Some remarks on invariants

TL;DR

This work analyzes the ring of invariants $S^{\mathfrak g}$ for a finite-dimensional Lie algebra ${\mathfrak g}$ acting on an irreducible module $V$ by recasting the problem in terms of an extended Lie algebra ${\mathfrak g}^+$. It develops criteria based on the finiteness or affine nature of ${\mathfrak g}^+$ to determine when the invariant ring is trivial, singly generated, or freely generated, and uses partition functions (Hilbert series) and resolutions (Tate and additive) to characterize the generator structure and relations. A broad collection of examples across ${\mathfrak g}=\mathfrak{sl}(n)$ and $\mathfrak{so}(n)$ illustrates how freeness and dimensionality correlate with the extended algebra, including notable cases where the ring is freely generated (e.g., certain 4-forms) and more intricate rings with many generators and relations (e.g., higher-form invariants). The paper then focuses on the physically motivated problem of classifying invariants of a self-dual 5-form in 10 dimensions, deriving a tensor-based and a spinor-based construction of low-order invariants (orders 4, 6, 8, 10, 12) and cataloging numerous explicit invariants, while highlighting substantial remaining challenges in achieving a complete higher-order classification. The results have potential implications for constructing higher-derivative interactions in string theory and duality-invariant field theories, where such invariants govern admissible non-linear terms.

Abstract

The demand to know the structure of functionally independent invariants of tensor fields arises in many problems of theoretical and mathematical physics, for instance for the construction of interacting higher-order tensor field actions. In mathematical terms the problem can be formulated as follows. Given a semi-simple finite-dimensional Lie algebra $\mathfrak g$ and a $\mathfrak g$-module $V$, one may ask about the structure of the sub-ring of $\mathfrak g$-invariants inside the ring freely generated by the module. We point out how some information about the ring of invariants may be obtained by studying an extended Lie algebra. Numerous examples are given, with particular focus on the difficult problem of classifying invariants of a self-dual 5-form in 10 dimensions.