# Sigma model

In physics, a sigma model is a physical system that is described by a Lagrangian density of the form:

${\displaystyle {\mathcal {L}}(\phi _{1},\phi _{2},\ldots ,\phi _{n})=\sum _{i=1}^{n}\sum _{j=1}^{n}g_{ij}\;\mathrm {d} \phi _{i}\wedge {*\mathrm {d} \phi _{j}}}$

Depending on the scalars in gij, it is either a linear sigma model or a non-linear sigma model. The fields ϕi, in general, provide a map from a base manifold called the worldsheet to a target Riemannian manifold of the scalars linked together by internal symmetries. (In string theory, however, that is often understood to be the actual spacetime.)

## Overview

The sigma model was introduced by Gell-Mann & Lévy (1960, section 5); the name σ-model comes from a field in their model corresponding to a spinless meson called σ, a scalar meson introduced earlier by Julian Schwinger.[1] The model served as the dominant prototype of spontaneous symmetry breaking of O(4) down to O(3): the three axial generators broken are the simplest manifestation of chiral symmetry breaking, the surviving unbroken O(3) representing isospin.

A basic example is provided by quantum mechanics which is a quantum field theory in one dimension. It's a sigma model with a base manifold given by the real line parameterizing the time (or an interval, or the circle, etc.) and a target space that is the real line.

The model may be augmented by a torsion term to yield the Wess–Zumino–Witten model.

## References

1. ^ Julian S. Schwinger, "A Theory of the Fundamental Interactions", Ann. Phys. 2(407), 1957.
• Gell-Mann, M.; Lévy, M. (1960), "The axial vector current in beta decay", Il Nuovo Cimento, 16: 705–726, Bibcode:1960NCim...16..705G, doi:10.1007/BF02859738 online copy