Classify indefinite-metric neural conformal algebras

Determine the appropriate type classification of the indefinite-metric algebras associated with neural network conformal theories, including whether a Krein-space analogue of the type III classification exists.

Background

The paper constructs an abstract ∗*-algebra from conformal generators, primary operators, descendants, and correlator relations. Because the correlator functional is not positive and the standard GNS construction does not apply, the resulting representation theory is expected to involve Krein spaces rather than Hilbert spaces.

The authors point out that the usual type III classification of local operator algebras belongs to the positive-metric Haag–Kastler framework, which is not directly available here. It remains unresolved whether an analogous classification can be formulated for these indefinite-metric algebras at all.

References

Even granting a $J$-representation, the appropriate ``type'' classification of these indefinite-metric algebras, i.e., a Krein-space analogue of the type III story, is unclear and unsure if such a structure even exists.

— Notes on algebraic perspective on neural network conformal theories  (2609.37074 - Leutheusser et al., 29 Sep 2026) in Section 6, subsection “Lessons for neural-network conformal theories”