I am reading M. Schwartz's book on QFT, equation (28.24)/(28.22). They say that a set scalar fields will transform as, where $g_L$ belongs to $SU(2)_{L}$ and $g_R$ belongs to $SU(2)_R$: $$\Sigma\rightarrow g_L\Sigma\ g_R^{\dagger}.\tag{28.22}$$ The Chiral Lagrangian is (spontaneous symmetry broken) $$L=tr[(\partial_\mu\Sigma)(\partial_\mu\Sigma^{\dagger})]+m^2tr[\Sigma\Sigma^{\dagger}]-\frac{\lambda}{4}tr[\Sigma\Sigma^{\dagger}\Sigma\Sigma^{\dagger}].$$ After Spontaneous symmetry breaking: $$\Sigma(x)=\frac{v+\sigma(x)}{\sqrt{2}}\exp\left(2i\frac{\pi^a(x)\tau^a}{F_\pi}\right),\tag{28.24}$$ where $\tau^a$ are Pauli matrices.
My question is that, were there any constraint on $\Sigma$? Because if it was four unconstrained complex scalar field, then there would be altogether 8 degrees of freedom. Why after symmetry breaking, there are only 4 left?