Exact solutions of domain wall junctions in arbitrary dimensions

Exact analytic solutions of static, stable, nonplanar Bogomol'nyi-Prasad-Sommerfield (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 SD+1 of degree D+1 spontaneously broken down to SD in vacua, admitting SD+1/SD 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 Wd for the d-dimensional subsimplexes.