  We confront the problem of giving a fundamental definition to perturbative
string theory in spacetimes with totally compact space (taken to be a torus for
simplicity, though the nature of the problem is very general) and non-compact
time. Due to backreaction induced by the presence of even a single string
quantum, the usual formulation of perturbative string theory in a fixed
classical background is infrared-divergent at all subleading orders in the
string coupling, and needs to be amended. The problem can be seen as a closed
string analogue of D0-brane recoil under an impact by closed strings (a
situation displaying extremely similar infrared divergences). Inspired by the
collective coordinate treatment of the D0-brane recoil, whereby the
translational modes of the D0-brane are introduced as explicit dynamical
variables in the path integral, we construct a similar formalism for the case
of string-induced gravitational backreaction, in which the spatially uniform
modes of the background fields on the compact space are quantized explicitly.
The formalism can equally well be seen as an ultraviolet completion of a
minisuperspace quantum cosmology with string degrees of freedom. We consider
the amplitudes for the universe to have two cross-sections with specified
spatial properties and string contents, and show (at the first non-trivial
order) that they are finite within our formalism.
