Non-Perturbative Regularization of 1+1D Anomaly-Free Chiral Fermions and Bosons: On the equivalence of anomaly matching conditions and boundary gapping rules

Understand

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.