Secondary invariants and non-perturbative states

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.

• The paper establishes that secondary invariants encode non-perturbative sectors, revealing deeper structure than primary invariants in finite-N matrix models.
• It employs invariant theory and Hironaka decomposition to systematically demonstrate how branch structures emerge through explicit matrix model examples.
• The study shows that integrating over invariant variables allows tractable computations of partition functions and correlators, matching Gaussian results.

Secondary Invariants and Non-Perturbative States: Algebraic and Physical Structure in Finite-$N$ Matrix Models

Overview and Motivation

This work systematically elucidates the algebraic structure underlying the space of gauge-invariant operators in finite-$N$ matrix models. At the core of the study is the observation that, in contrast to the traditionally considered large-$N$ limit where the invariant operator algebra is freely generated, at finite $N$ the ring of invariants develops nontrivial structure, governed by trace relations. Employing tools from invariant theory, specifically the Cohen–Macaulay property and Hironaka decomposition, the paper establishes that the gauge-invariant ring is a free module over its primary invariants, with secondary invariants forming a finite module basis.

This structure is shown to have direct physical implications. The authors propose, and provide evidence for, a correspondence in which primary invariants parametrize perturbative fluctuations, while secondary invariants label non-perturbative sectors—providing an algebraic fingerprint for global, non-perturbative features of the path integral and Hilbert space.

Algebraic Structure: Hironaka Decomposition and Its Physical Meaning

The central algebraic statement is that for classes of matrix models with linearly reductive gauge symmetry (notably $U(N)$), the invariant ring $\mathcal{R}$ of gauge-invariant polynomials can be decomposed as:

$$\mathcal{R} = \bigoplus_{\alpha=0}^{N_S-1} \mathcal{R}_P \, s_\alpha$$

where $\mathcal{R}_P$ is the polynomial ring generated by $N_P$ algebraically independent primary invariants, and $\{s_\alpha\}$ are secondary invariants. This forms a free module structure akin to the Hironaka decomposition for Cohen–Macaulay rings.

Physically, this decomposition implies that (i) the primary invariants serve as 'coordinates' for the continuous base of the quotient space, and (ii) the secondary invariants resolve discrete data associated with the fiber above a given primary configuration. The latter is interpreted as encoding non-perturbative sector information that is invisible from a perturbative perspective localized on a single branch.

Explicit Matrix Model Examples: Fiber Geometry and Branch Structure

The paper demonstrates these ideas concretely via a sequence of zero-dimensional $N$0 Hermitian matrix models with $N$1 matrices. For each case, the authors implement a systematic reduction from matrix elements to invariants:

• Two matrices ($N$2): The ring is generated by five primaries and the secondary structure is trivial. The full configuration space, after reduction, is simply parametrized by the primaries, reflecting a single-branch geometry.
• Three matrices ($N$3): Nine primaries and a single nontrivial secondary emerge. The quotient space exhibits a two-sheeted cover (sign of oriented volume), and the secondary resolves this discrete ambiguity.
• Four matrices ($N$4): Thirteen primaries and seven nontrivial secondaries structure the quotient as an eight-sheeted algebraic cover (quartic order for a diagonal trace invariant, doubled by orientation data).

In all cases, the number of algebraic branches of the cover matches the number of secondary invariants. The transformation to invariant variables is meticulously carried out, including careful gauge-fixing and Jacobian calculation. The positivity constraints on Gram matrices are imposed to ensure integration over the physical (Hermitian) slice rather than the complexified quotient. Notably, the reduction of the path integral to invariant variables manifests as a sum over branches, with branchwise contributions precisely matched to the structure predicted by the Hironaka decomposition.

For each case, the reduced measure is validated by reproduction of explicit Gaussian correlators in the invariant basis, reinforcing the correctness of the approach.

