Compactification in string theory

Compactification makes some target spatial dimensions compact and sufficiently small that lower-energy observations see an effective lower-dimensional spacetime. A simple geometry is a product of noncompact spacetime with a compact internal manifold; its momentum and winding sectors influence the effective spectrum. Reaching a four-dimensional model from a critical ten-dimensional superstring requires such additional geometric and field-theoretic data. Worldsheet supersymmetry alone does not choose the compactification or establish a realistic particle spectrum.

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 $d$-plane in a charge space of signature $(d,d+16)$, modulo lattice identifications. The continuous dimension is $d(d+16)$; 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.

A compactification with quantized background form-field strengths on internal cycles. Fluxes can generate potentials for moduli of a string compactification and affect supersymmetry and vacuum geometry. They must satisfy field equations, Bianchi identities and global charge constraints; choosing flux integers alone is not a complete consistency proof.