We discuss how to construct non-abelian chiral gauge theories with anomaly-free multiplets of Weyl fermions on the lattice without breaking the gauge invariance or violating any other fundamental principle. There are two approaches for this purpose:
1.To establish the integrability condition for the chiral determinant of overlap Weyl fermions
2.To gap out the mirror degrees of freedom of overlap Dirac fermions (Domain-wall fermions) with certain (boundary) interaction terms
In the former case, the integrability condition for the chiral determinant of overlap Weyl fermions can be formulated with five-, and six-dimensional lattice Domain-wall fermions. This formulation of the integrability condition is in parallel to the recent cobordism classification of global ‘t Hooft anomaly with the η-invariant based on the Dai-Freed theorem and the APS index theorem in the continuum theory. In the latter case, the required condition for the interaction terms is given by the complete cancellation of ‘t Hooft anomalies. We discuss the recent results on these two approaches.