Topological String Blowup Equations via Stable Pairs

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 \(\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}\), \(0\leq\ell\leq2\), 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.