This research aims to analyze the integrability condition of the chiral determinant of 4D overlap fermions and construct lattice chiral gauge theories.
$\quad$ We formulate the integrability condition with 5D and 6D lattice domain wall fermions. Our formulation parallels the recent cobordism classification of the global ‘t Hooft anomaly using the $\eta$-invariant based on the Dai-Freed theorem and the Atiya-Patodi-Singer index theorem in the continuum theory.
$\quad$ The necessary and sufficient condition for constructing a lattice chiral gauge theory comes down to the statement that "$\exp ( 2\pi i \eta ) = 1$ for any gauge configurations satisfying the admissibility condition in 5D lattice space.", where $\exp ( 2\pi i \eta ) $ is defined as the phase of the partition function of the 5D domain wall fermion.