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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.

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,Writing $U=\exp (i{\pi }^a{T}^a/v)$ 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.

For Hermitian $M$, the continuous transformations preserving the field space act by conjugation,with the central $U(1)$ acting trivially; there is also the discrete symmetry $M\mapsto -M$. The vacuum equation is $M^2=v^{2}1$, so every vacuum is unitarily conjugate toBecause the integer $p$ cannot change continuously, the vacuum manifold has disconnected componentsOn the $p$th component the unbroken continuous group is $U(p)\times U(N-p)$, and the Goldstone theorem givesGoldstone bosons. The discrete sign symmetry exchanges the components $p$ and $N-p$ but produces no Goldstone mode.

Solved by gpt-5.6-sol high.

A global symmetry is spontaneously broken when it preserves the action but does not preserve a chosen ground state. If $G$ is broken to the stabilizer $H$, the degenerate vacua form a vacuum manifold $G/H$. The Goldstone theorem states that a relativistic theory has one massless scalar mode for each broken continuous internal generator, so the standard counting gives $\dim G-\dim H$ Goldstone bosons.

Solved by gpt-5.6-sol high.

Spontaneous symmetry breaking occurs when the action and equations have a symmetry but a chosen ground state is invariant only under a proper subgroup. Acting with the broken symmetry generates a degenerate vacuum manifold.