Non-linear sigma model

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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 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 is given by:

$\mathcal{L}={1\over 2}g(\partial^\mu\Sigma,\partial_\mu\Sigma)-V(\Sigma)$

where here, we have used a + - - - metric signature and the partial derivative $\partial\Sigma$ 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,

$\mathcal{L}={1\over 2}g_{ab}(\Sigma) \partial^\mu \Sigma^{a} \partial_\mu \Sigma^{b} - V(\Sigma)$ .

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.

$\sqrt{\det g}\mathcal{D}\Sigma$