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The genus-one global mirror theorem for the quintic $3$-fold

Part of: Symplectic geometry, contact geometry Projective and enumerative geometry

Published online by Cambridge University Press:  30 April 2019

Shuai Guo  and
Dustin Ross
Shuai Guo
Affiliation:
School of Mathematical Sciences, Peking University, No 5, Yiheyuan Road, Beijing 100871, China email guoshuai@math.pku.edu.cn
Dustin Ross
Affiliation:
Department of Mathematics, San Francisco State University, Thornton Hall 941, 1600 Holloway Avenue, San Francisco, CA 94132, USA email rossd@sfsu.edu
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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 $3$-fold, 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.

Keywords

mirror symmetryGromov–Witten theoryFan–Jarvis–Ruan–Witten theory

MSC classification

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants 53D45: Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 53D37: Mirror symmetry, symplectic aspects; homological mirror symmetry; Fukaya category
Type
Research Article
Information
Compositio Mathematica , Volume 155 , Issue 5 , May 2019 , pp. 995 - 1024
Copyright
© The Authors 2019 

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