Topological String Blowup Equations via Stable Pairs

Source abstract

For the local Hirzebruch threefolds $Y_\ell=\operatorname{Tot}_{\mathbb F_\ell}K_{\mathbb F_\ell}$, $0\leq\ell\leq2$, we prove blowup equations for the two-variable, torus-equivariant symmetrized $K$-theoretic stable-pair series. After division by the fibre-class contribution, four-chart localization is identified coefficientwise with the equivariant Euler-characteristic series of $(\det\mathcal V)^\ell$ on moduli spaces of framed rank-two sheaves, with $\mathcal V$ tautological. The framed-sheaf blowup formulas then give the unity and vanishing equations; for $\ell=2$ this is an identity of localized indices. We state separately the two conjectures needed for the conditional local $\mathbb P^2$ equations, the general Huang--Sun--Wang conjecture, and a rational-elliptic specialization involving the $E_8$ lattice.