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The proper Landau-Ginzburg potential is the open mirror map
Algebraic Geometry · Submitted 26 Apr 2022
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero.
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The abstract, sentence by sentence
The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. | background 1.00 | 0.09 | 0.05 | asserted 1.00 |
| s1 | Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. | background 1.00 | 0.05 | 0.02 | asserted 1.00 |
| s2 | When $E$ is smooth, the superpotential is proper. | background 0.65 | 0.16 | 0.05 | asserted 1.00 |
| s3 | We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. | result 0.95 | 0.95 | 0.82 | asserted 1.00 |
| s4 | As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. | result 0.83 | 0.83 | 0.21 | asserted 1.00 |
| s5 | Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. | contribution 0.77 | 0.95 | 0.04 | asserted 1.00 |
| s6 | In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. | contribution 0.85 | 0.93 | 0.03 | asserted 1.00 |
| s7 | One of the two proofs that we give introduces birational modifications of a scattering diagram. | method 0.85 | 0.93 | 0.03 | asserted 0.99 |
| s8 | We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram. | result 0.66 | 0.89 | 0.05 | asserted 1.00 |
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The paper
Original abstract
The mirror dual of a smooth toric Fano surface $X$ equipped with an anticanonical divisor $E$ is a Landau-Ginzburg model with superpotential, W. Carl-Pumperla-Siebert give a definition of the the superpotential in terms of tropical disks using a toric degeneration of the pair $(X,E)$. When $E$ is smooth, the superpotential is proper. We show that this proper superpotential equals the open mirror map for outer Aganagic-Vafa branes in the canonical bundle $K_X$, in framing zero. As a consequence, the proper Landau-Ginzburg potential is a solution to the Lerche-Mayr Picard-Fuchs equation. Along the way, we prove a generalization of a result about relative Gromov-Witten invariants by Cadman-Chen to arbitrary genus using the multiplication rule of quantum theta functions. In addition, we generalize a theorem of Hu that relates Gromov-Witten invariants of a surface under a blow-up from the absolute to the relative case. One of the two proofs that we give introduces birational modifications of a scattering diagram. We also demonstrate how the Hori-Vafa superpotential is related to the proper superpotential by mutations from a toric chamber to the unbounded chamber of the scattering diagram.
Human review
How the literature uses it
6 arXiv papers cite this in 20 sentences
Papers citing it · 16 sentences without a confident label
Builds on it 1 paper
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The goal of this paper is to generalize the result of [ref] to all dimensions via a direct computation of two-point relative Gromov–Witten invariants.
The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map · Introduction · 1.00
Uses its data or values 1 paper
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gives the broken lines to compute the log Gromov-Witten invariant $R_1,5(ℙ^2,E)=25$ of conics meeting $E$ in a fixed point with multiplicity $1$ and another point with multiplicity $5$ as in [ref] , Appendix B.
Counting (tropical) curves via scattering.sage · Broken lines · 0.82
Cites it as background 2 papers
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This generating function of these open invariants is referred to as the open mirror map in [ref] .
The proper Landau--Ginzburg potential, intrinsic mirror symmetry and the relative mirror map · Toric varieties with a smooth, nef anticanonical divisor · 0.80
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Also in related work to appear in Advances in Math, in [ref] Gräfnitz, Ruddat, and Zaslow obtain an open mirror map for del Pezzo surfaces via wall-crossing of the Hori–Vafa LG potential.
Mirror Map for Landau-Ginzburg models with nonabelian groups · Introduction · 0.94
What it cites · 71 sentences without a confident label
Builds on 1 paper
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In this section, we generalize the theta function multiplication rule from its genus zero version in [ref] , assuming $B=2$, to arbitrary genus and then use the rule to prove relations among the two-point invariants that we introduced in the previous section.
Theta functions on varieties with effective anti-canonical class · Theta calculations · 1.00
Uses the method or tool of 3 papers
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Since this potential can be easily computed in the central chamber to agree with the Hori–Vafa superpotential, we can fix an order $k$ for the wall structure and relate that to the function $M(Q)$ mod $t^k$ by a finite series of mutations à la [ref] .
A tropical view on Landau-Ginzburg models · Steps of the proof and auxiliary results · 1.00
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For a trivalent vertex $V∈V(Γ)$ define, with $u_(V,E)∈ι_⋆Λ_h(V)$ as in [ref] , Definition 3.3, [equation] where $E_1,E_2$ are any two edges adjacent to $V$.
Tropical correspondence for smooth del Pezzo log Calabi-Yau pairs · Tropical curves · 0.84
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If two rays intersect with determinant $m$, the scattering diagram is locally isomorphic via a change of lattice (see [ref] , Proposition C.13) to the one obtained from the “standard” scattering diagram with function $1+atx^m$ and $1+bty^m$, for some $a,b∈ℤ$.
Canonical bases for cluster algebras · Steps of the proof and auxiliary results · 0.80
Compares with 3 papers
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The second difference is that in [ref] the general fiber has trivial log structure, while our general fiber $X$ carries the divisorial log structure from the divisor $E⊂X$.
The degeneration formula for stable log maps · Relating one point invariants to multi point invariants · 0.97
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The computations of [ref] agree with Mayr's calculations using differential equations [ref] , a method further developed in [ref] .
On N=1 Mirror Symmetry for Open Type II Strings · Steps of the proof and auxiliary results · 0.93
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As mentioned in Remark [ref], similar results were obtained for moment fibers, rather than Aganagic–Vafa outer branes, in [ref] — see their Remark 5.7.
SYZ mirror symmetry for toric Calabi-Yau manifolds · The case of $^2$: Picard–Fuchs equations and mirror maps · 0.87
Cites as background 14 papers
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Aganagic–Vafa studied a class of Lagrangian A-branes in a toric $K_X$ and their mirror B-branes: the open mirror map relates parameters for their common moduli space [ref] .
Mirror Symmetry, D-Branes and Counting Holomorphic Discs · Introduction and Summary · 0.96
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By “open mirror map,” we mean the relation between complex and symplectic open parameters defined by the logarithmic solution to the Picard–Fuchs differential equations of Lerche–Mayr [ref] — see Section [ref].
On N=1 Mirror Symmetry for Open Type II Strings · Introduction and Summary · 0.80
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The function $M(Q)$ originally appeared in the work of Aganagic–Klemm–Vafa [ref] as the open mirror map relating Aganagic–Vafa branes [ref] on the canonical bundle $K_^2$ and its mirror, and was used to determine open Gromov–Witten invariants in [ref] .
Disk Instantons, Mirror Symmetry and the Duality Web · Introduction and Summary · 0.89
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