Abstract
Abstract We consider a lattice implementation of the $\eta$-invariant, using the complex phase of the determinant of the simplified domain-wall fermion, which couples to an interpolating 5D gauge field. We clarify the relation to the effective action for chiral Ginsparg–Wilson fermions. The integrability, which holds true for anomaly-free theories in the classical continuum limit, is not assured on a lattice with finite spacing. A lattice expression for the 5D Chern–Simons term is obtained.
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