  We study vacuum stability in 1+1 dimensional Conformal Field Theories with
external background fields. We show that the vacuum decay rate is given by a
non-local two-form. This two-form is a boundary term that must be added to the
effective in/out Lagrangian. The two-form is expressed in terms of a
Riemann-Hilbert decomposition for background gauge fields, and its novel
"functional" version in the gravitational case.
