Fermion determinant for general background gauge fields

An exact representation of the Euclidean fermion determinant in two dimensions for centrally symmetric, finite-ranged Abelian background fields is derived. Input data are the wave function inside the field's range and the scattering phase shift with their momenta rotated to the positive imaginary axis and fixed at the fermion mass for each partial wave. The determinant's asymptotic limit for strong coupling and small fermion mass for square-integrable, unidirectional magnetic fields is shown to depend only on the chiral anomaly. The concept of duality is extended from one- to two-variable fields, thereby relating the two-dimensional Euclidean determinant for a class of background magnetic fields to the pair production probability in four dimensions for a related class of electric pulses. Additionally, the ``diamagnetic'' bound on the two-dimensional Euclidean determinant is related to the negative sign of $\ensuremath{\partial}\mathrm{Im}{S}_{\mathrm{eff}}/\ensuremath{\partial}{m}^{2}$ in four dimensions in the strong coupling, small mass limit, where ${S}_{\mathrm{eff}}$ is the one-loop effective action.