Vacuum Energies of String Compactified on Torus

Computation of the one-loop vacuum energy is attempted for closed bosonic string compactified on various tori. Modular invariance of the one-loop vacuum energy is shown. The divergent tachyon contribution forces us to employ a subtraction prescription. For one-dimensional torus, the affine Kac-Moody algebra for SU(2)×SU(2) is realized at the absolute minimum of the vacuum energy. For general r-dimensional torus, the algebra for SU(2)×SU(2)r is found to be an unstable saddle point. A detailed study of r=2 case shows that SU(3)×SU(3) has the lowest vacuum energy.