Moduli (physics)

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". The word moduli (Moduln in German) first appeared in 1857 in Bernhard Riemann's celebrated paper "Theorie der Abel'schen Functionen".

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.

In general quantum field theories, even if the classical potential energy is minimized over a large set of possible expectation values, generically once quantum corrections are included nearly all of these configurations cease to minimize the energy. The result is that the set of vacua of the quantum theory is generally much smaller than that of the classical theory. A notable exception occurs when the various vacua in question are related by a symmetry which guarantees that their energy levels remain exactly degenerate.

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