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

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”