Potentials between D-Branes in a Supersymmetric Model of Space-Time Foam

We study a supersymmetric model of space-time foam with two stacks each of eight D8-branes with equal string tensions, separated by a single bulk dimension containing D0-brane particles that represent quantum fluctuations. The ground-state configuration with static D-branes has zero vacuum energy, but, when they move, the interactions among the D-branes and D-particles due to the exchanges of strings result in a non-trivial, positive vacuum energy. We calculate its explicit form in the limits of small velocities and large or small separations between the D-branes and/or the D-particles. This non-trivial vacuum energy appears as a central charge deficit in the non-critical stringy $\sigma$ model describing perturbative string excitations on a moving D-brane. These calculations enable us to characterise the ground state of the D-brane/D-particle system, and provide a framework for discussing brany inflation and the possibility of residual Dark Energy in the present-day Universe.