Higher-Level Generalization of Conformal-Block Operators

Definitions and mathematical statement

The supplied record concerns operators described in its cleaned statement as conformal-block operators at level $\ell=3$. The original passage proposes a set of level-three operators and conjectures a generalization to all $\ell>3$. Neither the operators nor the meaning and admissible values of the level parameter $\ell$ are defined in the supplied excerpt.

Question (source-qualified formulation). For every admissible level $\ell>3$, does there exist a corresponding generalization of the operators proposed at level $\ell=3$, satisfying the intended defining properties of those operators?

This is an unresolved formulation rather than a fully specified mathematical conjecture: the excerpt omits the level-three construction, its underlying algebraic setting, the operators' domains and codomains, and the properties that a higher-level generalization must preserve. In particular, the original phrase "does exactly this" refers to preceding exposition that is not supplied. The cleaned statement describes the level-three operators as constructed, whereas the original passage describes them as proposed and the supplied literature assessment calls their pattern conjectural. Their established properties therefore cannot be inferred from this record.

Short English statement

The question asks whether the proposed level-three conformal-block operators have counterparts at every admissible level $\ell>3$. The excerpt does not specify the operators or what qualifies as a counterpart, so a complete mathematical statement requires the surrounding source context. The supplied record labels the problem open; that status is not independently verified here.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30006126$, source problem code catalogue entry, titled "Higher-Level Generalization of Conformal-Block Operators," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2024.
• Supplied report citation: Mini-Workshop: Infinite-Dimensional Kac-Moody Lie Algebras in Supergravity and M Theory (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/53, pp. 3055--3102. https://doi.org/10.4171/owr/2024/53. The record attributes the question to this workshop report but supplies no precise page or named contribution for the passage.
• Supplied literature source, "Kac--Moody OWR report": https://oa.tib.eu/renate/bitstreams/67fd7081-be76-40fb-956b-4f1194e2f4f8/download.
• Supplied literature source, "A string-like realization": https://pure.mpg.de/rest/items/item_3633535_2/component/file_3633536/content. Further bibliographic details are not supplied.
• Literature assessment dated 2026-08-22: the supplied curation describes the level-three operators as a conjectural pattern and their extension to every level greater than three as open. This is supplied dated curation, not an independently verified literature assessment; the background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked, the question isolated from surrounding exposition, and its notation repaired. These are supplied verification claims, not verification performed for this draft.