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

Abstract

By analysing the work of Campolattaro we argue that the second Seiberg-Witten equation over the Spin^c_4 manifold, i.e., F^+_{ij}=< M,S_ij M >, is the generalization of the Campolattaro&#39;s description of the electromagnetic field tensor F^{μν} in the bilinear form F^{μν}=\barΨ S^{μν}Ψ. It turns out that the Seiberg-Witten equations (also the perturbed Seiberg-Witten equations) can be well understood from this point of view. We suggest that the second Seiberg-Witten equation can be replaced by a nonlinear Dirac-like Equation. We also derive the spinor representation of the connection on the associated unitary line bundle over the Spin^c_4 manifold.