Chiral Determinant as an Overlap of Two Vacua

The effective action induced by chiral fermions can be written, formally, as an overlap of two states. These states are the Fock ground states of Hamiltonians for fermions in even dimensional space with opposite sign mass terms coupled to identical static vector potentials. A perturbative analysis of the overlap in the continuum framework produces the correct anomaly for Abelian gauge fields in two dimensions. When a lattice transfer matrix formalism is applied in the direction perpendicular to a domain wall on which chiral fermions live a lattice version of the overlap is obtained. The real part of the overlap is nonperturbatively defined and previous work indicates that the real part of the vacuum polarization tensor in four dimensions has the correct continuum limit for a chiral theory. The phase of the overlap represents the imaginary part of the chiral action and suffers from ambiguities.