Direct computation of the degree 4 Gopakumar–Vafa invariant on a Calabi–Yau 3-fold

In this work we compute the topological Euler characteristic of the moduli space of stable sheaves of Hilbert polynomial 4n+14n+1 on P2P2 to be 192, using tools of algebraic geometry. This Euler characteristic is equal up to sign to the degree 4 BPS (Gopakumar–Vafa) invariant of local P2P2, a (noncompact) Calabi–Yau 3-fold. This is a new result verifying an instance of conjecture motivated by physics.