Projected topological branes

Nature harbors crystals of dimensionality ($d$) only up to three. Here we introduce the notion \emph{projected topological branes} (PTBs): Lower-dimensional branes embedded in higher-dimensional parent crystals, constructed via a geometric cut-and-project procedure on Hilbert space lattice Hamiltonian. When such brane is inclined at rational or an irrational slope, either new periodicity quasicrystal emerges. The latter gives birth topoquasicrystals within landscape PTBs. As PTBs are shown inherit hallmarks, as bulk-boundary, bulk-dislocation correspondences and invariant, crystals. We exemplify these outcomes by focusing two-dimensional Chern insulators, leaving its signatures projected one-dimensional (1D) terms localized endpoint, dislocation modes local number. Finally, stacking 1D showcase imprints three-dimensional Weyl semimetals $d=2$, namely Fermi arc surface states bulk chiral zeroth Landau level, responsible for anomaly. Altogether, proposed open realistic avenue harness ($d>3$) phases laboratory.