This paper presents an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture by studying the partition function of N = 4 topologically twisted supersymmetric Yang-Mills theory on four-manifolds. The results reveal unexpected connections with two-dimensional rational conformal field theory. The N = 4 theory is believed to be exactly finite and conformally invariant, with a symmetry exchanging strong and weak coupling and electric and magnetic fields. This symmetry, originally proposed by Montonen and Olive, is extended to a full SL(2, Z) symmetry acting on a complex parameter τ = θ/(2π) + 4πi/g². The paper shows that the partition function of the N = 4 theory on certain four-manifolds, such as K3, CP², and ALE spaces, is the Euler characteristic of instanton moduli space. These results provide strong evidence for the S-duality conjecture, which is also related to string theory. The paper also discusses the implications of S-duality for string theory, including the possibility of unifying quantum mechanical and stringy corrections. The results are tested using modular properties of the partition function, and the paper concludes with a discussion of the relation to string theory and the potential for further generalizations.This paper presents an exact strong coupling test of the Montonen-Olive strong-weak duality conjecture by studying the partition function of N = 4 topologically twisted supersymmetric Yang-Mills theory on four-manifolds. The results reveal unexpected connections with two-dimensional rational conformal field theory. The N = 4 theory is believed to be exactly finite and conformally invariant, with a symmetry exchanging strong and weak coupling and electric and magnetic fields. This symmetry, originally proposed by Montonen and Olive, is extended to a full SL(2, Z) symmetry acting on a complex parameter τ = θ/(2π) + 4πi/g². The paper shows that the partition function of the N = 4 theory on certain four-manifolds, such as K3, CP², and ALE spaces, is the Euler characteristic of instanton moduli space. These results provide strong evidence for the S-duality conjecture, which is also related to string theory. The paper also discusses the implications of S-duality for string theory, including the possibility of unifying quantum mechanical and stringy corrections. The results are tested using modular properties of the partition function, and the paper concludes with a discussion of the relation to string theory and the potential for further generalizations.