The genus-one global mirror theorem for the quintic -fold

Abstract

We prove the genus-one restriction of the all-genus Landau-Ginzburg/Calabi-Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This gives the first evidence supporting the higher-genus Landau-Ginzburg/Calabi-Yau correspondence for the quintic threefold, and exhibits the first instance of the "genus zero controls higher genus" principle, in the sense of Givental's quantization formalism, for non-semisimple cohomological field theories.1 genus-zero correspondence through an explicit quantization procedure. While such a principle has been studied extensively and proved in many cases for semisimple cohomological field theories, for example [Tel12, Giv01b, BCR13, Zon16, CI14, HLSW15, IMRS16], this is the first significant evidence for such a principle in the non-semisimple case.1.1. Plan of the Paper. We begin in Section 2 by recalling the basic definitions in Gromov-Witten and Fan-Jarvis-Ruan-Witten theory. We recall some previously known results, including the genuszero mirror theorems, the genus-zero Landau-Ginzburg/Calabi-Yau correspondence, and the genusone mirror theorems. In Section 3, we discuss the Birkhoff factorization of the symplectomorphism U and recall Givental's quantization formulas in order to make Theorem 3.3 precise. We also apply the string and dilaton equations to reduce the main theorem to the one parameter 'small statespace'. In Section 4, we provide a proof of the genus-zero restriction of the quantization conjecture, mostly in order to set up notation for the genus-one correspondence. The proof of the genus-one correspondence occupies Sections 5, 6, and 7, where we carefully analyze the vertex-and loop-type graphs that appear in the quantization formula. 7

Higher-genus wall-crossing in Landau–Ginzburg theory

“…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.…”

Genus-one mirror symmetry in the Landau�Ginzburg model

“…In addition, while the twisted invariants are a necessary part of our proof, we expect that there is a way to circumvent the use of the twisted invariants and derive Theorem 1.5 directly from the localization relations obtained from the moduli spaces of dual-extended 5-spin curves. One benefit of our approach using twisted invariants is that the analysis of the twisted invariants that we carry out in this paper is also necessary for the arguments in the sequel [GR17].…”

“…This work builds the potential for several new directions in regards to the Landau-Ginzburg/Calabi-Yau correspondence. Most notably, we use Theorem 1.5 in the sequel [GR17] to prove the genus-one version of the Chiodo-Ruan formulation of the LG/CY correspondence [CR10]. Namely, we prove that the quantization of the genus-zero symplectic transformation computed in [CR10] identifies the genus-one FJRW potential with the genus-one Gromov-Witten (GW) potential.…”

“…Of particular interest is the case of the Fermat quintic threefold. The genus-zero Gromov-Witten invariants of the quintic were first computed by Givental [Giv98b] and Lian-Liu-Yau The main result of this paper verifies the genus-one mirror symmetry formula conjectured by Huang, Klemm, and Quackenbush. In the sequel to this paper [GR17], we use the genus-one mirror formulas contained here and in Zinger's work [Zin09] to prove the genus-one specialization of Chiodo and Ruan's higher-genus Landau-Ginzburg/Calabi-Yau correspondence. This provides the first evidence for the validity of the higher-genus quantization conjecture.…”

A new method toward the Landau–Ginzburg/Calabi–Yau correspondence via quasi-maps

“…Also in genus zero, Lee, Priddis and Shoemaker [32] establish a proof of LG/CY correspondence assuming the crepant transformation conjecture. In [24], Guo and Ross verified the Landau-Ginzburg/Calabi-Yau correspondence in genus one. The correspondence for higher genera remains open.…”