Clash of symmetries on the brane

Abstract

If our $(3+1)$-dimensional universe is a brane or domain wall embedded in a higher dimensional space, then a phenomenon we term the ``clash of symmetries'' provides a new method of breaking some continuous symmetries. A global ${G}_{\mathrm{cts}}\ensuremath{\bigotimes}{G}_{\mathrm{discrete}}$ symmetry is spontaneously broken to ${H}_{\mathrm{cts}}\ensuremath{\bigotimes}{H}_{\mathrm{discrete}},$ where the continuous subgroup ${H}_{\mathrm{cts}}$ can be embedded in several different ways in the parent group ${G}_{\mathrm{cts}},$ and ${H}_{\mathrm{discrete}}<{G}_{\mathrm{discrete}}.$ A certain class of topological domain wall solutions connects two vacua that are invariant under differently embedded ${H}_{\mathrm{cts}}$ subgroups. There is then enhanced symmetry breakdown to the intersection of these two subgroups on the domain wall. This is the ``clash.'' In the brane limit, we obtain a configuration with ${H}_{\mathrm{cts}}$ symmetries in the bulk but the smaller intersection symmetry on the brane itself. We illustrate this idea using a permutation symmetric three-Higgs-triplet toy model exploiting the distinct $I\ensuremath{-},$ $U\ensuremath{-}$ and $V\ensuremath{-}\mathrm{spin}$ U(2) subgroups of U(3). The three disconnected portions of the vacuum manifold can be treated symmetrically through the construction of a threefold planar domain wall junction configuration, with our universe at the nexus. A possible connection with ${E}_{6}$ is discussed.