All exact solutions of a1/4Bogomol’nyi-Prasad-Sommerfield equation

Abstract

We study a $1/4$ Bogomol'nyi-Prasad-Sommerfield equation containing configurations made of walls, vortices, and monopoles in the Higgs phase, using the supersymmetric $U({N}_{\mathrm{C}})$ gauge theories with eight supercharges with ${N}_{\mathrm{F}}$ fundamental hypermultiplets. We find the total moduli space to be the quotient of the space of holomorphic maps from the complex plane to the ${N}_{\mathrm{C}}\ifmmode\times\else\texttimes\fi{}{N}_{\mathrm{F}}$ matrix divided by the space of the holomorphic maps to $GL({N}_{\mathrm{C}},\mathbf{C})$. We obtain all possible solutions exactly in the strong coupling limit where the moduli space reduces to the space of all holomorphic maps from a complex plane to the wall moduli space found recently, the deformed complex Grassmann manifold. Monopoles in the Higgs phase are also found in $U(1)$ gauge theory.