Construct the Krein representation for large-N generalized free fields

Construct in closed form the Krein space, metric operator, and $J$-representation for the large-$N$ generalized free field with negative scaling dimension $4 orall n<0$, thereby providing a concrete realization of the proposed indefinite-metric structure.

Background

The paper identifies the large-NN generalized free field with scaling dimension $4 orall n<0$ as the cleanest test case for resolving the outstanding representation-theoretic questions. This theory is quasi-free and non-unitary, so its correlators may permit an explicit analysis of the state space without the complications of a general interacting neural network conformal theory.

A closed-form construction would test whether the assumed Krein positivity and partial-majorant properties can actually be realized, and would make explicit the metric operator and the representation of the correlator algebra. The problem is presented as a concrete route toward settling the broader existence questions for these structures.

References

The cleanest setting in which to make all of this explicit is the large-$N$ generalized free field of Section \ref{subsec:larg-N}, starting from a generalized free field with $\Delta = -n < 0$, which is a quasi-free non-unitary theory, and its Krein space, metric operator, and $J$-representation should be constructible in closed form, providing a concrete testbed for the structures conjectured above.

— 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”