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.
