Fractional $θ$-dependence from the sum over all instanton sectors

An instanton carries integer topological charge, seemingly at odds with the multibranched $θ$ dependence of the Yang-Mills vacuum. We show that summing over all instanton sectors resolves this mismatch in self-dual supersymmetric $\mathbb{C}P^{N-1}$ and $\mathrm{SU}(N)$ Yang-Mills theories with a chiral mass, in two and four dimensions. The path integral reduces to a sum of instanton moduli-space volumes, yielding the multibranched vacuum structure. In four dimensions, this extends Nekrasov's instanton counting for Seiberg-Witten theory to a QCD-like theory without scalars.