Compactification

Compactification makes some spatial dimensions compact, so modes along them acquire discrete momenta. At energies below the first nonzero compact-mode mass, a dimensional reduction to the zero modes gives a lower-dimensional effective field theory. The compact volume fixes the gravitational kinetic coefficient in an unwarped product background.

For an unwarped product metric with fixed compact volume $V_n$, integrating the higher-dimensional Einstein-Hilbert action gives $M_{\mathrm{Pl}}^2=M_{\ast }^{n+2}{V}_n$ when the action coefficient is $M_D^{D-2}/2$. The reduced Planck mass convention must be distinguished from the unreduced one, and stabilizing the internal metric is a separate dynamical issue.

A Kaluza-Klein mode is a coefficient in a field expansion in eigenfunctions on compact extra dimensions. On a circle of circumference $L$, momentum is $2\pi \ell /L$, and a higher-dimensional massless field has lower-dimensional squared mass $(2\pi \ell /L{)}^2$. Nonzero modes give exponentially suppressed static-potential corrections at distances much larger than $L$.

On three noncompact spatial dimensions times a circle of circumference $L$, the method of images gives ${\sum }_{j\in \mathbb{Z}}[r^2+j^2{L}^2{]}^{-1}=\pi \coth (\pi r/L)/(Lr)$ at equal compact coordinates. This interpolates from $1/r^2$ at short distances to $\pi /(Lr)$ at long distances. A gravitational circle-radius scalar can change the long-range coefficient if it remains unstabilized.

Dimensional reduction retains fields independent of chosen internal coordinates. In a flat-torus zero-mode reduction, a metric supplies a metric, internal-component vectors and internal metric scalars; a massless p-form gauge field supplies lower-rank fields with internal indices. Fluxes, twists or projections can alter the spectrum and preserved supercharges.