Non-Perturbative Superpotentials In String Theory

String theory needs to explain why the universe is not supersymmetric, and the usual route is a superpotential: an energy term that, once generated, can lift a would-be flat direction and break supersymmetry. Perturbation theory cannot generate one, so it has to come from instantons. Witten's setting is eleven-dimensional M-theory squeezed down to three dimensions on a Calabi-Yau four-fold. The instanton in question is an M-theory five-brane whose six-dimensional world-volume wraps a complex surface (a divisor) inside the four-fold. Witten shows that such a wrapping produces a superpotential only if the divisor passes a sharp arithmetic test: its arithmetic genus, a holomorphic index computed from its cohomology, must equal exactly one. When the divisor is rigid and has no holomorphic forms of degree one, two, or three, the test passes and the superpotential is generated. When every divisor in the four-fold fails the test, no superpotential appears and supersymmetry is unbroken. He then runs the test on concrete geometries. Complete intersections in projective space fail, so those models have unbroken supersymmetry. A four-fold obtained by blowing up a Z4 orbifold singularity passes, because the exceptional divisor is a projective three-space with the right cohomology. Hypersurfaces in products of projective spaces fail; hypersurfaces in certain projective bundles pass. Because a Calabi-Yau four-fold can be elliptically fibered, the same answers carry over to four-dimensional F-theory and, when the base is itself a fibration by spheres, to the heterotic string. There the two divisor types map onto the two known instanton sources: space-time instantons and world-sheet instantons.