Exact solutions of domain wall junctions in arbitrary dimensions

Exact analytic solutions of static, stable, non-planar BPS domain wall junctions are obtained in extended Abelian-Higgs models in $(D+1)$-dimensional spacetime. For specific choice of mass parameters, the Lagrangian is invariant under the symmetric group ${cal S}_{D+1}$ of degree $D+1$ spontaneously broken down to ${cal S}_D$ in vacua, admitting ${cal S}_{D+1}/{cal S}_D$ domain wall junctions. In $D=2$, there are three vacua and three domain walls meeting at a junction point, in which the conventional topological charges $Y$ and $Z$ exist for the BPS domain wall junctions and the BPS domain walls, respectively as known before. In $D=3$, there are four vacua, six domain walls, four junction lines on which three domain walls meet, and one junction point on which all the six domain walls meet. We define a new topological charge $X$ for the junction point in addition to the conventional topological charges $Y$ and $Z$. In general dimensions, we find that the configuration expressed in the $D$-dimensional real space is dual to a regular $D$-simplex in the $D$-dimensional internal space and that a $d$-dimensional subsimplex of the regular $D$-simplex corresponds to a $(D-d)$-dimensional intersection. Topological charges are generalized to the level-$d$ wall charge $W_d$ for the $d$-dimensional subsimplexes.