Decomposition of Spin(4) Gauge Potential and Determinant Equation for Twisting U (1) Potential in Seiberg-Witten Theory

The Seiberg-Witten equations are studied from the viewpoint of gauge potential decomposition. We find a determinant equation ∆Aμ = −λAμ for the twisting U(1) potential Aμ of the Seiberg-Witten theory, which is in itself an eigenvalue problem of the Laplacian operator, with the eigenvalue being the vacuum expectation value of the field function, λ = ‖Φ‖ /2. This establishes a direct relationship between the spectral theory of the Laplacian operator and the classification of the moduli space of the Dirac operator. Topological characteristic numbers of instantons in the self-dual SU(2)+ sub-space are also discussed.