By reading articles I ran several times into two terms, never being defined so I assume they must have well established definitions somewhere.
The first is semi-classical space-time. If I where to guess I would say spin 2 field quantized over a Minkowski space, or qft in curved space-time (but space-time is totally classical in this case).
The second is supersymmetric Minkowski space. Would it be a flat space-time with both commuting and anti-commuting coordinates? (From what I've read the interest in this cosmological solution, is the fact that the Hilbert space describing needs to be infinite dimensional. How is it related to the presence of supersymmetry?)