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Precompactness of quantum Riemannian 1-spaces with measure and spectral bounds

Prove that the space of isomorphism classes of quantum Riemannian 1-spaces with a measure, nonnegative Ricci curvature (in the Bakry–Émery sense), a uniform positive lower bound on the spectral gap, and dimension spectra contained in a fixed interval [a,b] is precompact in the topology defined via convergence of amplitudes and states.

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Background

Quantum Riemannian 1-spaces assign trace-class amplitudes to metrized graphs with a semigroup generator L and a compatible state τ; a Bakry–Émery CD(0,∞) condition encodes nonnegative Ricci curvature.

The conjecture parallels classical precompactness results for metric-measure spaces with curvature and diameter bounds, extending them to the noncommutative graph-field-theory framework with spectral constraints.

References

Conjecture The space of isomorphism classes of quantum Riemannian $1$-spaces\label{conjecture about precompactness of quantum spaces} with a measure, which have non-negative Ricci curvature, spectral gap bounded below by a given number $C$, and such that the corresponding spectral triples have dimensional spectrum belonging to a given interval $[a,b]$ (see [CoMar] for the definition of the dimension spectrum) is precompact in the topology defined in Section \ref{spaces with measure}.

— Moduli space of Conformal Field Theories and non-commutative Riemannian geometry (2506.00896 - Soibelman, 1 Jun 2025) in Section 5.4, Non-negative Ricci curvature for quantum 1-geometry (Conjecture)