### Nonlinear sigma model

In quantum field theory, a **nonlinear σ model** describes a scalar field Σ which takes on values in a nonlinear manifold called the target manifold T. The non-linear σ-model was introduced by Gell-Mann & Lévy (1960, section 6), who named it after a field corresponding to a spin 0 meson called σ in their model.
^{[1]}

## Description

The target manifold is equipped with a Riemannian metric g. Σ is a differentiable map from Minkowski space M (or some other space) to T.

The Lagrangian density in contemporary chiral form^{[2]} is given by:

- $\backslash mathcal\{L\}=\{1\backslash over\; 2\}g(\backslash partial^\backslash mu\backslash Sigma\_a,\backslash partial\_\backslash mu\backslash Sigma\_b)-V(\backslash Sigma)$

where here, we have used a + - - - metric signature and the partial derivative $\backslash partial\backslash Sigma$ is given by a section of the jet bundle of T×M and V is the potential.

In the coordinate notation, with the coordinates Σ^{a}, a=1,...,n where n is the dimension of T,

- $\backslash mathcal\{L\}=\{1\backslash over\; 2\}g\_\{ab\}(\backslash Sigma)\; \backslash partial^\backslash mu\; \backslash Sigma^\{a\}\; \backslash partial\_\backslash mu\; \backslash Sigma^\{b\}\; -\; V(\backslash Sigma)$.

In more than 2 dimensions, nonlinear σ models are nonrenormalizable. This means they can only arise as effective field theories. New physics is needed at around the distance scale where the two point connected correlation function is of the same order as the curvature of the target manifold. This is called the UV completion of the theory.

There is a special class of nonlinear σ models with the internal symmetry group G *. If G is a Lie group and H is a Lie subgroup, then the quotient space G/H is a manifold (subject to certain technical restrictions like H being a closed subset) and is also a homogeneous space of G or in other words, a nonlinear realization of G. In many cases, G/H can be equipped with a Riemannian metric which is G-invariant. This is always the case, for example, if G is compact. A nonlinear σ model with G/H as the target manifold with a G-invariant Riemannian metric and a zero potential is called a quotient space (or coset space) nonlinear σ model.

When computing path integrals, the functional measure needs to be "weighted" by the square root of the determinant of g

- $\backslash sqrt\{\backslash det\; g\}\backslash mathcal\{D\}\backslash Sigma.$

This model proved to be relevant in string theory where the two-dimensional manifold is named **worldsheet**. Proof of renormalizability was given by Daniel Friedan.^{[3]} He showed that the theory admits a renormalization group equation, at the leading order of perturbation theory, in the form

- $\backslash lambda\backslash frac\{\backslash partial\; g\_\{\backslash mu\backslash nu\}\}\{\backslash partial\backslash lambda\}=\backslash beta\_\{\backslash mu\backslash nu\}(T^\{-1\}g)=R\_\{\backslash mu\backslash nu\}+O(T^2).$

being $R\_\{\backslash mu\backslash nu\}$ the Ricci tensor.

This represents a Ricci flow having Einstein field equations for the target manifold as a fixed point. The existence of such a fixed point is relevant, as it grants, at this order of perturbation theory, that conformal invariance is not lost due to quantum corrections and one has a sensible quantum field theory. Further adding nonlinear interactions representing flavor-chiral anomalies results in the Wess–Zumino–Witten model,^{[4]} which
augments the geometry of the flow to include torsion, leading to an infrared fixed point as well, on account of teleparallelism ("geometrostasis").^{[5]}

## O(3) Non-linear Sigma Model

One of the most famous examples, of particular interest due to its topological properties, is the O(3) nonlinear sigma model in 1 + 1 dimensions, with the Lagrangian density

- $\backslash mathcal\; L=\; \backslash tfrac\{1\}\{2\}\backslash \; \backslash partial^\backslash mu\; \backslash hat\; n\; \backslash cdot\backslash partial\_\backslash mu\; \backslash hat\; n$

where $\backslash hat\; n=(n\_1,n\_2,n\_3)$ with the constraint $\backslash hat\; n\backslash cdot\; \backslash hat\; n=1$ and $\backslash mu=1,2$. This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning $\backslash hat\; n=\backslash textrm\{const.\}$ at infinity. Therefore in the class of finite-action solutions we may identify the points at infinity as a single point, i.e. that space-time can be identified with a Riemann Sphere. Since the $\backslash hat\; n$-field lives on a sphere as well, we have a mapping $S^2\backslash rightarrow\; S^2$, the solutions of which are classified by the Second Homotopy group of a 2-sphere. These solutions are called the O(3) Instantons.

## See also

- Sigma model
- Chiral model
- Little Higgs
- Skyrmion, a soliton in non-linear sigma models
- WZW model
- Fubini-Study metric, a metric often used with non-linear sigma models.
- Ricci flow
- Scale invariance

## References

## External links

- 'Nonlinear Sigma model' on Scholarpedia