Building a Better Racetrack

String theory compactifications typically leave dozens of scalar fields, called moduli, whose values the equations do not fix. A compactification with free moduli cannot describe our universe, because those fields would mediate long-range forces that are not observed. In 2003 Kachru, Kallosh, Linde and Trivedi proposed a way to fix the size moduli of a type IIB compactification: wrap branes on cycles of the internal space, let the resulting gauge theory or brane instanton generate a superpotential, and balance that against a small constant superpotential produced by fluxes. Their proposal was a recipe, not a worked example. This paper supplies worked examples. The authors first derive a geometric criterion: a divisor of the internal Calabi-Yau fourfold contributes to the superpotential only if it has arithmetic genus one, meaning it carries no extra light modes. They then search the classification of Fano threefolds and of projective-line bundles over toric surfaces for bases where enough such divisors exist to fix every modulus. Only six of the eighteen toric Fano threefolds qualify. They also argue that no model with a single size modulus can work, because the required divisor would have negative volume. They then compute divisor volumes and intersection numbers explicitly for three examples, solve the supersymmetry conditions numerically, and check that the resulting critical points are minima rather than saddle points. The catch is that the flux superpotential must be tuned to about 10 to the minus 30, and the paper's own counting of flux vacua suggests such tiny values are rare in the regime where the calculation is under control.