A parameter of an internal geometry, gauge background or dilaton configuration that becomes a scalar field in the lower-dimensional theory. A continuously undetermined parameter gives a flat direction and often a massless field. Fluxes and other effects can generate a potential, leading to moduli stabilization. Shape, size and coupling moduli affect low-energy observables.
For a toroidally compactified heterotic string, the torus metric, antisymmetric tensor and gauge Wilson lines determine a positive -plane in a charge space of signature , modulo lattice identifications. The continuous dimension is ; the coupling is an additional modulus. The three-torus case has fifty-seven shape and Wilson-line parameters.
Generating a potential with an acceptable stable minimum for previously undetermined compactification scalar fields. It can involve fluxes, branes and nonperturbative effects, with supersymmetry and consistency restrictions. A plausible phenomenological model must determine or suitably control these fields rather than leave arbitrary massless scalars and varying couplings.
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