Universal polynomials for Gromov-Witten invariants on toric surfaces

Abstract

We study the polynomiality of combined logarithmic and ordinary Gromov-Witten invariants with a Lambda class insertion on toric surfaces Δ(c,d). We show that the combined logarithmic Gromov-Witten invariants can be expressed in terms of universal relative node polynomials when the direction vector c and length vector d of the h-transverse lattice polygon Δ(c,d) are sufficiently large and spread out. In particular, when the normal fan of the h-transverse lattice polygon Δ(c,d) is smooth and its edges satisfy a length condition, we show that the combined Gromov-Witten invariants are equal to a universal node polynomial. We also show that the higher genus Block-Göttsche polynomials of smooth toric surfaces corresponding to these h-transverse lattice polygons, including toric blowups beyond Hirzebruch and toric del Pezzo surfaces, are equal to a generating function of higher genus Gromov-Witten invariants.

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BibTeX RIS

BibTeX

@misc{ding2026universal,
  title = {{Universal polynomials for Gromov-Witten invariants on toric surfaces}},
  author = {Yanqiao Ding and Dadi Ni},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35122},
  eprint = {2610.35122},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35122},
}

RIS

TY  - UNPB
TI  - Universal polynomials for Gromov-Witten invariants on toric surfaces
AU  - Yanqiao Ding
AU  - Dadi Ni
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35122
UR  - https://arxiv.org/abs/2610.35122
AB  - We study the polynomiality of combined logarithmic and ordinary Gromov-Witten invariants with a Lambda class insertion on toric surfaces Δ(c,d). We show that the combined logarithmic Gromov-Witten invariants can be expressed in terms of universal relative node polynomials when the direction vector c and length vector d of the h-transverse lattice polygon Δ(c,d) are sufficiently large and spread out. In particular, when the normal fan of the h-transverse lattice polygon Δ(c,d) is smooth and its edges satisfy a length condition, we show that the combined Gromov-Witten invariants are equal to a universal node polynomial. We also show that the higher genus Block-Göttsche polynomials of smooth toric surfaces corresponding to these h-transverse lattice polygons, including toric blowups beyond Hirzebruch and toric del Pezzo surfaces, are equal to a generating function of higher genus Gromov-Witten invariants.
ER  -

Abstract

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