发表机构
Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo（东京大学宇宙学起源研究前沿中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对特定局部Hirzebruch三维簇，本文证明双变量等变\boldsymbol{K}-理论稳定对级数的拓扑弦爆破方程，给出相关恒等式并陈述两个关键猜想。
AI 中文摘要
针对局部Hirzebruch三维簇\boldsymbol{Y_\tau=\text{Tot}_{\boldsymbol{\tau}}K_{\boldsymbol{\tau}}}，其中\boldsymbol{0\boldsymbol{\tau\boldsymbol{2}}}，我们证明了双变量、环面对称化\boldsymbol{K}-理论稳定对级数的爆破方程。在除以纤维类贡献后，四图定位在系数意义上与带标架的秩2层模空间上\boldsymbol{(\text{det}\boldsymbol{\tau})^\tau}的等变欧拉特征级数等同，其中\boldsymbol{\tau}为重言式。带标架层的爆破公式给出了单位和消失方程；对于\boldsymbol{\tau=2}，这是一个定位指标的恒等式。我们单独陈述了条件局部\boldsymbol{\boldsymbol{P}^2}方程所需的两个猜想：一般的Huang–Sun–Wang猜想，以及涉及\boldsymbol{E_8}格的有理椭圆特例。
英文摘要
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 \(\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}\), \(0\leq\ell\leq2\), as identities of torus-equivariant symmetrized \(K\)-theoretic stable pair invariants; for \(\ell=2\) 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 \((\det\mathcal V)^\ell\) on the moduli space of framed rank \(2\) sheaves on \(\mathbb P^2\). The blowup formulas of Nakajima--Yoshioka for framed sheaves then yield the unity and vanishing equations. For local \(\mathbb P^2\) 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.
