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

TL;DR

This paper compares the Hilbert series method with explicit operator construction for the $O(N)$ nonlinear sigma model, confirming the Hilbert series' effectiveness in classifying higher-order Lagrangians up to high dimensions.

Contribution

It demonstrates the agreement between the Hilbert series approach and explicit constructions for higher-order operators in the $O(N)$ model, supporting the Hilbert series conjecture.

Findings

Hilbert series accurately predicts operator counts up to dimension 16

Explicit constructions match Hilbert series results up to dimension 12

Supports the conjecture on co-closed but not co-exact forms in the Hilbert series

Abstract

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