Genus-One Mirror Symmetry in the Landau-Ginzburg Model

Guo, Shuai; Ross, Dustin

doi:10.48550/arxiv.1611.08876

Cited by 13 publications

(14 citation statements)

References 35 publications

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“…It is expected that the two phases are more closely related near ǫ = 0. This has been worked out in genus 0 and genus 1 [GR16,RR17].…”

Section: Introductionmentioning

confidence: 99%

Quasimap wall-crossing for GIT quotients

Zhou¹

2019

Preprint

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In this paper, we prove a wall-crossing formula for ǫ-stable quasimaps to GIT quotients conjectured by Ciocan-Fontanine and Kim, for all targets in all genera, including the orbifold case. We prove that adjacent chambers give equivalent invariants, provided that both chambers are stable. In the case of genus-zero quasimaps with one marked point, we compute the invariants in the left-most stable chamber in terms of the small I-function. Using this we prove that the quasimap J-functions are on the Lagrangian cone of the Gromov-Witten theory. The proof is based on virtual localization on a master space, obtained via some universal construction on the moduli of weighted curves. The fixed-point loci are in one-to-one correspondence with the terms in the wall-crossing formula. Contents 1. Introduction 1 2. Curves with entangled tails and calibrated tails 10 3. Quasimap invariants with entangled tails 22 4. The master space and its virtual cycle 25 5. The properness of the master space 28 6. Localization on the master space 37 7. The wall-crossing formula 47 Appendices 52 Appendix A. Intersection theory on inflated projective bundles 52 References 55

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“…It is expected that the two phases are more closely related near ǫ = 0. This has been worked out in genus 0 and genus 1 [GR16,RR17].…”

Section: Introductionmentioning

confidence: 99%

Quasimap wall-crossing for GIT quotients

Zhou¹

2019

Preprint

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“…Using wall-crossing Ross-Ruan proved the LG/CY correspondence in genus 0 [29]. In genus 1, Guo-Ross used the wall-crossing formula to compute the FJRW invariants of the quintic 3-fold explicitly [18] and verified the genus-1 LG/CY correspondence [19]. Our result generalizes their wall-crossing formulas to all genera.…”

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confidence: 51%

Higher-genus wall-crossing in Landau–Ginzburg theory

Zhou¹

2020

Advances in Mathematics

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For a Fermat quasi-homogeneous polynomial, we study the associated weighted Fan-Jarvis-Ruan-Witten theory with narrow insertions. We prove a wall-crossing formula in all genera via localization on a master space, which is constructed by introducing an additional tangent vector to the moduli problem. This is a Landau-Ginzburg theory analogue of the higher-genus quasi-map wall-crossing formula proved by Ciocan-Fontanine and Kim. It generalizes the genus-0 result by Ross-Ruan and the genus-1 result by Guo-Ross.It is easy to see that the isomorphism is compatible with the cosections.Then we consider the τ * T M term of the distinguished triangle (15). By Lemma 13, the fixed part is τ * T F 1/r J . Moreover, since pr 1 isétale, we have T F 1/r J ∼ = pr * 1 T M 1/r g,γ J . This finishes the proof that pr * 1 ([M 1/r,ϕ g,γJ ] vir loc ) = [F ϕ J ] vir loc .2 The C * -action on L is only well-defined up to µ µ µr. Nevertheless we can compose the action with the r-th power map C * → C * so that it is well-defined. 1/r 0,γ . However, we can pushforward everything to a point instead. For any non-negative integer c this gives (28) J ψ c φ a , µ + J (−ψ)|(φ bj ) j ∈J 0 0,2|(n−|J|) = µ B (z) z −c−1 ,

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“…There had been a few notable progress obtained by applying the localization of MSP relations (1.2) developed by this paper. In [GR1,GR2], Guo-Ross used Theorem 1.1 to package g = 1 FJRW invariants of the Fermat quintic, confirming the mirror symmetry prediction and verifying the g = 1 Landau-Ginzburg/Calabi-Yau correspondence. In [CGLZ18], Chang-Guo-Li-Zhou used Theorem 1.1 to obtain an explicit formula of g = 1 GW invariants of the quintic CY threefold, recovering Zinger's formula [Zi] (also recovered by [KL18]).…”

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confidence: 74%