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PreprintCC BY 4.0arXiv

Topological String Blowup Equations via Stable Pairs

  • Lutian Zhao
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Abstract

The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi--Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces Tot𝔽ℓK𝔽ℓ, 0≀ℓ≀2, as identities of torus-equivariant symmetrized K-theoretic stable pair invariants; for β„“=2 the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of (det𝒱)β„“ on the moduli space of framed rank 2 sheaves on β„™2. The blowup formulas of Nakajima--Yoshioka for framed sheaves then yield the unity and vanishing equations. For local β„™2 we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang--Sun--Wang conjecture for general local Calabi--Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the E8 lattice.

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Cite

BibTeXRIS
BibTeX
@misc{zhao2026topological,
  title = {{Topological String Blowup Equations via Stable Pairs}},
  author = {Lutian Zhao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.25397},
  eprint = {2608.25397},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.25397},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Topological String Blowup Equations via Stable Pairs
AU  - Lutian Zhao
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2608.25397
UR  - https://arxiv.org/abs/2608.25397
AB  - The blowup equations of Huang, Sun and Wang are bilinear relations satisfied by the refined topological string partition function of a local Calabi--Yau 3-fold. We prove the blowup equations for the local Hirzebruch surfaces Tot_𝔽_β„“K_𝔽_β„“, 0≀ℓ≀2, as identities of torus-equivariant symmetrized K-theoretic stable pair invariants; for β„“=2 the invariants are localized indices. The proof identifies the stable pair vertex sum, after division by the fibre contribution, with the equivariant Euler characteristic of (det𝒱)^β„“ on the moduli space of framed rank 2 sheaves on β„™^2. The blowup formulas of Nakajima--Yoshioka for framed sheaves then yield the unity and vanishing equations. For local β„™^2 we obtain the blowup equations conditionally on two explicitly stated conjectures. We also state the Huang--Sun--Wang conjecture for general local Calabi--Yau 3-folds in the language of stable pairs, and formulate stable pair conjectures for the local rational elliptic surface involving the E_8 lattice.
ER  -