CONSTRAINTS ON LOW-DIMENSIONAL STRING COMPACTIFICATIONS

We study the restrictions imposed by cancellation of the tadpoles for two-, three-, and four-form gauge fields in string theory, hi-theory a nd F-theory compactified to two, three and four dimensions, respective ly, For a large class of supersymmetric vacua, turning on a sufficient number of strings, membranes and three-branes, respectively, can canc el the tadpoles, and preserve supersymmetry, However, there are cases where the tadpole cannot be removed in this way, either because the ta dpole is fractional, or because of its sign. For M-theory and F-theory compactifications, we also explore the relation of the membranes and three-branes to the non-perturbative space-time superpotential.