Editing Kaluza–Klein theory (section)

=== Generalizations ===
In the above, the size of the loop <math>\Lambda</math> acts as a coupling constant between the gravitational field and the electromagnetic field.  If the base manifold is four-dimensional, the Kaluza–Klein manifold ''P'' is five-dimensional. The fifth dimension is a [[compact space]] and is called the '''compact dimension'''. The technique of introducing compact dimensions to obtain a higher-dimensional manifold is referred to as [[compactification (physics)|compactification]]. Compactification does not produce group actions on [[Chirality (physics)|chiral]] [[fermions]] except in very specific cases: the dimension of the total space must be 2 mod 8, and the G-index of the Dirac operator of the compact space must be nonzero.<ref>L. Castellani et al., Supergravity and superstrings, vol.&nbsp;2, ch.&nbsp;V.11.</ref>

The above development generalizes in a more-or-less straightforward fashion to general [[principal G-bundle|principal ''G''-bundles]] for some arbitrary [[Lie group]] ''G'' taking the place of [[U(1)]]. In such a case, the theory is often referred to as a [[Yang–Mills theory]] and is sometimes taken to be synonymous. If the underlying manifold is [[supersymmetric]], the resulting theory is a super-symmetric Yang–Mills theory.