All Exact Solutions of a 1/4 Bogomol'nyi-Prasad-Sommerfield Equation

Abstract

We obtain all possible solutions of a 1/4 Bogomol'nyi-Prasad-Sommerfield equation exactly, containing configurations made of walls, vortices and monopoles in the Higgs phase. We use supersymmetric U(N_C) gauge theories with eight supercharges with N_F fundamental hypermultiplets in the strong coupling limit. The moduli space for the composite solitons is found to be 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.

Figures (3)

• Figure 1: Surfaces defined by the same energy density $t_{\rm w}+t_{\rm v}=0.5c$: a) A vortex stretched between walls with $H_0(z)e^{Mx^3}=\sqrt{c}(e^{x^3},ze^{4},e^{-x^3})$. b) A vortex attached to a tilted wall with $H_0(z)e^{Mx^3}=\sqrt{c}(z^2e^{x^3},e^{-1/2z})$. Note that there are two surfaces with the same energy for each wall.
• Figure 2: Multi-vorices between multi-walls: Surfaces defined by the same energy density $t_{\rm w}+t_{\rm v}=0.5c$ with $H_0(z)e^{Mx^3}= \sqrt{c}((z-4-2i)(z+5+8i)e^{3/2x^3}, (z+8-i)(z-7+6i)e^{1/2x^3+12}, z^2e^{-1/2x^3+12},(z-6-5i)(z+6-7i)e^{-3/2x^3})$.
• Figure 3: A cat's-cradle soliton: Surfaces defined by the same energy density $t_{\rm w}+t_{\rm v}=0.5c$ with $H_0(z)e^{Mx^3}= \sqrt{c}(e^{x^3},(z-2-5i)(z-6+5i)e^{3/4z-1/2},e^{-x^3})$.