  Extending the work of Gaiotto, Itzhaki and Rastelli in hep-th/0304192, we
derive a general prescription for computing amplitudes involving a periodic
array of D-branes in imaginary time to arbitrary order. We use this
prescription to show that closed string amplitudes with b boundaries are
identical to closed string amplitudes with b additional insertions of a
particular physical closed string state. We perform an explicit computation for
the annulus, and argue on the basis of open and closed string field theory for
higher order amplitudes. We also discuss possible subtleties in the
prescription related to collisions of boundaries and insertions, and argue that
they are harmless. This verifies the proposal that a periodic array of D-branes
in imaginary time corresponds to a pure closed string background.
