Editing Moduli (physics) (section)

== Moduli spaces in quantum field theories ==
{{Redirect|Vacuum manifold|a similar term in engine mechanics|Manifold vacuum}}
In quantum field theories, the possible vacua are usually labeled by the vacuum expectation values of scalar fields, as Lorentz invariance forces the vacuum expectation values of any higher spin fields to vanish.  These vacuum expectation values can take any value for which the potential function is a minimum.   Consequently, when the potential function has continuous families of global minima, the space of vacua for the quantum field theory is a manifold (or orbifold), usually called the '''vacuum manifold'''.<ref>{{Cite journal |last=Teerthal |first=Patel |date=2022-01-16 |title=Kibble mechanism for electroweak magnetic monopoles and magnetic fields |journal=[[Journal of High Energy Physics]] |volume=2022 |issue=1 |publisher=[[Arizona State University]] |page=10 |doi=10.1007/JHEP01(2022)059 |arxiv=2108.05357 |bibcode=2022JHEP...01..059P |s2cid=256034831 }}</ref>  This manifold is often called the '''moduli space of vacua''', or just the moduli space, for short.

The term '''moduli''' is also used in [[string theory]] to refer to various continuous parameters that label possible [[string background]]s: the expectation value of the [[dilaton]] field, the parameters (e.g. the radius and complex structure) which govern the shape of the compactification manifold, et cetera.   These parameters are represented, in the quantum field theory that approximates the string theory at low energies, by the vacuum expectation values of massless scalar fields, making contact with the usage described above.  In string theory, the term "moduli space" is often used specifically to refer to the space of all possible string backgrounds.