The paper explores N = 2 supersymmetric four-dimensional gauge theories in an Ω-background, a deformation of spacetime that allows explicit calculation of the partition function. The partition function is represented as a sum over random partitions, a random curve ensemble, and a free fermion correlator. These representations enable the derivation of the Seiberg-Witten geometry, curves, differentials, and prepotential. The study includes pure N = 2 theories, matter hypermultiplets in fundamental or adjoint representations, and five-dimensional theories compactified on a circle. The partition function is analyzed in the Ω-background, leading to the prepotential, which is shown to correspond to the Seiberg-Witten prepotential of the low-energy effective theory. The paper also discusses the quasiclassical limit, the SW curve, and the relation to random matrix theory and Gromov-Witten theory. The results are connected to the periodic Toda lattice prepotential and the Plancherel measure on partitions. The study provides a rigorous derivation of the Seiberg-Witten geometry and its connection to random partitions and free fermions.The paper explores N = 2 supersymmetric four-dimensional gauge theories in an Ω-background, a deformation of spacetime that allows explicit calculation of the partition function. The partition function is represented as a sum over random partitions, a random curve ensemble, and a free fermion correlator. These representations enable the derivation of the Seiberg-Witten geometry, curves, differentials, and prepotential. The study includes pure N = 2 theories, matter hypermultiplets in fundamental or adjoint representations, and five-dimensional theories compactified on a circle. The partition function is analyzed in the Ω-background, leading to the prepotential, which is shown to correspond to the Seiberg-Witten prepotential of the low-energy effective theory. The paper also discusses the quasiclassical limit, the SW curve, and the relation to random matrix theory and Gromov-Witten theory. The results are connected to the periodic Toda lattice prepotential and the Plancherel measure on partitions. The study provides a rigorous derivation of the Seiberg-Witten geometry and its connection to random partitions and free fermions.