Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 1 / iv

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.