Non-linear sigma model

Example

Let $(\Sigma, h)$ be a Riemannian manifold of dimension $d$, the spacetime, and let $(M, g)$ be a Riemannian manifold, the target. The non-linear sigma model with target $M$ is the following field theory. Its fields are smooth maps $\phi \colon \Sigma \to M$, and its action is $$S(\phi) = \frac{1}{2 \lambda} \int_\Sigma h^{ij}(x) \, g_{ab}(\phi(x)) \, \partial_i \phi^a \, \partial_j \phi^b \, \mathrm{vol}_h,$$ where $\lambda > 0$ is a coupling constant and $\phi^a$ are local coordinates of $\phi$. Up to the factor $1/\lambda$, this is the Dirichlet energy of $\phi$, so the classical solutions are the harmonic maps from $\Sigma$ to $M$.

Check with the $O(N)$ model: take $M = S^{N-1} \subset \R^N$ with the round metric. A map is then a function $\phi \colon \Sigma \to \R^N$ with $\phi \cdot \phi = 1$, and $S(\phi) = \frac{1}{2\lambda} \int_\Sigma \abs{\nabla\phi}^2$. Adding a Lagrange multiplier for the constraint, the Euler–Lagrange equation is $\Delta\phi + \abs{\nabla\phi}^2 \phi = 0$, the harmonic map equation into the sphere. The action is invariant under the group $O(N)$ acting on $\R^N$, which gives the model its name. The model is non-linear because the constraint $\phi \cdot \phi = 1$ makes the space of fields non-linear, even though the action looks quadratic.