  A non-perturbative lattice regularization of chiral fermions and bosons with
anomaly-free symmetry $G$ in 1+1D spacetime is proposed. More precisely, we ask
"whether there is a local short-range quantum Hamiltonian with a finite Hilbert
space for a finite system realizing onsite symmetry $G$ defined on a 1D spatial
lattice, such that its low energy physics produces a 1+1D anomaly-free chiral
matter theory of symmetry $G$?" In particular, we propose that the
3$_L$-5$_R$-4$_L$-0$_R$ U(1) chiral fermion theory, with two left-moving
fermions of charge-3 and 4, and two right-moving fermions of charge-5 and 0 at
low energy, can be put on a 1D spatial lattice where the U(1) symmetry is
realized as an onsite symmetry, if we include properly designed multi-fermion
interactions with intermediate strength. In general, we propose that any 1+1D
U(1)-anomaly-free chiral matter theory can be defined as a finite system on a
1D lattice with onsite symmetry by using a quantum Hamiltonian with continuous
time, but without suffering from Nielsen-Ninomiya theorem's fermion-doubling,
if we include properly-designed interactions between matter fields. We propose
how to design such interactions by looking for extra symmetries via
bosonization/fermionization. We comment on the new ingredients and the
differences of ours compared to Ginsparg-Wilson fermion, Eichten-Preskill, and
Chen-Giedt-Poppitz (CGP) models, and suggest modifying CGP model to have
successful mirror-decoupling. We show a topological non-perturbative proof of
the equivalence between $G$-symmetric 't Hooft anomaly cancellation conditions
and $G$-symmetric gapping rules (e.g. Haldane's stability conditions for
Luttinger liquid) for multi-U(1) symmetry. We expect our result holds
universally regardless of spatial Hamiltonian or Lagrangian/spacetime path
integral formulation. Numerical tests are demanding tasks but highly desirable
for future work.
