Chiral matrix scalar model ($\Phi \mapsto e^{i\theta }{U}_L\Phi U_R^{†}$)

A theory of an $N\times N$ matrix of complex scalar fields with kinetic term $\mathrm{tr}(\partial {\Phi }^{†}\partial \Phi )$ and potential $V=\alpha \mathrm{tr}[({\Phi }^{†}\Phi {)}^2]/2+\beta [\mathrm{tr}({\Phi }^{†}\Phi ){]}^2/2+m^2\mathrm{tr}({\Phi }^{†}\Phi )$. It is invariant under global $SU(N{)}_L\times SU(N{)}_R\times U(1)$ transformations. For $\alpha ,\beta >0$ and $m^2<0$, minimizing the nonnegative eigenvalues of ${\Phi }^{†}\Phi$ gives a vacuum expectation value $C1$ with $C^2=-m^2/(\alpha +N\beta )$, up to a global symmetry transformation. The connected stabilizer is diagonal $SU(N)$, with $N^2$ broken generators. Expanding $\Phi =C1+(H+iP)/\sqrt{2}$ with Hermitian matrices gives $N^2$ massless Goldstone bosons in $P$ and squared masses $2C^2(\alpha +N\beta )$ for the singlet in $H$ and $2\alpha C^2$ for its traceless Adjoint representation. The faithful unbroken conjugation action factors through $SU(N)/{\mathbb{Z}}_N$.