Tunneling between fermionic vacua and the overlap formalism

The probability amplitude for tunneling between the Dirac vacua corresponding to different signs of a parity breaking fermionic mass $M$ in $2+1$ dimensions is studied, making contact with the continuum overlap formulation for chiral determinants. It is shown that the transition probability in the limit when $M \to \infty$ corresponds, via the overlap formalism, to the squared modulus of a chiral determinant in two Euclidean dimensions. The transition probabilities corresponding to two particular examples: fermions on a torus with twisted boundary conditions, and fermions on a disk in the presence of an external constant magnetic field are evaluated.