Hilbert series and higher-order Lagrangians for the O(N) model

We compare the Hilbert series approach with explicit constructions of higher-order Lagrangians for $O(N)$ nonlinear sigma model. use to find number and type operators up mass dimension 16, spacetime $D$ 12 $N$ 12, further classify into parity internal symmetry group $O(N)$. The construction is done both parities even 10 other three cases. results two methods are in full agreement. This provides evidence conjecture regarding co-closed but not co-exact $k$-forms, which takes account integration-by-parts relations.