  We fix the long-standing ambiguity in the 1-loop contribution to the mass of
a 1+1-dimensional supersymmetric soliton by adopting a set of boundary
conditions which follow from the symmetries of the action and which depend only
on the topology of the sector considered, and by invoking a physical principle
that ought to hold generally in quantum field theories with a topological
sector: for vanishing mass and other dimensionful constants, the vacuum
energies in the trivial and topological sectors have to become equal. In the
two-dimensional N=1 supersymmetric case we find a result which for the
supersymmetric sine-Gordon model agrees with the known exact solution of the
S-matrix but seems to violate the BPS bound. We analyze the nontrivial relation
between the quantum soliton mass and the quantum BPS bound and find a
resolution. For N=2 supersymmetric theories, there are no one-loop corrections
to the soliton mass and to the central charge (and also no ambiguities) so that
the BPS bound is always saturated. Beyond 1-loop there are no ambiguities in
any theory, which we explicitly check by a 2-loop calculation in the
sine-Gordon model.
