  Synthetic spaces allow physicists to bypass constraints imposed by certain
physical laws in experiments. Here, we show that a synthetic torus, which
consists of a ring trap in the real space and internal states of ultracold
atoms cyclically coupled by Laguerre-Gaussian Raman beams, could be threaded by
a net effective magnetic flux through its surface---an impossible mission in
the real space. Such synthetic Hall torus gives rise to a periodic lattice in
the real dimension, in which the periodicity of density modulation of atoms
fractionalizes that of the Hamiltonian. Correspondingly, the energy spectrum is
featured by multiple bands grouping into clusters with nonsymmorphic symmetry
protected band crossings in each cluster, leading to braidings of wavepackets
in Bloch oscillations. Our scheme allows physicists to glue two synthetic Hall
tori such that localization may emerge in a quasicrystalline lattice. If the
Laguerre-Gaussian Raman beams and ring traps were replaced by linear Raman
beams and ordinary traps, a synthetic Hall cylinder could be realized and
deliver many of the aforementioned phenomena.
