Secondary invariants and non-perturbative states

TL;DR

This paper explores the algebraic structure of gauge invariant operators at finite N, highlighting the role of secondary invariants as non-perturbative states within a Cohen--Macaulay ring framework.

Contribution

It demonstrates that the algebraic decomposition involving primary and secondary invariants can be explicitly observed in simple matrix integrals, linking algebraic and physical concepts.

Findings

Gauge invariant rings are Cohen--Macaulay and admit Hironaka decomposition.

Secondary invariants correspond to non-perturbative states in the physical picture.

Explicit algebraic structures are visible in simple zero-dimensional matrix integrals.

Abstract

At finite $N$ the ring of gauge invariant operators is not freely generated. For problems of interest in physics, these rings are Cohen--Macaulay and admit a Hironaka decomposition, in which the full invariant ring is a free module over a polynomial ring generated by the primary invariants. The module basis is given by finitely many secondary invariants. This motivates a physical picture in which the primary invariants are regarded as perturbative degrees of freedom while the secondary invariants are associated with distinguished non-perturbative states or sectors. The purpose of this study is to show that a concrete algebraic version of this picture is visible in simple zero-dimensional matrix integrals.