The general flop formula for Donaldson–Thomas invariants

Let $f$ be a flop between nonsingular projective $3$-folds, and let

$$and$$

denote the corresponding flop identities for the Donaldson–Thomas invariants and their contributions from the center of the flop. General flop formula conjecture. The formulae

$$and$$

should hold for general flops. The paper proves the formula for a disjoint union of $(-2)$-curves and relates it to BPS state counts; extending these identities to arbitrary flops is proposed as the remaining conjectural generalization.