Open–Closed Conformal-Field Structure in String Topology

Definitions and mathematical statement

String topology concerns algebraic operations associated with spaces of loops and paths in a target space. An open--closed topological conformal field theory involves open and closed sectors, with operations associated with suitable moduli spaces of surfaces and compatibility under sewing. These descriptions explain the terminology only; the supplied record does not specify the mathematical models needed to turn them into definitions for this problem.

The supplied question is:

Does string topology carry the structure of an open--closed topological conformal field theory?

The formulation remains unresolved at the level of hypotheses and intended structure. In particular, the record does not specify the target space or manifold, its required hypotheses, coefficient ring, chain-level or homology-level interpretation, open-sector boundary conditions, relevant moduli spaces, or sewing and unit requirements. It also does not state whether the assertion is intended for every target in some class or for a particular target. Consequently, the evidence supports the displayed question, but does not determine a fully quantified existence statement or a unique definition of the requested structure. These choices require source clarification and human mathematical review.

Short English statement

The question asks whether string topology admits compatible open and closed operations forming a topological conformal field theory. The supplied record leaves the target, hypotheses, and precise theory unspecified, so a fully precise formulation cannot be recovered from this evidence alone.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001270$, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year $2009$ and supplies no individual proposer.
• Supplied report citation: Strings, Fields and Topology (2009), Oberwolfach Reports, pp. 1547--1596, DOI 10.4171/owr/2009/28. Source URL: https://doi.org/10.4171/owr/2009/28. The same URL is supplied under the literature-source label "original OWR report." No problem-specific page, speaker attribution, or passage defining the intended structure is supplied.
• Literature assessment dated 2026-08-21: this is supplied dated curation, not an independently verified assessment. It labels the problem open because, according to that assessment, the extraction contains only the word "conjecture" and no mathematical statement, so no resolution could be verified. The background describes this as dated literature triage and requests independent verification of time-sensitive status and cited claims.
• The supplied statement metadata labels the statement corrected_verified and says that the mathematical question was preserved while open--closed typography was normalized. This supplied verification label and the supplied open-status label are not independent mathematical or source verification by this draft.