Reconstruction of Real Gromov-Witten Invariants for Semisimple Targets

Definitions and mathematical statement

Let $X$ be a real target in the category intended by the supplied report. Real Gromov--Witten theory concerns curve-counting invariants compatible with the real structures on the target and domain. The source assumes that $X$ is real-orientable and that its quantum cohomology is semisimple. The precise category of targets, real-orientation data, coefficient ring, and meaning of semisimplicity at the relevant quantum parameters are not specified in the supplied excerpt; these hypotheses must be recovered from the report before a complete formulation is possible.

Write $t,q$ for the parameters used in the source and $\psi$ for a cotangent-line class at a marked point. A descendant insertion $\psi^{k_i}\gamma_i$ consists of a nonnegative power of this class and a cohomology insertion $\gamma_i$. The excerpt places the proposed identity in the range \[ 1-g-n<0,\qquad \tau(t)=t, \] where $g$ is the genus and $n$ the number of displayed descendant insertions. The action denoted by $\tau$, the permitted insertions, and the parameter domains are not defined in the supplied record.

Conjecture 1, as recoverable from the supplied record. For every real-orientable target $X$ whose quantum cohomology is semisimple, there exists a suitable series $U_{t,q}(\psi)$ such that the reconstruction identity called Theorem 1 in the cited report holds for its real Gromov--Witten theory, throughout the stated range $1-g-n<0$ and at every permitted point $t$ fixed by $\tau$. The intended identity reconstructs the relevant real descendant invariants from integrals of $\psi$-classes over moduli spaces of curves and the genus-zero complex and real Gromov--Witten theory of $X$.

The excerpt describes a graph sum with weights $1/|\operatorname{Aut}(\Gamma)|$. Its vertex forms have $n_v+k$ arguments, indexed by incident half-edges, marked leaves, real leaves, and $k$ additional insertions. The insertion rules involve $R_{t,q}$, $S_{t,q}$, $E_{t,q}$, $T_{t,q}$, involution operators $\widehat\tau,\widehat\tau'$, and coefficients of $z^{-k_i-1}$. Transposes are taken with respect to the Poincaré pairing; the stated real-leaf insertion is $U_{t,q}(-\psi)^T$. However, the displayed theorem and several insertion formulas are corrupted in the supplied extraction. The graph class, normalization, operator definitions, and type and admissibility conditions of $U_{t,q}$ cannot be recovered reliably. Thus this is a source-dependent statement of the conjectured extension, not a complete reconstruction formula.

The supplied dated curation says that the report proves the identity for $(\mathbb P^{2N-1},\theta_N)$ with its stated involution, while the extension to every target satisfying the conjectural hypotheses remains the intended question. The involution $\theta_N$ is not defined in the supplied excerpt. This reported special case and the supplied status label are not independently verified here.

Short English statement

The conjecture asks whether a suitable series $U_{t,q}(\psi)$ extends the report's reconstruction formula from its stated real projective targets to every real-orientable target with semisimple quantum cohomology. The supplied range is $1-g-n<0$, with $\tau(t)=t$. Corruption of the extracted formula and missing definitions prevent a fully explicit identity; human review against the cited report is required.

Sources

• Local UnsolvedMath registry, permanent ID $30005943$, source problem code catalogue entry, title: Reconstruction of Real Gromov--Witten Invariants for Semisimple Targets. Supplied collection: Oberwolfach Reports Open Problems; proposed year: 2024. No individual proposer is supplied.
• Supplied report citation: Real and Logarithmic Enumerative Geometry (2024), Oberwolfach Reports, pp. 1285--1338, DOI 10.4171/owr/2024/23. Source URL and supplied literature reference labelled "Oberwolfach Report 23/2024": https://doi.org/10.4171/owr/2024/23.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem "partially solved" and states that the report proves the reconstruction identity for the real projective targets with the stated involution, whereas Conjecture 1 concerns all real-orientable targets with semisimple quantum cohomology. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was checked against the official report. That verification claim is supplied provenance; the report has not been consulted for this draft, and the supplied original-statement extraction remains incomplete and corrupted.