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Collapse limit to quantum Riemannian 1-spaces with CD(0,∞)

Establish that, after rescaling by the inverse minimal positive eigenvalue, a collapsing family of unitary CFTs converges to a quantum Riemannian 1-space whose semigroup generator L satisfies the curvature-dimension inequality CD(0,∞).

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Background

The double-scaling collapse considers rescaling by the spectral gap and analyzing amplitudes for surfaces built from long tubes. The anticipated limit is a graph-field-theory-like quantum 1-geometry.

The CD(0,∞) inequality is the Bakry–Émery curvature lower bound analog, corresponding to nonnegative Ricci curvature in smooth settings; proving it in the limit connects CFT collapse with metric-measure geometry.

References

Conjecture After rescaling $L_0+\overline{L}0$ by the factor $\lambda{min}{-1}$, there is a limit in the sense of Section \ref{spaces with measure} of the unitary CFTs, which is a quantum Riemannian $1$-space in the sense of Section \ref{quantum Riemannian 1-spaces}, such that the operator $L$ obtained as a limit of $(L_0+\overline{L}0)/\lambda{min}$ satisfies the curvature-dimension inequality $CD(0,\infty)$ from Section \ref{semigroups and CD inequalities}.

— Moduli space of Conformal Field Theories and non-commutative Riemannian geometry (2506.00896 - Soibelman, 1 Jun 2025) in Section 3.2, Segal’s axioms and collapse (Conjecture about the limit)