Dynamical chiral symmetry breaking on a brane in reduced QED

Reduced gauge theories are theories in which while gauge fields propagate in a bulk, fermion fields are localized on a brane. We study dynamical chiral symmetry breaking on a 2-brane and a 1-brane in reduced ${\mathrm{QED}}_{3+1},$ and on a 1-brane in reduced ${\mathrm{QED}}_{2+1}.$ Since, unlike higher dimensional gauge theories, ${\mathrm{QED}}_{3+1}$ and ${\mathrm{QED}}_{2+1}$ are well defined, their reduced versions can serve as a laboratory for studying dynamics in a higher dimensional brane world. The analysis of the Schwinger-Dyson (SD) equations in these theories reveals rich and quite nontrivial dynamics in which the conformal symmetry and its breakdown play a crucial role. Explicit solutions of the SD equations in the near-critical regime are obtained and the character of the corresponding phase transition is described.