Moduli (physics)

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This article is about "moduli" as the term is used in physics. For the mathematical term, see Moduli space.

In quantum field theory, the term moduli (or more properly moduli fields) is sometimes used to refer to scalar fields whose potential energy function has continuous families of global minima. Such potential functions frequently occur in supersymmetric systems. The term "modulus" is borrowed from mathematics, where it is used synonymously with "parameter".

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 parameters which label possible string backgrounds: the expectation value of the dilaton field, the values of various coupling constants, the parameters (e.g. the radius and complex structure) which govern the shape of the compactification manifold, the expectation values of Wilson lines of gauge fields around non-trivial cycles, et cetera. These parameters are represented, in string perturbation theory, by scalar fields on the string worldsheet, hence the dual usage. In string theory, the term "moduli space" is often used specifically to refer to the space of all possible string backgrounds.