R-module theory for descendants in leading OPE terms

Extend the KSW cyclic R-module construction to the one-point functions associated with descendant vectors when every pair attaining the minimal short-distance exponent occurs at positive descendant level.

Background

The leading short-distance coefficient of a torus n-point function is expressed as a finite sum of torus one-point functions associated with minimizing fusion channels and descendant levels. The KSW cyclic R-module machinery applies directly when a minimizing contribution comes from a lowest-weight torus-primary vector, but the paper does not provide an R-module description when all minimizing contributions occur at positive descendant level.

References

We also do not address the further question of an R-module home for {{\mathcal Y}\nu}(u{\nu,m},\tau) when every minimizing pair has m>0. KSW's{} machinery is built around lowest-weight vectors, and extending it to descendants is left open.

— Modular properties of affine \(\SLA{sl}{2}\) torus \(n\)-point functions  (2609.01496 - Zuevsky, 1 Sep 2026) in Remark 3.11 (Remark \ref{rmkEmin-caveats})

As shown in Figure~\ref{fig:exceptional}, this leading-order matching is a one-to-one correspondence, but we have not yet provided a criterion for determining which MHC amplitude relates to them.

— Massless-Massive Amplitude Correspondence III: Massive Amplitude Bases in the SMEFT  (2609.11767 - Ni et al., 10 Sep 2026) in Section 3, subsection “Massless-Massive Correspondence,” paragraph immediately preceding subsection “Current Decomposition for Massive Amplitudes”

A priori from a SCFT perspective it is unclear why we distinguish this particular superdescendant correlator.

— Super Liouville theory on the torus: bootstrap and the super Virasoro minimal string  (2610.01909 - Collier et al., 1 Oct 2026) in Section 2.1, paragraph titled “A distinguished superdescendant”