All-level BRST cohomology of flowed principal-series modules

Establish the complete positive-level BRST cohomology of the one-unit spectrally flowed Drinfel'd–Sokolov reduction of relaxed affine $\widehat{\mathfrak{sl}}(2)_k$ modules induced from the principal continuous series, and prove that for generic real $P$ it is a single non-degenerate Virasoro Verma module $\mathcal{V}_{h_P}$ with $h_P=(c-1)/24+P^2$.

Background

The paper directly analyzes the level-zero constraint and finds a one-dimensional cohomology class for fixed principal-series Casimir. It also reports checks of the cohomology through finite levels, with dimensions matching the Virasoro descendant count.

The authors do not provide an all-level proof for the flowed reduction, because standard unflowed highest-weight Drinfel'd–Sokolov theorems do not directly apply to the spectrally flowed relaxed modules relevant above the BTZ threshold. A complete result would rigorously justify the identification of principal-series Wilson-line sectors with individual heavy Virasoro primaries and their full descendant towers.

References

For the principal series the level-zero mechanism can nevertheless be analyzed directly: at fixed $s$ and fixed $\epsilon$, the cylinder constraint selects a single Virasoro primary class, as spelled out in Appendix~\ref{app:DS-reduction}. The positive-level BRST cohomology of the flowed discrete and relaxed modules will be discussed in .

— 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 1, “Strategy: Quantize-then-Reduce”; Section 2.7; Appendix B