  We study the quantum dynamics of N=1 supergravity in four dimensions with a
compact spatial circle. Supersymmetry ensures that the perturbative
contributions to the Casimir energy on the circle cancel. However, instanton
contributions remain. These render supersymmetric compactification on a circle
unstable and the background dynamically decompactifies back to four dimensions.
The calculation provides a testing ground for some old ideas in Euclidean
quantum gravity. In particular, we show that gravitational instantons are
associated to a new, infra-red scale which is naturally exponentially
suppressed relative to the Planck scale and arises from the logarithmic running
of the Gauss-Bonnet term. There are also some interesting technical details,
including the non-cancellation of bosonic and fermionic determinants around the
background of a self-dual gravitational instanton, despite the existence of
supersymmetry.
