U(1) Connection, Nonlinear Dirac-Like Equations, and Seiberg–Witten Equations

Abstract

By analyzing the work of Campolattaro we arguethat the second Seiberg–Witten equation over theSpin ₄ ^c manifold, i.e.,F_ij ⁺ = 〈M,S_ijM〉, is the generalization ofCampolattaro's description of the electromagnetic field tensor F^μυ in thebilinear form $F^{\mu v} = \bar \Psi S^{\mu v} \Psi$. It turns out thatthe Seiberg–Witten equations (also the perturbedSeiberg–Witten equations) can be well understoodfrom this point of view. We suggest that the secondSeiberg–Witten equation can be replaced by anonlinear Dirac-like equation. We also derive the spinorrepresentation of the connection on the associatedunitary line bundle over the Spin₄ ^c manifold.