Extra Dimensions and Compactification

Why dimensions become hidden

Superstring theories are naturally formulated in ten spacetime dimensions: one time dimension and nine spatial dimensions. We experience three large spatial dimensions because the remaining six could be compact, meaning they close back on themselves at a scale too small to resolve directly.

The simplest analogy is a garden hose. From far away it looks one-dimensional, but an ant close to it can move along the length and around the circumference. A compact internal dimension can generate a tower of heavier Kaluza-Klein modes. If its radius is small enough, those modes are too heavy to produce in current experiments.

Geometry becomes physics

The shape and size of the compact space affect the effective four-dimensional theory. They determine gauge symmetries, chiral matter, supersymmetry, coupling constants, and the spectrum of light scalar fields called moduli. A popular class of compactifications uses Calabi-Yau threefolds because they can preserve some supersymmetry and support rich cycles on which branes and fluxes can wrap.

Different geometries can have different topology, encoded partly by quantities such as the number of independent cycles. Those cycles provide places for flux, branes, and wrapped strings to live. The resulting four-dimensional fields are not arbitrary decorations: they arise from the higher-dimensional metric, form fields, and brane positions.

Stabilizing the moduli

A dangerous freedom is that a compact space can often change its size or shape at little energy cost. Such unfixed moduli would appear as long-range scalar fields and would make physical constants vary. Fluxes, non-perturbative effects, branes, and orientifold planes can contribute to a potential that fixes some or all moduli.

Stabilization is tied to the vacuum-energy problem. A compactification may produce anti-de Sitter, Minkowski, or de Sitter-like effective solutions depending on the ingredients and approximations. Constructing controlled, metastable de Sitter vacua remains one of the most debated technical areas.

The landscape

Because many internal geometries, flux choices, brane arrangements, and vacuum states may be possible, string theory appears to have a very large landscape of effective theories. The challenge is not merely finding a model resembling the Standard Model; it is explaining why one vacuum is selected and why its parameters take their observed values.

Read branes and dualities to see why compact cycles and extended objects are more than geometry.