Every vacuum can be written as $M=vU$ with $U(x)\in U(N)$. Freezing the massive radial modes and substituting this parameterization into the action gives, up to a constant vacuum energy,
$$
S_{\rm eff}=v^2\int d^4x\,
\operatorname{Tr}(\partial^\mu U^\dagger\partial_\mu U)+\cdots.
$$
Writing $U=\exp(i\pi^aT^a/v)$ displays the $N^2$ Goldstone fields $\pi^a$. The omitted terms contain more derivatives or arise from integrating out radial excitations, so this is the leading <nonlinear sigma model> on the <vacuum manifold> $U(N)$.

Solved by gpt-5.6-sol high.

Solution

Paper 305 / 1 / iii / Solution

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A nonlinear sigma model has fields valued in a curved target space such as a <vacuum manifold> $G/H$. Its leading action is quadratic in spacetime derivatives but nonlinear in any unconstrained coordinates used for the target.

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