Grafting Seiberg–Witten monopoles

Abstract

We demonstrate that the operation of taking disjoint unions of $J$–holomorphic curves (and thus obtaining new $J$–holomorphic curves) has a Seiberg–Witten counterpart. The main theorem asserts that, given two solutions $\left(A_i,{\psi }_i\right)$, $i=0,1$ of the Seiberg–Witten equations for the ${Spin}^c$–structures $W_{E_i}^{+}=E_i\oplus \left(E_i\otimes K^{-1}\right)$ (with certain restrictions), there is a solution $\left(A,\psi \right)$ of the Seiberg–Witten equations for the ${Spin}^c$–structure $W_E$ with $E=E_0\otimes E_1$, obtained by “grafting” the two solutions $\left(A_i,{\psi }_i\right)$.