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PreprintCC BY 4.0arXiv

From wall structures to closed mirror symmetry. The case of Kℙ2: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration

  • Michel van Garrel
  • Bernd Siebert
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Abstract

We recover closed Gromov-Witten invariants and renormalized mirror periods for Kℙ2 by the same operation on a wall function. This gives a direct passage from the Gross-Siebert construction of intrinsic mirror pairs to classical enumerative mirror symmetry. The link is a polynomiality theorem for punctured invariants. Assuming the expected identification with the normalized slab function, we also obtain a finite tree sum for closed Gromov-Witten invariants in terms of types of plane tropical curves. The mechanism is expected to extend to more general Calabi-Yau mirror pairs.

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Cite

BibTeXRIS
BibTeX
@misc{garrel2026from,
  title = {{From wall structures to closed mirror symmetry. The case of $K_\mathbbP^2$: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration}},
  author = {Michel van Garrel and Bernd Siebert},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04252},
  eprint = {2610.04252},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04252},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - From wall structures to closed mirror symmetry. The case of K_ℙ^2: Renormalized periods over the positive real locus, closed Gromov-Witten invariants from wall functions, and tropical enumeration
AU  - Michel van Garrel
AU  - Bernd Siebert
PY  - 2026
DA  - 2026/10/03
PB  - arXiv
DO  - 10.48550/arxiv.2610.04252
UR  - https://arxiv.org/abs/2610.04252
AB  - We recover closed Gromov-Witten invariants and renormalized mirror periods for K_ℙ^2 by the same operation on a wall function. This gives a direct passage from the Gross-Siebert construction of intrinsic mirror pairs to classical enumerative mirror symmetry. The link is a polynomiality theorem for punctured invariants. Assuming the expected identification with the normalized slab function, we also obtain a finite tree sum for closed Gromov-Witten invariants in terms of types of plane tropical curves. The mechanism is expected to extend to more general Calabi-Yau mirror pairs.
ER  -