BPS invariants of semi-stable sheaves on rational surfaces
Jan Manschot
TL;DR
This work develops a cohesive framework to compute BPS invariants for semi-stable sheaves on rational surfaces by combining extended Harder–Narasimhan filtrations, fibre-restriction generating functions, and the blow-up formula. It delivers explicit rank-2 and rank-3 results on Hirzebruch surfaces and derives P^2 invariants via the blow-up relation, with recursive strategies that extend to higher rank. The results are cross-validated against wall-crossing, integrality, and known Euler numbers, and they reveal modular structures encoded in eta- and theta-functions within the generating functions. Together, these methods provide a practical, modular toolkit for computing BPS invariants on rational surfaces and link the geometry of moduli spaces to physical BPS indices.
Abstract
BPS invariants are computed, capturing topological invariants of moduli spaces of semi-stable sheaves on rational surfaces. For a suitable stability condition, it is proposed that the generating function of BPS invariants of a Hirzebruch surface takes the form of a product formula. BPS invariants for other stability conditions and other rational surfaces are obtained using Harder-Narasimhan filtrations and the blow-up formula. Explicit expressions are given for rank <4 sheaves on a Hirzebruch surface or the projective plane. The applied techniques can be applied iteratively to compute invariants for higher rank.