Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 337 / 2 / f / Solution

Because $\tilde{\mathbf{n}}\cdot (\tilde{\mathbf{n}}\times {\partial }_t\tilde{\mathbf{n}})=0$, the component of $\mathbf{q}$ parallel to $\tilde{\mathbf{n}}$ is $-\mathbf{B}\cdot \tilde{\mathbf{n}}$. The Gaussian integral over $\lambda$ removes precisely this longitudinal component. ThusUsing the vector triple-product identity and $|\tilde{\mathbf{n}}|=1$, this becomes the $O(3)$ nonlinear sigma model in a background field:The magnetic field acts as the temporal component of an $O(3)$ background gauge connection.