SIGMA 22 (2026), 023, 24 pages      arXiv:2504.14273      https://doi.org/10.3842/SIGMA.2026.023

Elliptic Virtual Structure Constants and Gromov-Witten Invariants for Complete Intersections in Weighted Projective Space

Masao Jinzenji ^a and Ken Kuwata ^b
a) Department of Mathematics, Okayama University, Okayama, 700-8530, Japan
^b) Department of General Education, National Institute of Technology, Kagawa College, Chokushi, Takamatsu, 761-8058, Japan

Received May 01, 2025, in final form February 17, 2026; Published online March 09, 2026

Abstract
In this paper, we generalize our formalism of the elliptic virtual structure constants to hypersurfaces and complete intersections within certain weighted projective spaces possessing a single Kähler class.

Key words: mirror symmetry; elliptic Gromov-Witten invariants; weighted projective space; residue integral.

pdf (482 kb)   tex (25 kb)  

References

1. Bershadsky M., Cecotti S., Ooguri H., Vafa C., Holomorphic anomalies in topological field theories, Nuclear Phys. B 405 (1993), 279-304, arXiv:hep-th/9302103.
2. Getzler E., Intersection theory on $\overline{\mathcal M}_{1,4}$ and elliptic Gromov-Witten invariants, J. Amer. Math. Soc. 10 (1997), 973-998, arXiv:alg-geom/9612004.
3. Greene B.R., Superconformal compactifications in weighted projective space, Comm. Math. Phys. 130 (1990), 335-355.
4. Iano-Fletcher A.R., Working with weighted complete intersections, in Explicit Birational Geometry of 3-Folds, London Math. Soc. Lecture Note Ser., Vol. 281, Cambridge University Press, Cambridge, 2000, 101-173.
5. Jinzenji M., Mirror map as generating function of intersection numbers: toric manifolds with two Kähler forms, Comm. Math. Phys. 323 (2013), 747-811, arXiv:1006.0607.
6. Jinzenji M., Geometrical proof of generalized mirror transformation of projective hypersurfaces, Internat. J. Math. 34 (2023), 2350006, 17 pages, arXiv:2503.21280.
7. Jinzenji M., Kuwata K., Elliptic virtual structure constants and generalizations of BCOV-Zinger formula to projective Fano hypersurfaces, J. High Energy Phys. 2024 (2024), no. 12, 135, 30 pages, arXiv:2404.07591.
8. Jinzenji M., Saito H., Moduli space of quasimaps from ${\mathbb P}^1$ with two marked points to ${\mathbb P}(1,1,1,3)$ and $j$-invariant, J. Math. Soc. Japan 73 (2021), 995-1018, arXiv:1712.07409.
9. Jinzenji M., Shimizu M., Multi-point virtual structure constants and mirror computation of $CP^2$-model, Commun. Number Theory Phys. 7 (2013), 411-468, arXiv:1305.0999.
10. Klemm A., Theisen S., Considerations of one-modulus Calabi-Yau compactifications: Picard-Fuchs equations, Kähler potentials and mirror maps, Nuclear Phys. B 389 (1993), 153-180, arXiv:hep-th/9205041.
11. Przyjalkowski V., Hori-Vafa mirror models for complete intersections in weighted projective spaces and weak Landau-Ginzburg models, Cent. Eur. J. Math. 9 (2011), 972-977, arXiv:1003.5200.