An arXiv version-1 preprint dated 26 Aug 2026 reports a coefficient-by-coefficient match between the stable-pair series for local Hirzebruch threefolds and a determinant-power framed-sheaf Euler-characteristic series. Its central objective is to establish blowup equations for the two-variable, torus-equivariant, symmetrized K-theoretic stable-pair series.
The calculation uses torus-equivariant localization on four affine charts. It divides out the fibre-class contribution and compares the resulting sum over two partitions with a fixed-point formula. The comparison is coefficientwise, so the formal series are matched term by term. QB and QF are independent formal Novikov coordinates, handled as algebraic symbols rather than numerical inputs.
A result with a defined boundary
On the framed-sheaf side, the paper states rank-two blowup identities for every admissible pair (j,d) and for local Hirzebruch indices from 0 through 2, with an explicit coefficient in each case. The resulting formulas give unity and vanishing blowup equations, and the normalization follows the Huang-Sun-Wang convention.
That result has a different mathematical status at index 2. For indices 0 and 1, the full stable-pair moduli spaces are proper. For index 2, the series is defined coefficient by coefficient through vertex-edge localization, and the blowup statement is interpreted as a localized-index identity rather than a full proper moduli-space invariant.
The projective-plane case stays conditional
The local projective-plane part is less settled. Conjecture 6.8 supplies the vertex comparison, while Conjecture 6.18 supplies analytic continuation and the coefficientwise u=0 specialization used for the listed identities. Under Conjecture 6.8, an explicit contraction-series model is regular at u=0, and its constant term equals the local projective-plane series.
Assuming both conjectures, the normalized local projective-plane series satisfies blowup equations for r=1, r=-1 and r=3. The r=3 calculation removes a common Q_{1/3} factor before coefficient extraction. Those equations therefore remain conditional on the two conjectures.
The same assumptions yield a recursive determination: the coefficient sums are finite, and the three blowup equations together with A0=1 uniquely determine the full local projective-plane series. The uniqueness conclusion is conditional because the two inputs have not been established in the supplied analysis.
Beyond the proved cases
Beyond the local examples, the paper formulates a general Huang-Sun-Wang conjecture involving finite disjoint subsets and coefficientwise meromorphic lattice sums. It remains a proposed framework rather than an established result.
The rational-elliptic section also proposes root-indexed and zero-vector blowup identities for the E8 lattice for every k≥0, with the first equation indexed by 240 roots. A related contraction conjecture concerns a (-1)-curve: after division by the exceptional factor, taking the zero coefficient is supposed to yield the Huang-Sun-Wang equation on the contracted surface.
Taken together, the paper’s established contribution is the local Hirzebruch comparison and blowup result within its formal and localized framework. The projective-plane recursion and the broader Huang-Sun-Wang, rational-elliptic and contraction extensions remain conjectural. The work was supported by JSPS KAKENHI Grant Number JP25K17226.
Paper data and sources
Original title: Topological String Blowup Equations via Stable Pairs
Authors: Lutian Zhao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text