] and [19].…”

“…On the other hand, one can easily check that l χ ∪ l χ = 0, l χ 2 ∪ l χ 2 = 0. Also, it is easy to compute that Also, we need the following identity of modular forms, which is proved in [13]. 1 ) Z/3 ∼ = Jac(W ) Z/3 Note that Jac(W ) is generated by 8 elements 1, x 1 , x 2 1 , x 2 , x 2 2 , x 3 , x 2 3 , x 1 x 2 x 3 and Z/3 acts on variables by multiplication of 3rd root of unity.…”

“…Also, as shown in [18], this provides a natural setting to compare Poincaré duality pairing of symplectic manifolds and residue pairing of W L in complex geometry. More recently, we have shown in [13] that these two pairings are conformally equivalent with conformal factor given by the ratio of quantum volume form and classical volume form of L. The results of [13] was restricted to the case of toric manifolds and P 1 a,b,c due to the absence of general Kodaira-Spencer map. Therefore, the construction in this paper can be used to prove a generalization of the results in [13] for the cases of (M , L) satisfying Cardy relations.…”