In this paper, we review Teleman’s work on lifting Givental’s quantization of 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 , , . Via mirror symmetry, we prove generating functions of Gromov–Witten cycles for those orbifolds are cycle-valued (quasi)-modular forms.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: 26800003
Funding source: National Science Foundation
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.
 V. I. Arnol’d, S. M. Gusein-Zade and A. N. Varchenko, Singularities of differentiable maps. Vol. II: Monodromy and asymptotics of integrals, Monogr. Math. 83, Birkhäuser, Boston 1988. 10.1007/978-1-4612-3940-6Search in Google Scholar
 M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Comm. Math. Phys. 165 (1994), 311–428. 10.1007/BF02099774Search in Google Scholar
 W. Chen and Y. Ruan, Orbifold Gromov–Witten theory, Orbifolds in mathematics and physics (Madison 2001), Contemp. Math. 310, American Mathematical Society, Providence (2002), 25–85. 10.1090/conm/310/05398Search in Google Scholar
 A. Chiodo and Y. Ruan, A global mirror symmetry framework for the Landau–Ginzburg/Calabi–Yau correspondence, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 7, 2803–2864. 10.5802/aif.2795Search in Google Scholar
 T. Coates and H. Iritani, in preparation. Search in Google Scholar
 B. Dubrovin, Geometry of 2D topological field theories, Integrable systems and quantum groups (Montecatini Terme 1993), Lecture Notes in Math. 1620, Springer, Berlin (1996), 120–348. 10.1007/BFb0094793Search in Google Scholar
 M. X. Huang, A. Klemm and S. Quackenbush, Topological string theory on compact Calabi–Yau: Modularity and boundary conditions, Homological mirror symmetry, Lecture Notes in Phys. 757, Springer, Berlin (2009), 45–102. 10.1007/978-3-540-68030-7_3Search in Google Scholar
 E. Looijenga, On the semi-universal deformation of a simple-elliptic hypersurface singularity. I: Unimodularity, Topology 16 (1977), no. 3, 257–262. 10.1016/0040-9383(77)90006-4Search in Google Scholar
 Y. I. Manin, Frobenius manifolds, quantum cohomology, and moduli spaces, Amer. Math. Soc. Colloq. Publ. 47, American Mathematical Society, Providence 1999. 10.1090/coll/047Search in Google Scholar
 D. Maulik, R. Pandharipande and R. P. Thomas, Curves on surfaces and modular forms. With an appendix by A. Pixton, J. Topol. 3 (2010), no. 4, 937–996. 10.1112/jtopol/jtq030Search in Google Scholar
 K. Saito, On periods of primitive integrals. I, Preprint 212, Research Institute for Mathematical Sciences, Kyoto 1982. Search in Google Scholar
 K. Saito and A. Takahashi, From primitive forms to Frobenius manifolds, From Hodge theory to integrability and TQFT tt*-geometry (Augsburg 2007), Proc. Sympos. Pure Math. 78, American Mathematical Society, Providence (2008), 31–48. 10.1090/pspum/078/2483747Search in Google Scholar
 W. Zhang, Modularity of generating functions of special cycles on Shimura varieties, PhD thesis, Columbia University, 2009. Search in Google Scholar
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