Permutation Symmetry Example: $N$5-Invariant Boson Systems

To further probe the generality, the authors analyze an $N$6-boson model in two dimensions with diagonal $N$7 symmetry. The ring of $N$8-invariants admits $N$9 primaries and $N$0 secondaries, with the generic fiber structure forming an $N$1-sheeted cover. The change of variables from coordinates to invariants exposes the sheet structure, and the branch data is explicitly encoded in the secondary invariants, which distinguish different pairings of $N$2 and $N$3 roots. This example cements the link between secondary invariants and global, discrete sector data.

Secondary Invariants as Non-Perturbative Loci: Variational and Path Integral Perspective

The paper advances the physical significance of this algebraic picture by constructing effective auxiliary actions in invariant space, whose stationary points correspond to the algebraic branches uncovered in the previous sections. Under this formalism, the set of branches identified algebraically becomes manifest as a set of companion path integral saddles (or more generally, critical submanifolds in complexified invariant space). The analysis is explicit for the $N$4 cases, and for the $N$5 permutation model.

Importantly, the secondary invariants do not introduce additional perturbative (continuous) degrees of freedom; instead, they label the emergent non-perturbative sectors that would be inaccessible in a strictly local (perturbative) analysis. In the path integral, summing over these branches becomes essential for reconstructing the full, global partition function.

Theoretical and Practical Implications

Theoretical Implications

• Non-perturbative sector labeling: The paper provides strong evidence that secondary invariants algebraically encode non-perturbative sector information; this bridges invariant theory with semi-classical and global data (such as potential black hole microstates in holography).
• Finite $N$6 effects: The work highlights substantial departures from the infinite-$N$7/free algebra scenario, emphasizing the need for refined algebraic tools in understanding genuine finite-$N$8 quantum effects.

Practical Implications

• Invariant variable integration: Explicit reduction to invariant variables not only clarifies the geometry but also renders multidimensional integrals tractable—relevant, for example, to computations of partition functions or correlators at finite $N$9.
• Potential applications to dynamical systems: While the models treated are zero-dimensional, the authors propose that analogous decompositions and sector structures are expected to persist in quantum mechanical and quantum field theoretic matrix or tensor models.

Numerical and Structural Results

• Gaussian correlator computations performed entirely in the invariant, branch-resolved measure are shown to match traditional matrix integral computations exactly for all studied cases.
• In all concrete examples, the number of algebraic sheets matches the number of secondary invariants, and branches are explicitly parameterized in terms of both primaries and secondaries.

Bold Claims

• The identification between the number of secondary invariants and the number of global non-perturbative sectors/branches is formulated as a conjecture with robust supporting evidence from both algebraic and integral perspectives.
• The secondary invariants do not correspond to extra perturbative fields; their role is fundamentally discrete and non-oscillator-like.

Contradictory Insights

• The traditional emphasis on primary invariants as sufficient for the bulk of physical computation at large $N$0 is inadequate at finite $N$1. The structure and consequences of secondary invariants become both algebraically and physically indispensable.

Prospects for Further Research

Several avenues are indicated for future exploration:

• Extension of the algebraic-geometry analysis to higher $N$2, where the number of secondary invariants grows rapidly and enumerative/combinatoric problems become nontrivial.
• Investigation of matrix quantum mechanics and field theory analogs, including potential relationships between Hironaka structure, background geometry duals, and black hole microstate counting.
• Deeper study of the connection between algebraic branch structure, Lefschetz thimble decomposition, and resurgent trans-series analysis in non-perturbative matrix quantum field theory.
• Characterization of discriminant/branch loci where the fiber structure becomes nontrivial and sector transitions may occur.

Conclusion

This work rigorously establishes the centrality of the Hironaka decomposition in the physical and algebraic analysis of finite-$N$3 matrix models. The explicit construction of the invariant ring’s module structure immediately translates to a meaningful decomposition of the Hilbert space and path integral over discrete, non-perturbative sectors determined by secondary invariants. The invariant variable formalism not only clarifies the geometry and sector structure of the quotient but also serves as a robust tool for explicit computation. The conjectured equivalence between secondary invariants and global non-perturbative branches forms a foundational element for a physical interpretation of finite-$N$4 phenomena in matrix and gauge theories, with anticipated ramifications reaching into dynamical and holographic regimes.