This paper presents a strong coupling test of the Montonen-Olive strong-weak duality conjecture for the $N = 4$ topologically twisted supersymmetric Yang-Mills theory on four-manifolds. The authors study the partition function of this theory, which is shown to be related to the Euler characteristic of instanton moduli spaces. By examining the modular properties of the partition function on various manifolds, such as $K3$, $\mathbf{CP}^2$, and ALE spaces, they find unexpected links to two-dimensional rational conformal field theory. The results provide strong evidence for the validity of the $S$-duality conjecture, which suggests that the theory has a symmetry exchanging strong and weak coupling and electric and magnetic fields. The paper also discusses the extension of $S$-diality to string theory and its implications for the real world.This paper presents a strong coupling test of the Montonen-Olive strong-weak duality conjecture for the $N = 4$ topologically twisted supersymmetric Yang-Mills theory on four-manifolds. The authors study the partition function of this theory, which is shown to be related to the Euler characteristic of instanton moduli spaces. By examining the modular properties of the partition function on various manifolds, such as $K3$, $\mathbf{CP}^2$, and ALE spaces, they find unexpected links to two-dimensional rational conformal field theory. The results provide strong evidence for the validity of the $S$-duality conjecture, which suggests that the theory has a symmetry exchanging strong and weak coupling and electric and magnetic fields. The paper also discusses the extension of $S$-diality to string theory and its implications for the real world.