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Gromov–Witten theory and cycle-valued modular forms

  • Todor Milanov EMAIL logo , Yongbin Ruan and Yefeng Shen


In this paper, we review Teleman’s work on lifting Givental’s quantization of +(2)GL(H) action for semisimple formal Gromov–Witten potential into cohomological field theory level. We apply this to obtain a global cohomological field theory for simple elliptic singularities. The extension of those cohomological field theories over large complex structure limit are mirror to cohomological field theories from elliptic orbifold projective lines of weight (3,3,3), (2,4,4), (2,3,6). Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.

Award Identifier / Grant number: 26800003

Award Identifier / Grant number: DMS 1103368

Award Identifier / Grant number: DMS 1159265

Award Identifier / Grant number: DMS 1405245

Award Identifier / Grant number: DMS 1159156

Funding statement: The work of the first author is supported by Grant-In-Aid and by the World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan. The first author is partially supported by JSPS Grant-In-Aid 26800003. The second author is partially supported by NSF grant DMS 1103368, DMS 1159265, DMS 1405245. The third author is partially supported by NSF grant DMS 1159156.


The second and third authors would like to thank Hiroshi Iritani for interesting discussions on the convergence of Gromov–Witten theory. The third author would like to thank Emily Clader, Nathan Priddis and Mark Shoemaker for helpful discussions on Givental’s theory. Finally, three of us would like to thank Kavli IPMU for hospitality where the part of this work is carried out. We thank Arthur Greenspoon for editorial assistance.


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Received: 2015-2-2
Published Online: 2015-6-20
Published in Print: 2018-2-1

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