Completed $\Theta$-Twisted Volumes of Bordered-Curve Moduli

Definitions and mathematical statement

Let $g$ denote the genus and $n$ the number of boundary components of the bordered curves under consideration, and let $L_1,\ldots,L_n$ denote their boundary-length parameters. Write $\widehat V_{g,n}(L_1,\ldots,L_n)$ for the completed super Weil--Petersson volume polynomial referred to in the supplied record, and write $V^\Theta_{g,n}(L_1,\ldots,L_n)$ for the $\Theta$-twisted volume polynomial defined using the stated natural extension of the relevant bundle.

Intended conjecture. For every admissible pair $(g,n)$, the two polynomials agree: \[ \widehat V_{g,n}(L_1,\ldots,L_n) = V^\Theta_{g,n}(L_1,\ldots,L_n). \] The asserted equality is an identity of polynomials in the boundary-length parameters, hence an equality of their corresponding coefficients.

Unresolved qualifications. The supplied record does not specify the admissible range of $(g,n)$, the allowed boundary lengths, or the treatment of zero-length boundaries and exceptional topological types. It also does not define the completion, the class $\Theta$, the relevant bundle and its natural extension, the precise moduli spaces or compactification, or the volume normalizations. Consequently, the notation above records the intended conjecture but does not give independently usable definitions of its two sides. The background also mentions a natural Euler-class identity without stating it; no precise Euler-class assertion can be recovered from the supplied text.

Short English statement

The conjecture asks whether the completed super Weil--Petersson volume polynomial equals the $\Theta$-twisted volume polynomial obtained from the natural extension of the relevant bundle. The supplied record leaves essential constructions and parameter restrictions unspecified, so this statement-only draft requires source consultation and human mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete source record for permanent ID $30004711$, exact source problem code catalogue entry, titled "Completed $\Theta$-Twisted Volumes of Bordered-Curve Moduli." The record belongs to the Oberwolfach Reports Open Problems collection, gives proposed year 2021, and names no proposer.
• Supplied report citation: Enumerative Geometry of Surfaces (2022), Oberwolfach Reports, pp. 1463--1518, DOI 10.4171/owr/2021/28. https://doi.org/10.4171/owr/2021/28. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source labeled "Norbury": https://arxiv.org/abs/2312.14558. No title or pinpoint reference is supplied.
• Supplied literature assessment dated exactly 2026-08-22: the record labels the problem open and reports that later work constructs super Weil--Petersson measures and proves finiteness and recursion properties, while the specific natural Euler-class identity and equality of volumes remain unproved. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• The supplied statement status is corrected_verified, with the accompanying assertion that the standalone statement and its hypotheses were checked against the official Oberwolfach report. This is a supplied verification claim, not verification performed for this draft.