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

TL;DR

This paper explores the connection between Seiberg-Witten equations and nonlinear Dirac-like equations, providing a new perspective on gauge fields and spinor representations in four-dimensional manifolds.

Contribution

It introduces a novel interpretation of the second Seiberg-Witten equation as a nonlinear Dirac-like equation and derives the spinor representation of the connection.

Findings

Seiberg-Witten equations can be understood via bilinear forms of spinors.

The second Seiberg-Witten equation can be replaced by a nonlinear Dirac-like equation.

Derived the spinor representation of the connection on the line bundle.

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's description of the electromagnetic field tensor F^{\mu\nu} in the bilinear form F^{\mu\nu}=\bar{\Psi} S^{\mu\nu}\Psi. 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.