Exact spectrum and multiplicities of heavy primaries in holographic CFTs

Determine the exact set and multiplicities of heavy Virasoro primary states present in a given holographic conformal field theory, beyond the universal continuum approximation supplied by the modular S-kernel.

Background

The paper constructs bulk microstates associated with individual heavy Virasoro primaries by means of principal continuous-series Wilson lines in SL(2,R)SL(2,\mathbb{R}) Chern–Simons theory followed by flowed Drinfel'd–Sokolov reduction. The modular S-kernel determines a universal asymptotic continuum density of such primaries in the Cardy regime, allowing the authors to derive the smooth BTZ saddle and its fluctuations.

However, the exact spectrum of a particular holographic CFT, including which heavy primaries occur and with what multiplicities, is theory-specific and is not determined by the universal modular description. Resolving this problem would provide a non-perturbative microscopic specification of the states whose ensemble produces black-hole thermodynamics.

References

Such states in the CFT correspond in the bulk to BTZ black holes, and the question what are the states in the boundary CFT that describe them is essentially the question what are the microstates of these black holes. This question remains unresolved as of this writing.

— Interpolating between single and double trace $T\bar T$ by moving LST into the bulk  (2609.10823 - Chu et al., 9 Sep 2026) in Section 2.3, The boundary theory

A thermal ensemble of microstates is useful as a statistical system if we have some understanding about which of these microstates are present in a given theory. In most situations that are of interest in statistical mechanics, particularly when strong chaos and thermalization are present, we do not know an exact answer to this question.

— Building a Quantum Black Hole Microstate: A Bulk Path Integral for a Heavy Virasoro Primary  (2608.14541 - Krishnan et al., 14 Aug 2026) in Section 5, Section 5.1, “From the Microstate to the Ensemble”

Identifying such bulk contributions remains open.

— Large gaps and BTZ entropy in modular spectra with positive integer degeneracies  (2609.39930 - Chang et al., 30 Sep 2026) in Section 6, Discussion, paragraph “Bulk interpretation of the spectral completion”