Growth of spinors in the generalized Seiberg–Witten equations on R⁴ and R³

The classical Seiberg–Witten equations in dimensions three and four admit a natural generalization within a unified framework known as the generalized Seiberg–Witten (GSW) equations, which encompasses many important equations in gauge theory. This article proves that the averaged L²-norm of any spinor with non-constant pointwise norm in the GSW equations on R⁴ and R³, measured over large-radius spheres, grows faster than a power of the radius, under a suitable curvature decay assumption. Separately, it is shown that if the Yang–Mills–Higgs energy of any solution of these equations is finite, then the pointwise norm of the spinor in it must converge to a non-negative constant at infinity. These two behaviors cannot occur simultaneously unless the spinor has constant pointwise norm. This work may be seen as partial generalization of results obtained by Taubes [25], and Nagy and Oliveira [16] for the Kapustin–Witten equations.