Moduli (physics) - Moduli Spaces in Quantum Field Theories

Moduli Spaces in Quantum Field Theories

In quantum field theories, the possible vacua are usually labelled 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. 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 which label possible string backgrounds: 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.