A simple construction of fermion measure term in U(1) chiral lattice gauge theories with exact gauge invariance

In the gauge invariant formulation of U(1) chiral lattice gauge theories based on the Ginsparg-Wilson relation, the gauge field dependence of the fermion measure is determined through the so-called measure term. We derive a closed formula of the measure term on the finite volume lattice. The Wilson line degrees of freedom (torons) of the link field are treated separately to take care of the global integrability. The local counter term is explicitly constructed with the local current associated with the cohomologically trivial part of the gauge anomaly in a finite volume. The resulted formula is very close to the known expression of the measure term in the infinite volume with a single parameter integration, and would be useful in practical implementations.