Journal of Differential Geometry

LG/CY correspondence for elliptic orbifold curves via modularity

Yefeng Shen and Jie Zhou

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We prove the Landau–Ginzburg/Calabi–Yau correspondence between the Gromov–Witten theory of each elliptic orbifold curve and its Fan–Jarvis–Ruan–Witten theory counterpart via modularity. We show that the correlation functions in these two enumerative theories are different representations of the same set of quasi-modular forms, expanded around different points on the upper-half plane. We relate these two representations by the Cayley transform.

Article information

J. Differential Geom., Volume 109, Number 2 (2018), 291-336.

Received: 14 April 2016
First available in Project Euclid: 23 May 2018

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Zentralblatt MATH identifier

Primary: 11Fxx: Discontinuous groups and automorphic forms [See also 11R39, 11S37, 14Gxx, 14Kxx, 22E50, 22E55, 30F35, 32Nxx] {For relations with quadratic forms, see 11E45} 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]


Shen, Yefeng; Zhou, Jie. LG/CY correspondence for elliptic orbifold curves via modularity. J. Differential Geom. 109 (2018), no. 2, 291--336. doi:10.4310/jdg/1527040874.

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