Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

Cho, Cheol-Hyun; Hong, Hyunkee; Lau, Siu Cheong

doi:10.4310/jdg/1493172094

Cited by 37 publications

(110 citation statements)

References 50 publications

Supporting

Mentioning

110

Contrasting

Order By: Relevance

“…where φ(q) and ψ(q) are series in q, and − log q is the area of the minimal XY Z triangle (see [CHL,Section 6.1], where q α there is denoted by q here). By a change of coordinates on (x, y, z), W can be rewritten as…”

Section: Enumerative Meaning Of Mirror Maps Of Elliptic Curve Quotientsmentioning

confidence: 99%

Lagrangian Floer potential of orbifold spheres

Cho

Hong

Kim

et al. 2017

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov-Witten potential, which serves as the quantum-corrected Landau-Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov-Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.

show abstract

Section: Enumerative Meaning Of Mirror Maps Of Elliptic Curve Quotientsmentioning

confidence: 99%

Lagrangian Floer potential of orbifold spheres

Cho

Hong

Kim

et al. 2017

Advances in Mathematics

Self Cite

View full text Add to dashboard Cite

show abstract

“…Cho, Hong, and Lau [4] described open Gromov-Witten potentials for elliptic orbifolds (and homological mirror symmetry). Explicit expressions for these were computed by Cho, Hong, Kim, and Lau [3].…”

Section: Intorduction and Statement Of Resultsmentioning

confidence: 99%

Indefinite theta functions arising in Gromov–Witten theory of elliptic orbifolds

Bringmann

Kaszian

Rolen

2018

Cambridge Journal of Mathematics

View full text Add to dashboard Cite

In this paper, we consider natural geometric objects coming from Lagrangian Floer theory and mirror symmetry. Lau and Zhou showed that some of the explicit Gromov-Witten potentials computed by Cho, Hong, Kim, and Lau are essentially classical modular forms. Recent work by Zwegers and two of the authors determined modularity properties of several simpler pieces of the last, and most mysterious, function by developing several identities between functions with properties generalizing those of the mock modular forms in Zwegers' thesis. Here, we complete the analysis of all pieces of Cho, Hong, Kim, and Lau's functions, inspired by recent work of Alexandrov, Banerjee, Manschot, and Pioline on similar functions. Combined with the work of Lau and Zhou, as well as the aforementioned work of Zwegers and two of the authors, this affords a complete understanding of the modularity transformation properties of the open Gromov-Witten potentials of elliptic orbifolds of the form P 1 a,b,c computed by Cho, Hong, Kim, and Lau. It is hoped that this will provide a fuller picture of the mirror-symmetric properties of these orbifolds in subsequent works.

show abstract

“…In this section, we recall the formalism of localized mirror functors developed in [6] and [7]. Later we will use this to understand Kapustin-Li pairing in terms of Lagrangian Floer theory.…”

Section: Localized Mirror Functorsmentioning

confidence: 99%

“…In fact, in the above work in progress, they discuss all cases P 1 a,b,c with bulk deformation by twisted sectors. In [6], it is shown that…”

Section: 2mentioning

confidence: 99%

“…For closed string mirror symmetry, we consider the Kodaira-Spencer map ks (as in [17]). For open string mirror symmetry, we use the homological mirror functor F L from localized mirror construction of [6] and [7]. There an explicit homological mirror functor has been constructed by studying formal deformation space of a particular (immersed) Lagrangian submanifold L, which is called reference Lagrangian.…”

mentioning

confidence: 99%

See 1 more Smart Citation

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

Cho¹,

Lee²,

Shin³

2019

Commun. Math. Phys.

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Localized mirror functor for Lagrangian immersions, and homological mirror symmetry for $\mathbb{P}^1_{a,b,c}$

Cited by 37 publications

References 50 publications

Lagrangian Floer potential of orbifold spheres

Lagrangian Floer potential of orbifold spheres

Indefinite theta functions arising in Gromov–Witten theory of elliptic orbifolds

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

Contact Info

Product

Resources

About